I.E.Irodov by Ashutosh Kumar

H. E. H poAoB 3A,ZI, ANH

FIO

EPI mH3HHE OBI.U,

H3Aa.Te\177bCTBO,=Hay\177ca\177

MOCKBa

I. E. Irodov

Problems in General Physics

Mir Publishers Moscow)))

Translated from

PREFACE

the Russian by Yuri Atanov

This book of problemsis intended as a textbookfor students at advanced coursein physics. studying higher educational institutions of number of the b ecause simple problemsit may be used Besides, great

First published 1981 Secondprinting 1983 Third printing 1988 Revised from the 1979Russian edition

students studying a general coursein physics. The book contains about 1900 problemswith hints for solving the ones. most complicated For students' convenienceeach chapter openswith a time-saving summary of the principal formulas for the relevant area of physics.As a rule the formulas are given without detailedexplanations sincea student, starting solving a problem,is assumedto know the meaning of the quantities appearing in the formulas. Explanatory notes are only given when misunderstanding may arise. in those cases All the formulas in the text and answersare in SI system, exceptin Part Six,where the Gaussiansystem is used.Quantitative data and with the rules of approximation and answers are presentedin accordance numerical accuracy. The main physical constants and tablesare summarised at the end of by

the

book.

The Periodic System of Elements is printed at the front end sheet and the Table of Elementary Particlesat the back sheet of the book. and a number In the presentedition, somemisprints are corrected, of problemsare substituted by new ones,or the quantitative data in and them are changed or refined (1.273, 3.97,4.194 2.189, 3.249,

5.78).

1.361,

the author wants to expresshis deep gratitude to colIn conclusion, leaguesfrom MIPhI and to readerswho sent their remarkson someprob-

lems , helping thereby to improve the

Printed in the Union

of Soviet

ISBN 5-03-000800-4

Socialist Republics

]\1773\177aTe2I]bCTBO

(\177Hay\177{a>>,

(\177)n.\177l\177l{o-\177aTeMa\177,\177ecKofi

English

translation,

F\177aBna\177

pexaKm\177

\177nTepaTyp\177L

Mir

Publishers,

1979

198\177

book.

I.E.Irodov)))

CONTENTS

Preface

................. .......... ................ .... .............. ....... .............. ............ ........... ........ ............. ............ ........ ..... ............. ....... ........... ............ .......... ........ ...... ............. ........ ...... ........ ........... ............ ........... ......... ............. ........ ........... ...........

A Few Hints Notation

for Solving the

ONE. PHYSICAL

PART

5 9 t0

Problems

FUNDAMENTALS

OF MECHANICS

t.t. Kinematics of Dynamics Fundamental t.2.The Equation Laws of Conservation of Energy, Momentum, t.3.Momentum t.4. Universal Gravitation of a Solid Body 1.5. t.6.Dynamics Elastic Deformations of a Solid Body

and

Angular

t.8.Relativistic Mechanics TWO. THERMODYNAMICS

2.t.Equation The First

of the

58 62 67

AND MOLECULAR

...

PHYSICS

Gas State. Processes

Law of Thermodynamics. Heat Capacity 2.2. 2.3.Kinetic Theory of Gases. Boltzmann's Law and Maxwell's Distribution 2.4.The SecondLaw of Effects Thermodynamics. Entropy 2.5.Liquids. Capillary 2.6.Phase Transformations Phenomena

2.7.Transport

PART

3.3.Electric Capacitance. Energy of an Electric Field 3.4.Electric Current Field. 3.5.Constant MagneticInduction.Magnetics 3.6.Electromagnetic ParticlesMaxwell's Equations in Electric and

PART

Magnetic

Charged

AND FOUR. OSCILLATIONS

4.t.MechanicalOscillations Oscillations

WAVES

4.2.Electric Waves. 4.3.Elastic

Acoustics 4.4. Electromagnetic Waves. Radiation PART

FIVE.

OPTICS

5.t. hotometry and of 5.2.PInterference

GeometricalOptics

Light of Light Polarization of Light

5.3.Diffraction 5.4.

75 75 78

88 93 96 t00 t05

THREE. ELECTRODYNAMICS

3.t.Constant Electric Field in Vacuum 3.2.Conductors and Dielectricsin an Electric Field 3.7.Motion

t.7.Hydrodynamics

PART

tl tt

t05

1tt t18 125 136

Fields

t47

t60

t66 t66

180 t88 t93

t99 t99

210 2t6 226)))

H. E. H poAoB 3A,ZI, ANH

FIO

EPI mH3HHE OBI.U,

H3Aa.Te\177bCTBO,=Hay\177ca\177

MOCKBa

I. E. Irodov

Problems in General Physics

Mir Publishers Moscow)))

NOTATION

PART ONE

PHYSICALFUNDAMENTALS OF MECHANICS Vectorsare written in boldface upright type, e.g.,r, F; the same lettersprinted in lightfaceitalic type (r, F) denotethe modulus of a vector. Unit vectors i, j, k are the unit vectorsof the Cartesiancoordinates x, y, (sometimesthe unit vectors are denotedas ex, %, %, ez are the unit vectorsof the cylindricalcoordinates p, % z,

t.t. KINEMATICS Average

\342\200\242

e\177),

are the unit vectors of a normal and a tangent. Mean values are taken in angle brackets( ), (v), (P). Symbols A, d, and 5 in front of quantities denote: A, the finite increment of a quantity, e.g.Ar = r 2 =AU --

\177,t

where

\342\200\242

the

Velocity

(t.ta)

\342\200\242

U2-

r\177;

dv ' w=--. dt

v\177-dt \342\200\242

Acceleration of a

normal to a

trajectory:

point

wz-where R

operations:

\177

the

\177).

\342\200\242

\177

\342\200\242

is the

modulus of the velocity Angular velocity and angular

[o\177r],

(l.tc)

\177

the given

point.

v dt,

(l.te)

do\177

dq\177

and

w\177

(tAd)

vector of a point. accelerationof a solid body:

d---[\"

t/elation between linear

body:

and the

Wn='-\177',

of the trajectory at

radius of curvature

o\177=

any multiplicity are denotedby a single sign and differ only by the integrationelement:dV, a volume element,dS, a surface element,and dr, a line element.The sign denotesan integralover a closedsurface,or around a closedloop.

the tangent

Distancecoveredby a point:

\303\227E,

Integralsof

dv\177

S where v

the gradient of (grad \177-E, the divergenceof E (div E), the curl of E (curlE). V\177,

\342\200\242

(lAb)

expressedin projectionson

U\177,

d, the differential (infinitesimal increment),e.g.dr, dU, 6, the elementary value of a quantity, e.g.6A, the elementarywork. Timederivative of an arbitrary function ] is denoted by dJ/dt, or by a dot over a letter, VectoroperatorV (\"nabla\.") It is used to denote the following

\177

accelerationof a point:

(v)=-\177-, (w>= displacementvector (an increment of a radius vector). and accelerationof a point:

e.g.,

and

\177.

e,\177,

vectors of velocity

[\177=

angular o\177R,

quantities I

u,.\177

[

for

rotating

solid

(t.t.f)

\177R,

\177

where r is the on the rotation

radius vector of the consideredpoint relative to axis, and R is the distancefrom the rotation

an arbitrary

axis.

point

1.1.A motorboatgoingdownstream overcamea raft at a point A; = 60min later it turned back and after some timepassed the raft at a distancel = 6.0km from the point A. Find the flow velocity assuming the duty of the engineto be constant. 1.2.A point traversed half the distancewith a velocity The remainingpart of the distancewas coveredwith velocity for half the time, and with velocity for the other half of the time. Find the mean velocity of the point averaged over the whole time of mo\342\200\242

\177o.

\177

tion.)))

1.3.A

w first with acceleration car starts moving rectilinearly, and is then initial uniformly, zero), velocity equalto (the at the samerate w, comesto a stop.The total finally, decelerating The average velocity during that 25 time of motion equals timeis equal to (v/ = 72 km per hour. How long does the car move uniformly? shows in one direction.Fig. A point moves rectilinearly \177-

5.0m/s

\177

\342\200\242

t.t

1.4.

of his walking if both swimmers reachedthe destination simultaneously? The streamvelocity vo =- 2.0km/hour and the velocity v' of each swimmer with respectto water equals 2.5km per hour. Two boats,A and B,move away from a buoy anchoredat the middleof a river along the mutually perpendicular straight lines: the boat A alongthe river, and the boat B acrossthe river.Having moved off an equal distance from the buoy the boats returned. Find the ratio of timesof motionof boats %t/\177 if the velocity of each boat with respectto water is ---- timesgreaterthan the

1.8.

t.2

\1771

stream velocity. 1.9.A boat moves relativeto water with a velocity which is n ----

2.0timeslessthan

----

the river

flow

velocity.At what angle to the

stream directionmust the boat move to minimizedrifting? 1.t0.Two bodieswere thrown simultaneouslyfrom the same point: one, straight up, and the other, at an angleof 0 60 to the horizontal.The initial velocity of each body is equal to Vo =- 25 m/s. Neglectingthe air drag, find the distancebetween the bodiest = t.70s later. \302\260

\177--

\177

1.11. Two particlesmove in a uniform gravitationalfield with an acceleration g. At the initial moment the particleswere locatedat

3.0m/s and 4.0m/s horizontally in oppositedirections.Find the distancebetween the particlesat the moment when their velocity vectors becomemutually perpendicular. 1.12.Three points are locatedat the vertices of an equilateral triangle whose sideequalsa. They all start moving simultaneously with velocity v constant in modulus,with the first point heading continually for the second,the secondfor the third, and the third for the first. How soon will the points converge? 1.13. Point A moves uniformly with velocity v so that the vectorv is continually \"aimed\" at point B which in its turn moves rectilinearly and uniformly with velocity u < v. At the initialmoment of time v_J_.u and the points are separatedby a distancel. How soon will the points converge? one point and moved

Fig.

1.1.

the distances traversed by the point as a function of the time t. Usingthe plot find: (a) the average velocity of the point during the time of motion; (h) the maximum velocity; (c) the time moment to at which the instantaneousvelocity is equal to the mean velocity averaged over the first to seconds.

1.5.Two particles,1 and 2, move with

constantvelocitiesvl and

At the initial moment their radius vectors are equal to r 1 and Howmust these four vectors be interrelatedfor the particlesto colr\177.

v\177.

lide?

1.6.A

to the

the

east with velocity

equator along ship 30kin/hour.The southeasternwind blows at an angle 60 Vo to the equatorwith velocity v --= 15km/hour.Find the wind velocity v' relative to the ship and the angle between the equator and the wind directionin the referenceframe fixed to the ship. moves

\302\260

\177

\177'

1.7.Two swimmersleavepoint A

on one bank of the river to reach the other bank. One of them crosses the river along the straight line AB while the other swims at right anglesto the stream and then walks the distancethat he has been carriedaway by the streamto get to point B.What was the velocity u

point B lying right acrosson

A train 1.14.

with

of length l

,\177elocities vl

\177

v\177

350m startsmoving rectilinearly with

constant acceleration 3.0. t0 m/s:;t----30s after the start the locomotiveheadlight is switched on (event 1),and = 60s after that event the tail signallight is switched on (event 2).Find the distancebetween these events in the referenceframes fixed to the train and to the Earth.How and at what constantvelocity V relative to the Earth must a certainreference frame K move for the two events to occurin it at the samepoint? 1.15.An elevator car whose floor-to-ceiling distance is equal to 2.7 m starts ascendingwith constant accelerationt.2 m/s\177; 2.0s after the start a bolt beginsfalling from the ceilingof the car.Find: free fall time; (a) the bolt's and the distancecoveredby the bolt during (b) the displacement the free fall in the reference frame fixed to the elevatorshaft. w

\177

-\177

\342\200\242

13)))

i.16.

Two particles, and 2, move with constant velocitiesvl and v2 along two mutually perpendicular straight linestoward the intersection point O.At the moment t ----0 the particleswere located at the distances l z and l 2 from the point O.Howsoonwill the distance between the particlesbecomethe smallest?What is it equal to? From point A locatedon a highway (Fig.1.2) one has to get to point B locatedin the field at a distancel by caras soonas possible from the highway. It is known that the car moves in the field timesslower than on the highway. At what distancefrom point D one must turn off the highway? A point travels along the x axiswith a velocity whose projectionvx is presentedas a function of time by the plot in Fig.

1.17.

\1771

1.18.

1.3.

first

4.0and

8.0s; draw the approximateplot s (t). The velocity of a particlemoving in the positive direction 1.22. where a is a positive constant. of the x axis varies as v ave,, = \177

Assumingthat at the moment point x 0, find:

0 the particlewas locatedat

the

\177

of the velocity and the acceleration of the (a) the timedependence particle; (b) the mean velocity of the particleaveraged over the time that the particletakesto cover the first s metresof the path. A point moves rectilinearly 1.23. with deceleration whosemodulus ---the v of the as w on velocity particle aV\177 where a is a depends

positive constant.At the initial moment the velocity of the point is equal to vo. What distancewill it traverse before it stops? What time will it take to cover that distance? i.24.A radius vector of a point A relativeto the originvaries with time t as r = ati bt2j, where a and b are positive constants,and i and j are the unit vectors of the x and y axes.Find: trajectory y (x); plot this function; (a) the equationof the point's w vec(b) the timedependenceof the velocity v and acceleration tors, as well as of the moduli of these quantities; of the anglea betweenthe vectors w and v; (c)the timedependence (d) the mean velocity vector averaged over the first t secondsof motion, and the modulus of this vector. A 1.25. moves in the plane xy accordingto the law x at, = at point y at), where a and a are positive constants, and t is

Fig.

t.2.

Fig.

t.3.

x--

of the point 0 at the moment t ----0, Assuming the coordinate draw the approximate timedependenceplots for the acceleration wx, the x coordinate, and the distancecovered s. 1.19. A point traversed hall a circleof radius R = t60cm during

t0.0s. Calculatethe following quantities averaged over that time: (a) the mean velocity (v); (b) the modulus of the mean velocity vector I(v)l; (c)the modulus of the mean vectorof the total accelerationI(w) if the point moved with constanttangent acceleration. A radius vector of a particlevaries with time t as r at at), where a is a constantvectorand a is a positive factor.

time interval

\177

1.20.-(! Find:

\177

-----

w of the particleas functions (a) the velocity v and the acceleration of time; (b) the timeinterval At taken by the particleto return to the initial points, and the distances coveredduring that time. 1.21. At the moment t ----0 a particleleavesthe origin and moves in the positive directionof the x axis. Its velocity varies with time -- t/T), where Vo is the initial velocity vector whose as v Vo modulus equals Vo t0.0cm/s; 5.0s. Find: (a) the x coordinateof the particleat the moments of time 6.0, t0, and 20 s; (b) the momentsof timewhen the particleis at the distancei0.0cm from the origin; \177

\177

\342\200\242

\177

(itime. Find:

-\177

(a) the equationof the point's trajectory y (x); plot this function; w of the point as functions (b) the velocity v and the acceleration

time; to at which the velocity vectorforms an anglen/4 with the acceleration vector. A moves in the plane xy accordingto the law x |.26. point -a sin a cos where a and are positive constants. y (t Find: of

\177

-----

\177ot,

-\177

\177o

\177ot),

(a) the distances traversed by the point during the time vectors. velocity and acceleration (b) the anglebetweenthe point's A particlemovesin the plane xy with constant acceleration 1.27. w \177;

directedalong the negative y axis. The equationof motion of the particlehas the form y ax -- bx where a and b are positive constants.Find the velocity of the particleat the originof coordinates. A smallbody is thrown at an angle to the horizontalwith 1.28. -\177

\177,

the initial velocity vo. Neglectingthe air drag, find: of the body as a function of time r (t); (a) the displacement (b) the mean velocity vector (v/ averaged over the first t seconds and over the total time of motion. A body is thrown from the surface of the Earth at an anglea

i.29.

!5)))

to the horizontalwith

the

initialvelocity vo. Assumingthe

air drag

to be negligible, find: (a) the timeof motion; (b) the maximum height of ascent and the horizontalrange; at what value of the anglea they will be equal to eachother; (c) the equationof trajectoryy (x),where y and x are displacements of the body along the vertical and the horizontalrespectively; (d) the curvature radii of trajectoryat its initial point and at its

it covered the n-th

= 0.t0)fraction

circleafter i.38.A point moves with decelerationalongthe circleof radius R sothat at any moment of timeits tangentialand normal accelerations

merit when

the beginningof

motion.

of the

peak.

1.30. Usingthe conditionsof the foregoing problem,draw the approximatetimedependenceof moduliof the normal wn and tangent acceleration vectors,as well as of the projectionof the total acceleration vector wv on the velocity vector direction. 1.3i.A ball starts falling with zero initial velocity on a smooth inclinedplane forming an angle a with the horizontal.Having falloff the inclinedplane. en the distanceh, the ball rebounds elastically At what distancefrom the impactpoint will the \"ball rebound for the secondtime? i.32.A cannon and a target are 5.t0km apart and locatedat the same level.Howsoonwill the shelllaunchedwith the initialvelocity 240 m/s reach the target in the absenceof air drag? 1.33.A cannon fires successivelytwo shellswith velocity vo = = 250 m/s; the first at the angle01= 60 and the secondat the angle 02 55 to the horizontal,the azimuth being the same.Neglecting the air drag, find the time interval between firings leading to the w\177

\302\260

-----

of the shells. collision

1.34.A balloonstarts rising from the surface of the Earth. The ascensionrate is constantand equal to vo. Due to the wind the balloon gathers the horizontal velocity componentvx = ay, where a is a constantand y is the height of ascent.Find how the following quantitiesdepend on the height of ascent: (a) the horizontaldrift of the balloonx (y); of the balloon. (b) the total,tangential, and normal accelerations 1.35. A particlemoves in the plane xy with velocity v = ai bxj, where i and j are the unit vectorsof the x and y axes,and a and b are constants.At the initial moment of timethe particlewas located \177-

point x = y ----0. Find: the trajectoryy (x); (a) equationof the particle's (b) the curvature radius of trajectory as a function of x. A particleA moves in one directionalonga given trajectory with a tangentialacceleration = aT, where a is a constant vector coincidingin directionwith the x axis(Fig.t.4),and is a unit vector coincidingin directionwith the velocity vector at a given point. Find how the velocity of the particledepends on x provided that its velocity is negligibleat the point x = 0. A point moves along a circlewith a velocity v = at, where a 0.50m/s2. Find the total accelerationof the point at the moat the

1.36.

w\177

\342\200\242

1.37.

-\177

t6\"

Fig.

i.4.

are equal in moduli.At the initialmoment t = 0 the velocity of the point equalsvo. Find: (a) the velocity of the point as a function of timeand as a function

of the distancecovereds; of the point as a function of velocity and (b) the totalacceleration the distancecovered. A point moves alongan arcof a circleof radius R. Itsvelocity dependson the distancecoveredsas v = a]/\177, where a is a constant. Find the angle a between the vector of the total acceleration and the vector of velocity as a function of A particlemoves alongan arcof a circleof radius//according to the law l ----a sin (ot, where l is the displacement from the initial positionmeasuredalong the arc,and a and (o are constants.Assum0.80m, and (o 2.00rad/s,find: m, a ing R = 1.00 of the particleat the (a) the magnitudeof the total acceleration points l = 0 and l q--a; (b) the minimum value of the totalacceleration w\177n and the cor-

1.39.

1.40.

-\177

-----

\177

respondingdisplacement lm. A point moves in the planeso that its tangentialacceleration

1.4!.

and its normal acceleration = bt where a and b are positiveconstants,and t is time.At the moment t -= 0 the point was at rest.Find how the curvature radius R of the point's trajectoryand the total acceleration w depend on the distancecovereds. A particlemoves alongthe plane trajectoryy (x) with veloof the parcity v whose modulus is constant.Find the acceleration ticle at the point x = 0 and the curvature radius of the trajectory at that point if the trajectoryhas the form (a) of a parabolay----axe; (b) of an ellipse (x/a) -4- (y/b) = a and b are constantshere. A particleA moves along a circleof radius R = 50 cm so that its radiusvectorr relativeto the point O (Fig.t.5)rotateswith the constant angular velocity (o = 0.40rad/s.Find the modulus of the velocity of the particle,and the modulus and directionof its w\177

\177

w,\177

\177,

1.42.

\177

1.43.

\177

total acceleration. 2--9451

t7)))

|.44. wheel rotatesaround a stationary axisso that A

the

rotation

at 2, where a ----0.20rad/s2. Find the angle varies with time as w of the point A at the rim at the moment t totalacceleration 2.5s if the linearvelocity of the point A at this moment v 0.65m/s. A shell acquiresthe initial velocity v = 320m/s, having made n ----2.0turns inside the barrelwhose length is equal to l ----2.0m. Assuming that the shell moves inside the barrelwith a uniform acceleration,find the angular velocity of its axial rotation at the moment when the shell \177

\177

(\177

\177--

|.45.

\177

escapesthe barrel. |.46.A solidbody rotatesabout a stationary axis according to the law (\177----at-

w (and the zero initial velocity), (b) with a constantacceleration with a constantangularvelowhile the discrotatescounterclockwise city i.52.A point A is locatedon the rim of a wheel of radius R 0.50m which rolls without slipping along a horizontalsurface with velocity v = t.00m/s. Find: vectorof the (a) the modulus and the directionof the acceleration A; point (b) the total distances traversed by the point A between the two successivemomentsat which it touchesthe surface. \302\242o.

\177

i.53.

= t0.0cm rolls without slipping down A ball of radius an inclinedplaneso that its centremoves with constantacceleration /\177

-- bt 3, where a 6.0rad/s and b 2.0 rad/s2. Find: (a) the mean values of the angular velocity and angular acceleration averaged over \177

\177

the time interval between t-\177 0 and the completestop; at the moment (b) the angularacceleration

body stops. |.47. solidbody starts rotating about a stationary axiswith an -2 ---A

when the

2.0.

t0 rad/s3. How soon angular acceleration at, where a after the beginningof rotation will the total acceleration vector of an arbitrary point of the body form an anglea ----60 w.ith its velo\177

\177

Fig.

t.6.

Fig.

\302\260

city vector?

1.48.A solidbody rotateswith decelerationabout a stationary axis with an angular deceleration o(:\177 where co is its angular velocity.Find the mean angularvelocity of the body averaged over the whole timeof rotationif at the initialmoment of timeits angular \177

velocity was equal to coo. 1.49. A solid body rotatesabout a stationary axisso that its angular velocity depends on the rotation angle as co coo where coo and a are positive constants.At the moment t--\177 0 the angle 0. Find the time dependenceof \177

\177

a(\177,

i.7.

----2.50cm/s\177; $ 2.00s after the beginningof motion its position to that shown in Fig. Find: corresponds of the pointsA, B, and O; (a) the velocities of these points. (b) the accelerations A cylinder rolls without slipping over a horizontal plane. The radius of the cylinder is equal to r. Find the curvature radiiof trajectoriestracedout by the points A and B (seeFig, Two solidbodiesrotate about stationary mutually perpenw

\177-

i.7.

|.54.

1.7).

|.55. dicular

intersectingaxes with constant

velocities

(a) with a constantvelocity v, while the discstartedrotating counterclockwise with a constantangularacceleration (the initialangular velocity is equal to zero);

----3.0rad/s and cos 4.0rad/s.Find the angular angularvelocity and angular accelerationof one body relativeto the other. A solidbody rotateswith angular velocity (a = ati bt2j, where a = 0.50rad/s b 0.060rad/sa, and i and j are the unit vectorsof the x and y axes.Find: (a) the moduliof the angularvelocity and the angularacceleration at the moment t ----t0.0s; (b) the angle between the vectorsof the angularvelocity and the angular accelerationat that moment. A round conewith half-anglea 30 and the of the base 5.0cm rollsuniformly and without slippingradius over a horizontal plane as shown in Fig. The cone apexis hinged at the point O which is on the samelevel with the point C, the cone base centre.The velocity of point C is v ---- cm/s.Find the moduliof

--\177

\177

(a) the rotation angle; (b) the angular velocity. A solid 1.50. body startsrotating about a stationary axiswith an cos where is a constantvectorand (p angular acceleration is an angle of rotation from the initial position. Find the angular velocity of the body as a function of the angle Draw the plot of this dependence. A rotating disc(Fig.t.6)moves in the positive directionof the x axis. Find the equation y (x) describing the position of the instantaneousaxis of rotation,if at the initial moment the axis C \177-

I\177

I\177o

\177,

I\177o

(\177.

i.5i.

of the

discwas locatedat

the point O after which it moved \177

\177

\302\242o\177

\177

|.56.

\177

\177,

i.57. /\177

\302\260

\177

i.8.

i0.0

t9)))

(a) the vectorof the angularvelocity of the coneand the angle it forms with the vertical; of the cone. (b) the vector of the angularacceleration A solidbody rotateswith a constantangularvelocity t.58. = -= 0.50rad/s about a horizontal axis AB. At the moment t = 0

Two touchingbars 1 and 2 are placedon an inclinedplane 1.61. forming an angle a with the horizontal(Fig.t.t0).The massesof to m aud and the coefficientsof friction bethe bars are equal

m2,

\177o

axisAB startsturning about the vertical with a constantangular acceleration =-0.i0rad/ss. Find the angular velocity and of the body after t 3.5s. angular acceleration the

\1770

-----

i.2.THE

FUNDAMENTAL

EQUATION

tween the inclinedplane and these bars are equal to

spectively,with

k\177

k 2.

Find:

and k 2 re-

of motion; (a) the force of interactionof the bars in the process (b) the minimum value of the anglea at which the bars start slid-

ing down. DYNAMICS

1.62.A smallbody was launchedup an inclinedplane set at an the coefficient of friction, anglea = i5 againstthe horizontal.Find ---2.0timeslessthan the if the time of the ascentof the body is \302\260

\342\200\242

ond

The fundamental

law):

of dynamics of a mass

equation

m \342\200\242

(t.2a)

The same equation expressedin projectionson point's trajectory: dv\177 \177--

(Newton's sec_

dv F. \177----

normal of the

m-.\177-

point

,. \177

\177,

the tangent

and

the

\177

m-.\177.-=

F n.

(t.2b)

\1771

time of its descent. 1.63.The following parametersof the arrangementof Fig.t.tt are available:the angle a which the inclinedplane forms with the horizontal, and the coefficient of friction k between the body and the inclined plane.The massesof the pulley and the threads, as well as the friction in the pulley, are negligible. Assuming both bodiesto be motionlessat the initial moment,find the mass ratio ras/m at which the body

-----

K' frame.

1.59.An aerostatof mass m starts comingdown with a constant accelerationw. Determinethe ballastmass to be dumped for the aerostatto reach the upward acceleration of the samemagnitude. The air drag is to he neglected. 1.60.In the arrangementof Fig.t.9 the masses too, and of bodiesare equal, the massesof the pulley and the threads are negligible,and there is no friction in the pulley. Find the accelerationw with which the body mo comesdown, and the tension of the thread binding togetherthe bodies and m2, if the coefficient of friction between these bodiesand the horizontalsurface is equal to k. Consider possiblecases. m\177,

m\177

(a) starts comingdown; (b) starts goingup; (c) is at rest. The inclinedplane of Fig.t.tt forms an anglea = 30 with 1.64. the horizontal.The mass ratio ms/m -= = 2/3.The coefficient of friction between the body m 1 and the inclinedplane is equal to k ----0.t0. The massesof the pulley and the threads are negligible. of the body m Find the magnitudeand the directionof acceleration when the formerly stationary system of massesstarts moving. \177

The equation of dynamics of a point in the non-inertial referenceframe K' which rotates with a constant angular velocity to about an axis translating with an accelerationwo: F -- mw o q- mto\177R -f- 2m Iv'to], mw' (l.2e) where R is the radius vector of the point relative to the axis of rotation of the \342\200\242

\302\260

\177

\1771

A plank of mass ml with a bar of mass m s placedon it lieson a smoothhorizontalplane.A horizontalforce growing with time t as F = at (a is constant)is appliedto the bar.Find how the accelerations of the plank wl and of the bar w s depend on t, if the coefficient of frictionbetween the plank and the bar is equal to k. Draw the approximateplotsof these dependences. A smallbody A startsslidingdown from the top of a wedge (Fig. t.t2)whose base is equal to l---=2.t0m. The coefficient of friction between the body and the wedge surface is k = 0.t40.At

1.65.

1.66.

2i)))

what value of the angle will the timeof slidingbe the least? What will it be equal top A bar of mass m is pulledby meansof a thread inclined plane forming an angle with the horizontal(Fig. up'anThe coefc\177

1.67.

i.13).

c\177

as greatas that of the other body, startsmoving with a constantaccelerationw. How soon will the bodiescollide? A pulley fixed to the ceilingof an elevator car carriesa thread whose ends are attached to the loadsof masses mz and wo. Assumingthe masses The carstartsgoingup with an acceleration of the pulley and the thread, as wellas the friction,to be negligible

i.7i.

find:

of the load rnz relativeto the elevatorshaft (a) the acceleration

relativeto

and

the

car;

(b) the force exertedby the pulley on the ceilingof the scar. w of body 2 in the arrangementhown Find the acceleration i.72. as great as the mass of bar 1and is times in Fig.t.t5,if its mass \177l

Fig.

1.12.

ficient of friction is equal to k. Find the angle which the thread must form with the inclinedplane for the tension of the thread to be minimum.What is it equal top \1773

t.68.At

the moment t 0 the force F at isappliedto a small body of mass m restingon a smooth horizontalplane(a isa constant). \177

--\177

Fig.

t.16.

the anglethat the inclinedplane forms with the horizontalis equal to a. The massesof the pulleys and the threads, as well as the fric-

tion,are assumed to

be negligible.Look into possiblecases.

t.t6the bodieshave masses i.73.In the arrangementshown in Fig. massesof the pulleys and the is absent, the friction m0, m2, of the body m 1. the threads are negligible.Find the acceleration cases. Look into possible i.74.In the arrangementshown in Fig.t.t7the mass of the rod M exceedsthe mass m of the ball.The ball has an openingpermitting m\177,

Fig.

i.i3.

The permanentdirectionof this force forms an angle with the horizontal (Fig. t.t4).Find: (a) the velocity of the body at the moment of its breakingoff the \177

plane;

(b) the distancetraversed by the body up to this moment. t.69.A bar of mass m restingon a smooth horizontalplane starts moving due to the forceF ragC of constant magnitude.In the processof itsrectilinearmotionthe angle between the directionof this force and the horizontalvaries as as, where a is a and s is the distancetraversed by the bar from itsinitialconstant, position. Find the velocity of the bar as a function of the angle \177

\177z

|.70.A horizontalplane with

two

bodies: a bar

-\177

the coefficient of friction k supports

and an electricmotorwith a battery on a block. A thread attached to the bar is wound on the shaft of the electric motor.The distancebetweenthe bar and the electricmotoris equal to l. When the motoris switched on, the bar, whose massis twice 22

\177I/I//////,\177

Fig. i.i7.

\177z.

Fig. t.t8.

Fig.

i.19.

slidealongthe thread with somefriction.The mass of the pulley At the initial moment the and the friction in its axleare negligible. ball was locatedopposite the lower end of the rod.When set free, it to

23)))

both bodiesbegan moving with constant accelerations. Find the friction force between the ball and the thread if t seconds fter the beginningof motionthe ball got oppositethe upper end of athe rod. The rod length equals In the arrangementshown in the mass of ball is ---- timesas great as that of rodFig. 2. The of the latter is length ---l t00cm. The massesof the pulleys and the threads, as as the friction, are negligible. The ball is set on the samelevelwell lower end of the rod and then released.How soon will the as the ball be oppositethe upper end of the rod? t.76.In the arrangementshown in body is = 4.0timesas great as that of Fig. Thethe mass of body height h = 20cm. The massesof the pulleys and the threads, as well as the friction, arenegligible. At a certainmoment body 2 is released and the arrangement set in motion.What is the maximum that height body 2 will go up to? 1.77.Find the accelerations of rod A and wedge B in the arrangement shown in Fig.1.20 if the ratioof the mass of the wedge to that of the rod equals and the friction between

t.75. i.8

l.i8

\177

t.i9 2.

\177

t.78.In the arrangementshown

1.21the

massesof the Fig. wedge M and the body rn are known. The friction exists appreciable in

1.82.

all contactsurfaces is

\177,

negligible.

t.St.

Prism of mass ra1 and with anglea (seeFig.i.23)restson Bar2 of mass me is placedon the prism.Assuma horizontalsurface. of the prism. find the acceleration ing the frictionto be negligible, In the arrangementshown in Fig.t.24the massesra of the bar and M of the wedge,as well as the wedge angle are known.

Fig.

i.22.

massesof the pulley and the thread are negligible.Thefriction of the wedge M. the acceleration i.83.A particleof mass m moves alonga circleof radius B.Find the modulus of the averagevectorof the force actingon the particle over the distanceequalto a quarter of the circle,if the particlemoves (a) uniformly with velocity v; (b) with constant tangential acceleration the initial velocity beingequal to zero.

is absent. Find

t.20.

only betweenthe

wedgeand the body ra, the friction being equal to k. The massesof the pulley and the threadcoefficient are negligible. Find the acceleration of the body m relativeto the horizontalsurface on which the wedge slides.

i.79.What is the minimum acceleration with which barA (Fig.i.22) should be shifted horizontally to keep bodies 1 and 2 stationary relativeto the bar? The massesof the bodiesareequal, and the coefficient of friction between the bar and the bodiesis to k. The equal massesof the pulley and the threads are negligible,the friction in the pulley is absent. i.80.Prism1with bar 2 of massm placedon it gets a horizontal

accelerationw directedto the left (Fig. 1.23). At what maximum value of this acceleration will the bar be stillstationary relativeto the prism,if the coefficient of friction between them k\177 cot \302\242z?

i.23.

The

An 1.84.

w\177,

Fig.

\177

500m

aircraft

with a

loopsthe loopof radius constant velocity

= 360km per hour. Find the weight of the Fig. t.24. = in the ra 70 of mass lower,upper, kg flyer and middlepoints of the loop. |.85.A smallsphereof mass ra suspendedby a thread is first taken asideso that the thread forms the right anglewith the vertical and then released.Find: of the sphereand the thread tension as (a) the total acceleration a function of 0,the angleof deflectionof the thread from the vertical;

(b) the threadtensionat the moment when the vertical component of the sphere'svelocity is maximum; (c) the angle0 between the thread and the vertical at the moment vectorof the sphereis directedhorizonwhen the total acceleration tally.

i.86.A

ball suspended by a thread swings in a vertical plane so values in the extremeand the lowest position acceleration are equal.Find the thread deflection angle in the extremeposition. |.87.A smallbody A startsslidingoff the top of a smooth sphere to the point of radius B.Find the angle 0 (Fig.t.25)corresponding at which the body breaks off the sphere,as well as the break-off velocity of the body. consistsof a smooth L-shaped rod locatt.88.A device(Fig.1.26) ed in a horizontalplaneand a sleeveA of mass m attachedby a weight-

that its

25)))

lessspringto a point B.The springstiffnessis equal to The whole system rotateswith a constantangularvelocity co about a vertical axispassingthrough the point O.Find the elongationof the spring. How is the result affected by the rotationdirection? t.89.A cyclist ridesalongthe circumference of a circularhorizontal plane of radius the friction coefficient beingdependentonly on \303\227.

/\177,

Find the magnitudeand directionof the force actingon 1.95.

the

particleof mass m duringitsmotion in the planexy accordingto the bcoscot, where a, b, and co are constants. law x asincot, y \177

\177

i.96oA

body of mass m is thrown at an angle to the horizontal with the initialvelocity v0. Assuming the air drag to be negligible,

find:

(a) the momentum incrementAp that the body acquiresover the t secondsof motion; (b) the modulus of the momentum incrementAp during the total time of motion. 1.97.At the moment t -= 0 a stationary particleof mass ra expeforce F at riencesa time-dependent t), where a is a constant vector, is the time during which the given force acts.Find: (a) the momentum of the particlewhen the actionof the force disfirst

-\177

(\177

\177

Fig.

distancer

i.25.

Fig.

\177

i.26.

from the centreO of the plane as k0 (t--r//\177), where is a constant.Find the radius of the circlewith the centre at the point alongwhich the cyclist can ride with the maximum velocity. What is this velocity? 1.90. A car moves with a constant tangential acceleration 0.62m/s2 along a horizontal surface circumscribinga circleof radius -----40m. The coefficient of sliding friction between the What distancewill wheels of the car and the surface is k ----0.20. the carridewithout sliding if at the initial moment of timeits velocity is equal to zero? t.91.A car moves uniformly along a horizontalsine curve y The coefficient of a sin (x/a), where a and a arecertainconstants. frictionbetweenthe wheelsand the road is equal to k.At what velocity will the carride without sliding? |.92.A chain of mass ra forming a circleof radius/\177 is slippedon a smooth round conewith half-angle0. Find the tensionof the chain if it rotateswith a constant angularvelocity co about a vertical axis coincidingwith the symmetry axis of the cone. 1.93.A fixed pulley carriesa weightlessthread with massesra1 and m2 at its ends.Thereis friction between the thread and the pulley. It is suchthat the thread startsslippingwhen the ratiom2/ml ko

-----

w\177

\177

/\177

-\177

\177

-\177

----

\1770.

Find:

(a) the friction coefficient; of the masseswhen m\177/ml =.\177l \177 (b) the acceleration 1.94.A particleof mass m moves along the internal smooth surface of a vertical cylinder of radius R. Find the force with which the particleacts on the cylinder wall if at the initial moment of time its velocity equals v0 and forms an angle a with the horizontal. \177lo.

continued: (b) the distancecovered by the particlewhile the force acted. i.98.At the moment t 0 a particleof mass rn startsmoving due to a force F -= F0 sin cot, where F0 and co are constants.Find the distancecoveredby the particleas a function of t. Draw the approximate plot of this function.

1.99. At the moment t ----0 a particleof mass ra startsmoving due

F ----F 0 coscot, where F0 and co areconstants.How long will it be moving until it stops for the first time?What distancewill it traverseduring that time? What is the maximum velocity of the particleover this distance? t.t00.A motorboatof mass ra moves alonga lakewith velocity v0. At the moment t 0 the engineof the boat is shut down. Assuming the resistance of water to be proportionalto the velocity of the boat to a force

find: F--\177--rv, (a) how long the motorboat moved

with the shutdown engine; as a function of the distancecov(b) the velocity of the motorboat eredwith the shutdown engine,as well as the total distancecovered

till the completestop; the mean

(c) velocity of the motorboatover the time interval (beginningwith the moment t 0), during which its velocity decreases times. Having gone through a plank of thickness h, a bullet changed its velocity from v0 to v. Find the time of motion of the bullet in the plank,assumingthe resistanceforce to be proportional to the square of the velocity. t.i02.A smallbar starts sliding down an inclinedplane forming an angle a with the horizontal.The friction coefficient dependson the distancex coveredas k ax, where a is a constant.Find the distancecoveredby the bar tillit stops,and its maximum velocity \177

i.10|. \1771

\177

distance. tion coefficient k. At the moment to it,which varies with timeas F over this

t.i03.A body of mass ra restson a horizontalplanewith 0

the

fric-

t a horizontalforce is applied ----at, where a is a constant vector. 27)))

after Find the distancetraversed by the body during the first t seconds

the forceaction began.

|,|04.body of mass is thrown straight up with velocity v0. Find the velocity v' with which the body comesdown if the air drag m

equalskv 2, where k is a constantand v is the velocity of the body. A particleof mass m t.105.

certainplane P due

moves

to a force F whose magnitude is constant and whosevectorrotatesin that planewith a constant angularvelocity a).Assuming the particle to be stationary at the moment t ----0, find: (a) its velocity as a function of in a

the friction coefficient

hi2.

-\177

tan

\177-

and at the

initial moment

chain of length l is placedon a smooth sphericalsurface of radius with one of its ends fixed at the top of the sphere.What of the chainwhen itsupper w of eachelement will be the acceleration end is released?It is assumed that the length of the chain l /\177

i.i08.A

smallbody is placedon the top of a smooth sphereof we Then the sphereis imparted a constant acceleration radius in the horizontaldirectionand the body beginssliding down. Find: (a) the velocity of the body relativeto the sphereat the moment of /\177.

break-off; (b) the angle 0o between the verticaland the radius vector drawn -----

centreof the sl)hereto the

break-off point;

calculate0o

l.i09.A

of a force particlemoves in a plane under the actionand to the particle's velocity depends which isalways perpendicular on a distanceto a certain point on the plane as i/r where n isa constant.At what value of n will the motion of the particlealong the circlebe steady? A sleeveA can slidefreely alonga smooth rod bent in the Thesystem is set in rotaof radius/\177 (Fig.1.28). a of shape half-circle tion with a constant angular velocity a) about a verticalaxis00'. to the steady positionof the sleeve. Find the angle(} corresponding A rifle was aimed at the verticalline on the target located preciselyin the northerndirection,and then fired.Assumingthe air \177,

|.ii{).

t.i||.

direcdrag to be negligible,find how much off the line,and in what tion,will the bullet hit the target.Theshot was fired in the horizontal

\177-

\302\242o

i.i07.A

from the for wo

l.|i2.

|.i|3.

(b) the distance covered by the Fig. t.27. particlebetween two successivestops, and the mean velocity over this time. i.i06.A smalldiscA is placedon an inclined plane forming an an initial is a nd horizontal the a with imparted (Fig. 1.27) angle velocity v0. Find how the velocity of the discdependson the angle -\177

\302\260,

\177

-----

time;

directionat the latitude ----60 the bullet velocity v ----900m/s, i.0kin. and the distancefrom the target equalss A horizontaldiscrotateswith a constant angular velocity 0) \177-6.0 rad/s about a verticalaxis passingthrough its centre.A smallbody of mass m ----0.50kg moves alonga diameterof the discwith a velocity v' = 50 cm/s which is constantrelativeto the disc.Find the force that the disc exerts on the body at the moment when it is locatedat the distance r 30 cm from the rotation axis. A horizontal smooth rod AB rotates with a constantangular velocity 2.00rad/s about a vertical axis passing through its end A. A freely slidingsleeveof mass m = 0.50kg \177'\177\\.

moves alongthe rod from the point A with the m/s. Find the Coriolis initial velocity v 0 1.00 -----

forceacting on the sleeve(in the reference frame

fixed to the rotating rod) at the moment when Fig. t.28. the sleeveis locatedat the distancer ----50 cm from the rotation axis. A horizontaldiscof radius R rotateswith a constantangular velocity a) about a stationary verticalaxis passingthrough its of the disca particleof mass m moves edge.Along the circumference At the moment with a velocity that is constantrelativeto the disc. when the particleis at the maximum distancefrom the rotationaxis, the resultant of the inertialforces Fin acting on the particlein the referenceframe fixed to the discturns into zero.Find: W' of the particlerelativeto the disc; (a) the acceleration the o dependence f Fin on the distancefrom the rotationaxis. (b) A smallbody of mass m ----0.30 kg startsslidingdown from the top of a smooth sphereof radiusR t.00m. The sphererotates with a constant angular velocity a)-\177 6.0rad/s about a vertical axispassingthrough its centre.Find the centrifugalforce of inertia and the Coriolis force at the moment when the body breaksoff the surface of the spherein the reference frame fixed to the sphere. A train of mass m ----2000 tons moves in the latitude ---= 60 North. Find: (a) the magnitudeand directionof the lateralforce that the train exertson the railsif it moves alonga meridianwith a velocity v ----

t.1t4.

1.1t5.

\177

t.tt6.

\177

\302\260

= 54 km per hour; (b) in what directionand with what velocity the train should move for the resultant of the inertialforces acting on the train in the referenceframe fixed to the Earth to be equal to zero. 1.tt7.At the equator a stationary (relative to the Earth) body falls down from the height h 500m. Assuming the air drag to be negligible,find how much off the vertical,and in what direction, \177-

the body will deviatewhen it hits the ground.

29)))

i.3.LAWS

ANGULAR

Increment of

\342\200\242

where A

particle.

AND

\342\200\242

(t.3a)

kinetic energy of a particle: T2 T1 = A,

forcesof a

of a particle in

of all

the

is equal to field: U2.

forcesacting

on the

decreaseof

the

(t.3c)

Relationship between the forceof a field and the potential energy of a the field: F= VU, (t.3d) the force is equal to the antigradient of the potential energy. Increment of the total mechanical energy of a particle in a given potential field: E 1 Aextr (t .3e) where Aextr is the algebraicsum of works performed by all extraneous forces, that is, by the forcesnot belonging to those of the given field. Increment of the total mechanical energy of a system:

i.e.

\342\200\242

ge--

\177

\342\200\242

E\177

--E 1= Aex t

\177_

A intnOncons,

E= T U, and U is the inherent potential Law of momentum variation of a system:

dp/dt where F

the resultant of Equation of motion

\342\200\242

F is the Kinetic

resultant energy

c \177t

\177

\342\200\242

the

system.

(t.3g)

: F,

(t.3h)

of all external forces. of a system T=\177-\177

where

energy

all external forces. of the system'scentre of inertia: m

where

(t .3f)

is its kinetic energy in the system of centre of inertia. of dynamics of a body with variahle mass:

Equation

dv F m-\177-= +--\177-u, where u

the velocity of the

considered. 3O

(t.3i)

separated(gained) substancerelative to

(t.3j) the body

=N'

(t.3k) N

is the

total moment

momentum of a system: M

field

the given A U1

particle in

\342\200\242

Angular

of a system:

momentum of the system, and

\342\200\242

where

momentum variation

angular

where M is the angular all external forces. \342\200\242

the

Law of

dM dt

the work performed by the resultant

energy

MOMENTUM,

SFdr= SFsds, P= Fv.

Work performed by the

\342\200\242

potential

ENERGY,

forceF:

Work and power of the

\342\200\242

CONSERVATION MOMENTUM

\1771

+ [rcP],

(t.31)

is its angular momentum in the system of the centre of inertia, rC is where the radius vector of the centre of inertia, and p is the momentum of the system. 1\177

sometrajectory in the planexy t.tt8.A particlehas shifted along point 1 whose radius vector r 1 = i + 2j to point 2 with the the

from

radius vector r 2 ----2i3j.During that time particleexperi+ encedthe action of certain forces,one of which being F-----3i Find the work performedby the force F. (Hererl,r2, and F are given in SI units). t.tt9.A locomotiveof mass m starts moving so that its velocity varies accordingto the law v = ave,, where a is a constant, and s is the distancecovered.Find the total work performed by all the forceswhich are acting on the locomotiveduring the first t seconds after the beginningof motion. t.t20.The kineticenergy of a particlemoving along a circleof radius B dependson the distancecovereds as T ----as2, where a is a constant.Find the force acting on the par-

ticle as a

function of

i.12i. A body of mass ra was slowly hauled a force F which at

up the hill (Fig.t.29)by each point was directedalonga tangent to the trajectory.Find the work performed by this force,if the height of the hill is h, the length of its basel, and the coefficient of friction k. t.t22.A discof mass m = 50 g slideswith the zero initial velocity down an inclined

Fig.

t.29.

horizontal; plane setat an anglea = 30 to the = l 50 cm alongthe horizontalplane, the distance traversed having the discstops.Find the work performedby the friction forces over the whole distance,assuming the friction coefficient k = 0.t5for both inclined and horizontalplanes. i.i23.Two bars of massesrnl and rn2 connectedby a non-deformed light spring rest on a horizontalplane.The coefficient of friction .betweenthe bars and the surfaceis equal to k. What minimum constant to the bar of mass force has to be appliedin the horizontaldirection in orderto shift the other bar? m = 0.80kg and length l = 1.5m rests i.i24.A chain of tmass on a rough-surfaced ableso that oneof its ends hangs over the edge. The chain startsslidingoff the tableall by itself provided the over\302\260

hangingpart equals

\177 \177l

1/3of the chain length. What

will be the 3t)))

total work performed by the friction forces acting on the chain by off the table? the moment it slidescompletely t.|25.A body of mass m is thrown at an anglea to the horizontal with the initialvelocity v0. Find the mean powerdevelopedby gravity over thewholetimeof motionof the body, and the instantaneouspower of gravity as a function of time. of radius R with t.i26.A particleof mass rn moves alonga circle = as at 2, where a is time with a normal acceleration wn varying of the power developedby all a constant.Find the timedependence the forcesacting on the particle,and the mean value of this power averaged over the first t secondsafter the beginningof motion. i.i27.A smallbody of mass m is locatedon a horizontalplane at the point O. The body acquiresa horizontalvelocity re. Find: the (a) the mean power developedby the friction force during m = 1.0 whole timeof motion,if the friction coefficient k ----0.27, kg,

and v0 = m/s; (b) the maximum instantaneouspower=developedby the friction force,if the friction coefficient varies as k ctx, where is a constant, and x is the distancefrom the point O. A smallbody of mass m = 0.t0kg moves in the reterence frame rotatingabout a stationary axiswith a constant angularveloc= 5.0rad/s. What work doesthe centrifugalforce of inertia ity path perform during the transfer of this body along an arbitrary r1 ----30 cm from point to point 2 which arelocatedat the distances and r2 = 50 cm from the rotation axis? A system consistsof two springs connectedin seriesand Find the minimum work the stiffness coefficientsk I and having to be performed in orderto stretchthis system by Al. A body of mass m is hauled from the Earth's surface by applying a force F varying with the heightof ascenty as F 2 (ay t) mg, where a is a positive constant.Find the work performed by this force and the incrementof the body's potentialenergy in the gravitationalfield of the Earth over the first half of the ascent. The potentialenergy of a particlein a certainfield has the form U = air b/r, where a and b are positive constants,r is the distancefrom the centreof the field. Find: to the equilibriumpositionof the (a) the value of r0 corresponding

axi -{- by|, where i and j are the unit vectorsof the x and y axes, Find out whether thesefields arepotential. and a and b areconstants. A body of mass rn is pushed with the initial velocity vo up an inclinedplaneset at an anglecc to the horizontal.The friction coefficient is equal to k. What distancewill the body cover before it stopsand what work do the friction forces perform over this dis\177

t.i34.

tance?

1.135.

A smalldiscA slides down with initial velocity equal to zero from the top of a smooth hill of height H having a horizontal What must be the height of the horizontalporportion (Fig. tion h to ensurethe maximum distances covered by the disc?What

1.30).

is it equal top

A smallbody A starts sliding from the height h down an 1.136. inclinedgroove passinginto a half-circleof radius h/2 (Fig.1.31).

1.5

c\302\242

i.128.

\177,//////////////////,\177\177-\177\"\"

o\177

i.i29.

k\177.

1.t30.

-----

i.i31. --

Fig.

1.30.

Fig.

1.31.

find the velocity of the body Assumingthe friction to be negligible, at the highestpoint of its trajectory(after breakingoff the groove). i.i37.A ballof mass rn is suspendedby a thread of length l.With what minimum velocity has the point of suspensionto be shifted for the ball to move alongthe circleabout in the horizontaldirection that point?What will be the tensionof the thread at the moment it will be passingthe horizontalposition? i.i38.A horizontalplane supports a stationary verticalcylinder of radius R and a discA attached to the cylinder by a horizontal thread AB of length l0 (Fig.1.32, top view).An initial velocity v0

\177

is steady; particle;examinewhether this positionattraction force;draw the (b) the maximum magnitude of the the radius vecforce on of the and Fr (r) (the projections plotsU (r) tor r). the potential i.i32.In a certaintwo-dimensionalfield2 of forcewhere and cox form U has the a of particle energy are positive constants whose magnitudesare different. Find out: (a) whether this field is central; of the (b) what is the shape of the equipotentialsurfaces and also surfaces for which the magnitudeof the vector of force F = const. t.t33.There are two stationary fields of force F ----ayi and F =

is impartedto the disc as shown in the figure. How long will it move alongthe plane until it strikesagainstthe cylinder? The friction is assumed to be absent.

3--9451

-----

\177-

c\302\242

\177y\177,

\177

Fig.

t.32.

Fig.

t.33.

33)))

t.t39.A

smooth rubber cord of length l whose coefficient of elasticity is k is suspendedby one end from the point O (Fig.i.33). The other end is fitted with a catch B. A smallsleeveA of mass m starts falling from the point O.Neglectingthe massesof the thread and the catch,find the maximumelongationof the cord. A smallbar A restingon a smooth horizontal plane is attachedby threadsto a point P (Fig.t.34)and, by meansof a weightless pulley, to a weight B possessingthe samemass as the bar itself.

i.i40.

i.i45.A closedchainA

of mass m ----0.36kg is attachedto a verand rotateswith ticalrotatingshaft by meansof a thread (Fig.1.37), a constantangularvelocity ----35rad/s.The thread forms an angle 0 = 45 with the vertical.Find the distancebetween the chain's centreof gravity and the rotation axis,and the tension of the thread. A round coneA of mass m ----3.2kg and halfrollsuniformly and without slipping angle alonga round conicalsurfaceB so that its apexO remains stationary (Fig.t.38).The centreof gravity of the coneA is at the samelevel as the point O and at a The cone'saxis moves distancel t7 cm from ml \302\2420

\302\260

: \302\260

it.

angularvelocity a). Find: (a) the static friction force acting on the coneA,

with o\177

t.0tad/s;

(b) at what values of

o\177

sliding,if the coefficient to k

Fig.

1.34.

Fig.

t.35.

\177.

i.i46. a----l0

0.25.

Fig.

1.36.

the coneA will roll without of friction between the surfaces is equal

i.i47.In the referenceframe

K two particlestravel along the x axis,one of mass m 1 with velocity vl, and the other of mass with Find: velocity frame K' in which the cumulative (a) the velocity of the reference kinetic energy of these particlesis minimum; (b) the cumulativekineticenergy of these particlesin the K\" frame. i.i48.Thereferenceframe, in which the centreof inertiaof a given system of particlesis at rest, translateswith a velocity V relative rn\177

v\177.

the bar is alsoattachedto a point O by meansof a light nonBesides, deformed spring of length l 0 = 50 cm and stiffness = 5 mg/lo, where m is the mass of the bar. The thread PA having been burned, the bar starts moving.Find its velocity at the moment when it is \303\227

breakingoff the plane. A horizontal plane supports a plank with a bar of mass m = 1.0 kg placedon it and attached by a light elastic non-deformed cordof length l 0 = 40 to a point O (Fig,.t.35). The coefficient of friction between the bar and the plank equals k = 0.20. The plank is slowly shifted to the right until the bar starts sliding over it. It occursat the moment when the cord deviatesfrom the verticalby an angle0 = 30 Find the work that has been performed by that moment by the friction force acting on the bar in the referenceframe fixed to the plane. i.142.A smooth light horizontalrod AB can rotate about a vertical axispassingthrough itsend A. The rod is fitted with a small sleeveof mass m attachedto the end A by a weightlessspringof length l o and stiffness What work must be performed to slowly get this system goingand reachingthe angularvelocity i.i43.A pulley fixed to the ceilingcarriesa thread with bodiesof massesml and m2 attachedto itsends.The massesof the pulley and the thread are negligible,friction is absent.Find the acceleration we of the centreof inertiaof this system. Two interactingparticlesform a closedsystem whose centre i.144. of inertia is at rest.Fig.1.36 illustratesthe positions of both particlesat a certainmoment and the trajectoryof the particleof mass ml. = ml/2. Draw the trajectoryof the particleof mass rn2 if

|.i4i.

\302\245

c\177n

\302\260.

Fig.

\303\227.

1.37.

Fig.

i.38.

frame K.The mass of the system of particles inertialreference total and the energyof the system in the frame of the centre equalsm, Find the total energy E of this system of is to of inertia equal frame K. in reference the particles t.i49.Two smalldiscsof massesrn 1 and interconnectedby a weightlessspring rest on a smooth horizontalplane.The discsare set in motionwith initialvelocitiesvl and va whose directionsare to an

\177.

m\177

rn\177

35)))

and liein a horizontalplane.Find the total mutually perpendicular of this system in the frame of the centreof inertia. energy t.iSO.A system consistsof two smallspheresof massesm I and interconnected by a weightlessspring.At the moment t 0 the \177

m\177

spheresareset in motionwith the initialvelocitiesVl and after which the system startsmoving in the Earth'suniform gravitational field.Neglectingthe air drag,find the time dependence of the total momentum of this system in the process of motionand of the radius vector of its centreof inertiarelativeto the initial positionof the centre. Two bars of massesml and connectedby a weightless spring of stiffness (Fig. i.39)rest on a smooth horizontalplane. v\177

i.i51.

m\177

\177\302\242

\177

m\177

;\177

///////#,\177'/.//././///////.//// .//\177/////////////////////// Fig.

t.39.

Fig.

t.40.

Bar2 is shifted a smalldistancex to the left and then released.Find the velocity of the centreof inertiaof the system after bar 1breaks oft the wall. i.i52.Two bars connectedby a weightlessspring of stiffness and length (in the non-deformedstate) l 0 rest on a horizontalplane. A constanthorizontalforce F startsactingon oneof the bars as shown Find the maximumand minimum distancesbetween the in Fig.i.40. bars during the subsequentmotion of the system, if the massesof the bars are: (a) equal; (b) equal to ml and m2, and the force F is appliedto the bar of mass

m\177.

t.t53.A system consistsof two identicalcubes,each of mass m, linked together by the compressed weightlessspring of stiffness (Fig.i.4i).The cubes are also connectedby a thread at a certainmoment.Find: isburned

which

through

(a) at what values of Al, the initial compression of the spring, the lower cube will bounceup after the thread has been burned through: (b) to what height h the centreof gravity of this of the spring system will riseif the initial compression

[\177]m

Fig. t.4i. one man in each move without friction due to inertia along the parallelrails toward eachother.When the buggiesget opposite each other,the men exchangetheir placesby jumping in the directo the motion direction.As a consequence, tion perpendicular buggy Al

-\177

i.t54.Two identicalbuggies1 and 2 with mg/\303\227.

the samedirection,with its veI stops and buggy 2keepsmovingthein initialvelocities of the buggies

locity becomingequalto v. Find if the mass of each buggy (without a man) equalsM and and the mass of each man m. Two identicalbuggiesmove oneafter the otherdue to inertia with the samevelocity v0. A man of mass m rides (with\177)ut friction) the rear buggy. At a certainmoment the man jumps into the front Knowing that the buggy with a velocity u relativeto his buggy. with which the mass of eachbuggy is equal to M, find the velocities that. after move buggieswill Two men, eachof mass m, stand on the edgeof a stationary be negligible,find the of buggy mass M. Assuming the friction to off with the samehorimen both after jump velocity of the buggy zontal velocity u relativeto the buggy: (l) simultaneously; (2) one after the other.In what casewill the velocity of the buggy be greater v\177

v\177

i.|55.

\342\200\242

|.i56.

and how many times? A chainhangs on a thread and touchesthe surface of a table through, by its lower end.Showthat after the thread has beenburned the force exertedon the tableby the fallingpart of the chain at any moment is twice as great as the force of pressureexertedby the part already resting on the table. = 50 g falls from the height h A steelball of mass m -- m on the horizontalsurface of a massive slab.Find the cumulative momentum that the ball imparts to the slab after numerous the velocity of the ball T1 ---bounces,if every impact decreases

i.i57.

i.i58. i.0

1.25

times. A raft of massM with a man of mass m aboard stays motionlesson the surface of a lake. The man moves a distancel' relative

|.i59.

Assumingthe water to the raft with velocity v'(t) and then stops. f ind: resistanceto be negligible, of the raft l relativeto the shore; (a) the displacement of the force with which the man acted horizontal the c omponent (b) the motion. on the raft during A stationary pulley carriesa ropewhose one end supports a ladderwith a man and the other end the counterweightof mass M. The man of mass m climbsup a distance with respectto the ladder and then stops. Neglectingthe mass of the ropeand the friction in I of the centreof inertia of the pulley axle,find the displacement

i.i60.

this system.

|.i6i.

A cannon of mass M starts sliding freely down a smooth inclinedplane at an anglea to the horizontal.After the cannon coveredthe distancel, a shot was fired, the shellleavingthe cannon in the the horizontaldirectionwith a momentum p. As a consequence, to be shell of the mass negligible, cannon stopped. Assuming the to that of the cannon, determinethe duration of the as compared shot. in a body i.i62.A horizontally flying bullet of mass mofgets stuck l t hreads identical two M (Fig. mass of length suspendedby 37)))

threadsswervethrough an angle0.Assumlngm << M, (a) the velocity of the bullet before strikingthe body; initialkineticenergy that turned (b) the fraction of the bullet's

As a result,the

find:

into

heat.

t.t63.A

placedon

body of mass M (Fig.t.43)with a smalldiscof mass m restson a smooth horizontalplane.The disc is set in

Fig.

t.42.

Fig.

t.43.

of the kineticenergy of the closedsystem i.i67.Find the increment due to their of masses and

comprisingtwo spheres v\177.

v\177

m\177.

\302\260.

motion in the horizontaldirectionwith velocity v. To what height (relativeto the initiallevel)will the discrise after breakingoff the body M? The friction is assumed to be absent. i.i64.A smalldiscof mass ra slidesdown a smooth hill of heighth without initial velocity and gets onto a plank of mass M lying on

\177

\1771

Fig.

t.44.

process.

(2) Can it be stated that the result obtained does not depend on

choiceof the

referenceframe? |.t65.A stone falls down without

-----

]/\177-\177

to the Earth with a constant velocity v0. A particleof mass g moving with velocity vl 3.0i a perfectly inelasticcollisionwith another par2.0jexperiences ticleof mass 2.0g and velocity v2 6.0k.Find the velocity of the formed particle (both the vector v and its modulus),if the componentsof the vectorsvl and v2 are given in the SI units.

i.0

\177

\302\260

m\177

initial velocity from a heighth onto tho Earth'ssurface.The air drag assumed to be negligible, the stonehits the ground with velocity v0 relativeto the Earth. Obtain the sameformula in termsof the referenceframe \"falling\"

--i.i66.

\1771

the horizontalplane at the base of the hill(Fig.i.44). Due to friction between the discand the plank the discslows down and, beginning with a certainmoment,moves in one piecewith the plank. (i) Find the total work performed by the friction forcesin this the

perfectly

m\177

of the sphereswere equal if the initialvelocities inelasticcollision, and to a perfectly elasticcolt.i68.A particleof mass ml experienced What fraction of the mass of a with lision stationary particle kineticenergy doesthe striking particlelose,if (a) it recoilsat right anglesto its original motion direction; is a head-onone? (b) the collision with Particle I experiencesa perfectly elastic collision i.i69. if mass t heir Determine ratio, a stationary particle2. the particlesfly apart in the opposite (a) after a head-oncollision directionswith equal velocities; (b) the particlesfly apart symmetrically relativeto the initial motion directionof particleI with the angleof divergence0 = 60 elasticallywith t.t70.A ball moving translationally collides of impact moment the A t samemass. of the another,stationary, ball the angle between the straight line passing through the centresof the ballsand the directionof the initial motion of the striking ball is equal to a 45% Assumingthe ballsto be smooth,find the fraction of the kineticenergyof the strikingball that turned into potential energy at the moment of the maximum deformation.into three 500 m/s bursts A shellflying with velocity v 1.17t. identicalfragments so that the kineticenergy of the system increases = t.5 times. What maximum velocity can one of the fragments obtain? experienced i.i72.Particle1 moving with velocity v 2i0ofm/s the samemass. a head-oncollisionwith a stationary particle the kineticenergy of the system decreased As a result of the collision, -and directionof the velocity the F ind i.0%. magnitude by collision. of particleI after the with a stationary i.i73.A particleof mass m anhaving collided the particleM whereas a/2 M deviated of mass angle by particle ---initial motion of the the direction 30 0 to recoiledat an angle of the particlem. How much (in per cent) and in what way has the if M/m kineticenergy of this system changedafter the collision, = 5.0? consistsof two particlesof masses |.i74.A closedsystem which move at right anglesto eachother with velocities and Find: and momentum of each particleand the (a) (b) the total kineticenergy of the two particlesin the reference frame fixed to their centreof inertia. elasticallywith a stationary i.i75.A particleof mass in1 collides maximum angle through the Find particleof mass in2 (in1 in2). which the strikingparticlemay deviateas a result of the collision. reston a smooth and C (Fig.1.45) i.i76.ThreeidenticaldiscsA,is B, with in motion set A velocity v after disc The horizontalplane.

4.0j-

v\177

m\177

v\177.

\177

39)))

which it experiences an elasticcollision simultaneously with the discsB and C. The distancebetween the centres of the latterdiscs is timesgreaterthan the diameterof eachdisc. priorto the collision Find the velocity of the disc A after the collision. At what value of will the disc \177l

recoilafter the collision; stop;move on? |.|77. moleculecollides another, moleculeof the same mass. \177l

stationary,

Demonstratethat

the angle of divergence is ideally (a)equals90 when the collision

\302\260

elastic;

(b) differs from

is inelastic.

i.i78.A

relativeto stratethat

90 \302\260

when the

collision

Fig.

\"\177,

t.45.

rocketejectsa steady jet whose velocity is equal to u rocket.The gas dischargerate equalsp kg/s.Demonrocketmotionequationin this casetakesthe form

the the

-----

F- pu,

where m is the mass of the rocketat a given moment,w is its acceleration,and F is the externalforce. 1.179. A rocketmoves in the absence of externalforces by ejecting a steady jet with velocity u constant relativeto the rocket. Find the velocity v of the rocketat the moment when its mass is the mass and equal to m, if at the initial moment it possessed itsvelocity was equal to zero.Make use of the formula givenmoin the

foregoingproblem.

1.180. Find the law accordingto which the mass of the rocket varies with time, when the rocketmoves with a constant acceleration w, the externalforcesare absent, the gas escapeswith a constant velocity u relativeto the rocket,and its mass at the initial

|.|8|.

moment equals m o.

spaceshipof mass m0 moves in the absenceof external forces with a constantvelocity v0. To changethe motiondirection, a jet engineis switched on.It startsejectinga gas jet with velocity u which is constant relativeto the spaceshipand directedat right anglesto the spaceshipmotion.The engineis shut down when the mass of the spaceship decreases to m. Through what angle did the motiondirectionof the spaceshipdeviatedue to the jet engine eration? |.182.A cartloadedwith sand moves alonga horizontalplane due to a constantforce F coincidingin directionwith the cart'svelocity vector.In the process,sand spills through a hole in the bottom with a constantvelocity p kg/s.Find the acceleration and the velocity of the cart at the moment t, if at the initial moment t 0 the cart with loadedsand had the mass m0 and its velocity was equal to zero. The friction is to be neglected. I.|83.A flatcar of mass mo starts moving to the right due to a constant horizontalforce F (Fig.'1.46). Sand spillson the flatcar A

\302\242z

\177

negligibly small. i.i84.A chain AB of length l is locatedin a smooth horizontal tube so that its fraction of length h hangs freely and touchesthe

surface of the tablewith its end B (Fig.t.47).At a certainmoment

with

from a stationary hopper.The velocity of loading is constant and equal to p kg/s. Find the time dependenceof the velocity and the acceleration of the flatcar in the processof loading.The friction is

Fig.

t.46.

Fig.

i.47.

the end A of the chain is setfree.With what velocity will this end of the chain slipout of the tube? The angularmomentum of a particlerelativeto a certain O varies with time as M = a q-bt2, where a and b are conpoint stant vectors,with Find the force moment N relativeto the point O actingon the particlewhen the anglebetweenthe vectors N and M equals 45 A ball of mass m is thrown at an anglea to the horizontal with the initial velocity v0. Find the time dependenceof the magnitude of the ball'sangularmomentum vectorrelativeto the point from which the ball is thrown. Find the angular momentum M at the highestpoint of the trajectoryif m = 130g, = 25 m/s.The air drag is to be neglected. A discA of mass m slidingover a smooth horizontalsurface with velocity v experiences a perfectly elasticcollision with a smooth The stationary wall at a point O (Fig.1.48). anglebetweenthe motion directionof the disc and the normal of the wall is equalto Find: (a) the points relativeto which the angular momentum M of the disc remainsconstantin

t.t85.

a/b.

\302\260

|.t86.

1.|87.

this process; (b) the magnitude of the incrementof the

vectorof thedisc's angularmomentum relative to the point O\" which is located in the plane of the disc's motion at the distancel from the point O.

A smallball of mass m suspended Fig. t.48. from the ceilingat a point O by a thread of length l moves alonga horizontalcirclewith a constant angular velocity co. Relative to which points doesthe angularmomentum M of the ball remainconstant?Find the magnitude of the increment

t.t88.

4't)))

of the vector of the bali'sangularmomentum relativeto the point O pickedup duringhalf a revolution. A ball of mass m falls down without initial velocity from a height h over the Earth'ssurface.Find the incrementof the angularmomentum vectorpickedup during the timeof falling(relaframe moving translationallyin tive to the point O of the reference a horizontal directionwith a velocity V). The ball starts falling from the point O. The air drag is to be neglected. A smooth horizontaldiscrotateswith a constant angular velocity co about a stationary verticalaxispassingthrough its centre, the point O. At a moment t ----0 a discis set in motion from that

1.189.

bali's

1.|90.

1.195.

A uniform sphere of mass m and radius R starts rolling without slipping down an inclinedplane at an angle a to the horizontal.Find the time dependenceof the angular momentum of the sphererelativeto the point of contact at the initialmoment.How will the obtainedresult changein the caseof a perfectly smooth

inclined plane?

i.i96.

A certainsystem of particlespossesses a total momentum p and an angularmomentum M relativeto a point O.Find its angular momentum M' relativeto a point O' whose positionwith respectto the point O is determined by the radius vector r 0. Find out when the angular momentum of the system of particlesdoesnot depend on the choiceof the point O. Demonstrate that the angularmomentum M of the system of particlesrelative to a point O of the reference frame K can be re-

i.i97.

presented as

+ [rcp],

l\177

where is its properangular momentum (in the referenceframe moving translationallyand fixed to the centreof inertia), r c is the radiusvectorof the centreof inertia relativeto the point O, p is the total momentum of the system of particlesin the reference frame K. A ball of mass m moving with velocity Vo experiences a with one of the spheresof a stationary head-on elastic collision The mass of eachsphereequals rigid dumbbell as whown in Fig. m/2, and the distancebetween them is Disregardingthe sizeof the spheres,find the properangularmomentum 2\177/of the dumbbell after the collision, the angularmomentum in the reference frame moving translationally and fixed to the dumbbell'scentre of inertia. Two smallidenticaldiscs, eachof mass m, lie on a smooth horizontalplane.The discs are interconnectedby a light non-deformed springof length l o and stiffness At a certainmoment oneof the discsis set in motion in a horizontal directionperpendicular to the springwith velocity v0. Find the maximum elongationof the spring in the processof motion, if it is known to be considerably lessthan unity. l\177

Ffg.

i.49.

Fig.

1.50.

point with velocity vo. Find the angular momentum M (t) of the discrelativeto the point O in the referenceframe fixed to the disc. Make sure that this angular momentum is caused by the Coriolis

force.

1.191.

A particlemoves along a closedtrajectory in a central field of force where the particle's potentialenergy U kr 2 (k is a O positiveconstant,r is the distanceof the particlefrom the centre of the field).Find the mass of the particleif its minimum distance from the point O equalsr x and its velocity at the point farthest from O \177

equals

|.192. l.

A smallball is suspendedfrom a point O by a light thread of length Then the ball is drawn asideso that the thread deviates through an angle 0 from the vertical and set in motion in a horizontal directionat right anglesto the vertical plane in which the What is the initialvelocity that has to be imparted thread is located. to the ball so that it coulddeviate through the maximum angle in the processof motion? A smallbody of mass m tied to a non-stretchablethread moves over a smooth horizontalplane.The other end of the thread is being drawn into a hole O (Fig.t.49)with a constantvelocity. Find the thread tension as a function of the distancer between the body and the hole if at r ----r o the angularvelocity of the thread is equal to coo. thread is wound on a massive fixed A light non-stretchable of mass m is tied to the free end pulley of radius H. A smallbody = 0 the system is releasedand starts t moment At a thread. of the momentum relativeto the pulley axle as its Find angular moving. a function of time t.

i.i98.

t.50. l.

i.e.

i.i99.

\303\227.

1,4.UNIVERSAL

1.193.

1.194.

\342\200\242

Universal

GRAVITATION gravitation

law

F =?

--. ?nlm2

(t .4a)

The squaresof the periodsof revolution of any two planets around the are proportional to the cubesof the major semiaxesof their orbits (Kepler): a (i.4b) field of a mass point: G and potential of the gravitational Strength \342\200\242

Sun

T\177\302\242\302\242

\177.

\342\200\242

\302\242p

\342\200\242

Orbital and escapevelocities: 43)))

\1772

t.200.A

of mass M moves along a circlearound the Sun planet ---34.9km/s (relative to the heliocentricreference v with velocity the Find periodof revolutionof this planet around the Sun. frame). The Jupiter'speriodof revolution around the Sun is t2 1.201. times that of the Earth.Assumingthe planetary orbitsto be circular,

find:

(a) how many timesthe distancebetween the Jupiterand the Sun exceedsthat between the Earth and the Sun; of Jupiterin the heliocentric (b) the velocity and the acceleration

referenceframe.

A planet of mass M moves around the Sun alongan ellipse 1.202. the maxiitsminimum distancefrom the Sun is equalto r and of its

so that

laws, find period distanceto R. Making use of Kepler's Sun. the around revolution A smallbody starts falling onto the Sun from a distance 1.203. initialvelocity of the equal to the radius of the Earth'sorbit.The r eference frame.Making heliocentric in the zero to is body equal use of Kepler's laws, find how long the body will be falling. system scaled 1.204. Supposewe have made a modelof the Solar mean the same of of materials density as but ratio in the down the actual materialsof the planetsand the Sun. Howwill the orbital modelschangein this case? periodsof revolutionofisplanetary around the t.205.A doublestar a system of two starsmoving centreof inertia of the system due to gravitation.Find the distance betweenthe componentsof the doublestar, if its totalmass equalsM and the periodof revolution T. Find the potentialenergy of the gravitationalinteraction 1.206. locatedat a distancer (a) of two mass pointsof massesml and from each other; rod of mass M (b) of a mass point of mass m and a thin uniform at a distancea line a located are if along straight and length l, they from each other;alsofind the force of their interaction. around the Sun A planet of mass m moves alongan ellipse 1.207. Sun areequal the f rom minimum and distances maximum so that its M of this momentum the Find angular to rl and respectively. the Sun. planet relativeto the centreof 1.208. Using the conservationlaws, demonstratethat the total mechanicalenergy of a planet of mass m moving around the Sun axis a. Find this along an ellipsedependsonly on its semi-major a. of as a function energy orbitaround the Sun. A planet A moves alongan elliptical 1.209. the Sun itsvelofrom the distance at ro At the moment when it was

Find the minimum disrelativeto the centreof the Sun (Fig.i.5i). tance by which this body will get to the Sun. A particleof mass rn islocatedoutsidea uniform sphereof mass M at a distancer from its centre.Find: of the particle (a) the potentialenergy of gravitationalinteraction and the sphere; (b) the gravitationalforce which the sphereexertson the particle. |.2i2.Demonstratethat the gravitationalforce acting on a particleA inside a uniform sphericallayer of matteris equal to zero. t.2i3.A particleof massm was transferred from the centreof the baseof a uniform hemisphereof mass M and radius/\177 into infinity.

|.211.

mum

\1771

rn\177

r\177

city 'was equal to vo and the angle between the radiusvectorr o and the velocity vectorvo was equal to a. Find the maximumand minithat will separatethis planet from the Sun during mum distances

its orbital motion.

1.2t0.

A cosmic body A moves to the Sun with velocity vo (when far from the Sun) and aimingparameterl the arm of the vector vo 44

work was performed in the processby the gravitationalforce on the by the hemisphere? is a uniform sphereof mass M and radius R. Find t.214.Thereparticle

What

\177xerted

the strength G and the potential of the gravitationalfield of this sphere as a function of the distancer from its centre(with r R and r R). Draw the approximateplotsof the functions G (r) and (r). Insidea uniform sphere of density p there is a spherical cavity whose centreis at a distancel from the centreof the sphere. Find the strength G of the gravitationalfield inside the cavity. A uniform sphere has a mass M and radius R. Find the pressurep inside the sphere,causedby gravitational compression, as a function of the distancer from its centre.Evaluate p at the centreof the Earth, assuming it to be a uniform sphere. Find the properpotentialenergy of gravitationalinteraction of matterforming (a) a thin uniform sphericallayer of mass m and radius (b) a uniform sphereof massm and radiusR (makeuse of the answer to Problemt.2t4). move in a common plane along cirTwo Earth'ssatellites cular orbits.The orbitalradius of one satelliter = 7000km while that of the other satelliteis hr ----70 km less.What time interval to eachother over separatesthe periodicapproachesof the satellites the minimum distance? the Calculatethe ratios of the following accelerations: acceleration ml due to the gravitationalforce on the Earth'ssurface,

\302\242p

1.2t5. \302\242p

t.216.

1.217.

/\177;

1.218.

t.219.

45)))

the acceleration due to the centrifugalforce of inertia on the w s causedby the Sun to the Earth's equator,and the acceleration w\177

bodieson the Earth. i.220.At what height over the Earth's polethe free-fall acceleration decreases by one per cent; by half? velocity vo 1.22i.On the poleof the Earth a bodyofistheimparted Earth and the freedirectedvertically up. Knowing the radius on its surface,find the height to which the body fall acceleration T he air drag is to be neglected. will ascend. satelliteis launchedinto a circularorbitaround An artificial i.222. frame moving transthe Earth with velocity v relativeto the reference Find the distance axis. rotation Earth's fixed to the and lationally to the Earth'ssurface.The radius of the Earth and from the satellite on its surface are supposed to be known. the free-fall acceleration orbit of a stationary i.223.Calculatethe radius of the circular with respectto its surwhich remainsmotionless Earth's satellite, in the inertialreference face.What are its velocity and acceleration of the Earth? the centre to moment a at fixed frame given orbit of rai.22A.A satelliterevolving in a circularequatorial certain a o ver east to west point 2.00-t0km from dius R appears at the equator every = ii.6hours. Using these data, calculate the mass of the Earth. The gravitationalconstantis supposed to be known. revolves from eastto west in a circularequatorial A satellite 1.225. t0s km around the Earth.Findthe velocity orbitof radiusR ----t.00. frame fixed to of the satellitein the reference and the acceleration the Earth. must move in the equatorialplane of the Earth A satellite 1.226. in the Earth'srotation directionor against either closeto its surface kineticenergy of the satellitein the it. Find how many timestheformer frame fixed case(in the reference lattercaseexceedsthat in the to the Earth). of the Moon revolves in a circular An artificialsatellite 1.227. the orbit whose radius exceedsthe radius of the Moon times.In d ue to resistance a satellite experiencesslight processof motionthe cosmicdust.Assumingthe resistanceforce to depend on the velocity of the satelliteas F = av where a is a constant,find how long the satellitewill stay in orbit until it falls onto the Moon'ssurface. Calculatethe orbital and escapevelocitiesfor the Moon. 1.228. Compare the resultsobtained with the correspondingvelocitiesfor the Earth. the Moon alonga parabolictrajecA spaceship 1.229. approaches moment tory which is almost tangent to the Moon'ssurface.At the time short for a fired w as brakerocket the of the maximumapproach of circularorbit a into transferred was the and interval, spaceship increased modulus the Find how a Moon satellite. spaceshipvelocity in the processof braking. is launchedinto a circularorbit closeto the A spaceship 1.230.

Earth'ssurface.What additionalvelocity has to be imparted to the spaceshipto overcomethe gravitationalpull? At what distancefrom the centreof the Moon is the point 1.231. at which the strength of the resultant of the Earth's and Moon's gravitationalfields is equal to zero?The Earth'smass is assumed to be -= 8i timesthat of the Moon, and the distancebetween the centresof theseplanetsn = 60timesgreaterthan the radiusof the Earth R. \1771

i.232.What

is the minimum work that has to be performed to a bring spaceshipof mass m -= 2.0-t03 kg from the surfaceof the Earth to the Moon? the the third cosmicvelocity v3, Find approximately minimum velocity that has to be impartedto a body relativeto the Earth'ssurface to drive it out of the Solarsystem. The rotation of the Earth about itsown axis is to be neglected.

i.e.

1.233.

1.5.DYNAMIES \342\200\242

Equation

\177

---=

SOLIDBODY

of dynamics of a solid body rotating about a statiortary axis z: (t.5a) z Nz, algebraicsum of the moments of external forcesrelative to the -----

I\1773

\342\200\242

where

axis. \342\200\242

\342\200\242

is the

According

Steiner'stheorem:

I = Ic

Kinetic energy of a solid

\177-

ma2.

body rotating t

about a

stationary

(t.5b) axis:

Ira2.

(t .5c)

-\177

\342\200\242

Work performed by axis: stationary

about a

external forcesduring A

the rotation

of a solid body

(1.5d)

Nz dip. 1\"

\342\200\242

Kinetic energy of a solid body

T-- Ic(o* 2

\1771

\177,

\342\200\242

its

[

plane motion: mv\177

(1.5e)

the angular velocity (a' of gyroscopeprecession, equal to I(o, and the moment N of the externalforces: [re'M]= N. (t.5f)

Relationship between momentum M

angular

i.234.A thin

uniform rod AB of mass m = t.0kg moves translaacceleration w = 2.0m/s2 due to two antiparallel forces tionally F1and (Fig.t.52).The distancebetween the pointsat which these forces are appliedis equal to a = 20 cm. Besides, it is known that 5.0N. Find the length of the rod. i.235.A force F = Ai -4-Bj is appliedto a point whose radius vector relativeto the origin of coordinates O is equal to r ai + A- bj, where a, b, A, B are constants,and i, j are the unit vectorsof with

F\177

\177-

\177

47)))

the x and y axes.Find the moment N and the arm l of the force F relativeto the point O. A forceF1= Aj is appliedto a point whose radiusvector 1.236. a force Fs Bi is appliedto the point whose radius r1 ai,while vector rs ----bj. Both radius vectors are determinedrelativeto the O, I and j are the unit vectorsof the x and y origin of coordinates -\177

-\177

discequals rn = 7.3kg. Find the moment of inertia of such a disc relativeto the axispassingthrough its centreof inertia and perpendicularto the plane of the disc.

|.242.Usingthe formula

for the moment of inertia of a uniform the moment of inertia of a thin sphericallayer of mass sphere, relative to the axis passingthrough its centre. m and radius 1.243. A light thread with a body of mass m tiedto its end is wound on a uniform solidcylinder of mass M and radius R (Fig. At a moment t = 0 the system is set in motion. the Assumingthefrictionintheaxleof der to be negligible, find the time dependence of (a) the angular velocity of the cylinder; (b) the kineticenergy of the whole system. The ends of thin threads tightly 1.244. wound on the axleof radius r of the Maxwell discare attached to a horizontal bar. When the discunwinds, the bar is raisedto keepthe Fig. 1.54. discat the sameheight. The mass of the disc with the axle is equal to m, the moment of inertiaof the arrangementrelativeto its axisis Find the tension of each thread and the acceleration of the bar. 1.245. A thin horizontaluniform rod AB of mass m and length can rotate freely about a vertical axis passingthrough its end A. At a certainmoment the end B starts experiencing a constantforce find

/\177

1.55).

cylin-\177\177

Fig.

i.52.

Fig.

axes,a, b, A, B are constants.Find the arm relativeto the point O.

i.53.

l of the

resultant force

Three forcesare appliedto a square plate as shown in 1.237. t.53. Find the modulus,direction,and the point of application Fig. of the resultant force,if this point istaken on the sideBC.

Find 1.238.

the moment of inertia rod relativeto the axiswhich is perpendicular a uniform of thin (a) to the rod and passesthrough its end, if the mass of the rod is m and

itslength l;

(b) of a thin uniform rectangularplaterelativeto the axispassing to the plane of the plate through one of its vertices, perpendicular of the plateare equal to a and b, and its mass is m. if the sides 1.239. Calculatethe moment of inertia (a) of a copperuniform discrelativeto the symmetry axisperpen-

mm dicularto the plane of the disc,if itsthicknessis equal to b=2.0 ----t00mm; and its radius to (b) of a uniform solidcone relativeto itssymmetry axis, if the mass of the cone is equal to m and the radius of its baseto that in the caseof a thin plate of arbitrary Demonstrate 1.240. is the following relationshipbetween the moments of shape there inertia:I1 Is I3,where subindicest, 2, and 3 define threemuaxespassingthroughone point, with axest and tually perpendicular 2 lying in the plane of the plate.Using this relationship,find the moment of inertia of a thin uniform round discof radius/tand mass rn relativeto the axis coincidingwith one of its diameters. = 20 cm has a round cut as A uniform disc of radius 1.241. The massof the remaining(shaded)portionof the shown in Fig.t.54.

Fig.i.56.

/\177

/\177.

\177

-----

/\177

F which is always perpendicularto the originalposition of the stationary rod and directedin a horizontalplane.Find the angularvelocity of the rod as a function of its rotationangle counted relative to the initial position. In the arrangementshown in Fig.1.56 1.246. the mass of the uniform solidcylinder of radius R is equal to m and the masses of two bodiesare equal to and The thread slippingand the friction in the axleof the cylinder are supposedto be absent.Find the angular acceleration of the cylinder and the ratio of tensions of the verticalsectionsof the thread ia the processof motion. q\177

rn\177

rn\177.

T\177/T\177

4--9451

49)))

i.247.In the system shown in Fig.i.57the masses of the bodies are known to be ml and m2, the coefficient of friction betweenthe body rn 1 and the horizontalplane is equal to k, and a pulley of mass m is assumed to be a uniform disc.The thread does not slipover the Assumpulley.At the moment t 0 the body m2 starts descending. ing the mass of the thread and the friction in the axleof the pulley find the work performedby the friction forces acting to be negligible, over the first t secondsafter the beginningof motion. on the body A uniform cylinder of radius//is spinned about itsaxisto 1.248. and then placedinto a corner (Fig. the angular velocity \177

of the (a) the tensionof each thread and the angularacceleration

cylinder;

(b) the timedependenceof the instantaneouspower developedby the gravitational force. Thin threads are tightly wound on the ends of a uniform solidcylinder of mass m.Thefree ends of the threadsare attachedto

i.254.

m\177

1.58).

\177o0

Fig. Fig.

i.57.

The coefficient of friction between the cornerwalls and the cylinder before is equal to k. How many turns will the cylinder accomplish it stops? velocity i.249.A uniform discof radius R is spinned to the angular and then carefully placedon a horizontalsurface.How long will the discbe rotatingon the surface if the friction coefficient is equal to k? The pressureexertedby the discon the surface can be regarded \177o

as uniform.

i.250.A flywheel with the initial angularvelocity 0 decelerates due to the forceswhose moment relative to the axisis proportional to the square root of its angular velocity. Find the mean angular time. velocity of the flywheel averaged over the total deceleration rotatefreei.251.A uniform cylinder of radiusR and massM can A thin cord of ly about a stationary horizontalaxisO (Fig.i.59). a in on the is wound layer.Find mass m single cylinder length l and of the cylinder as a function of the length the angular acceleration x of the hangingpart of the cord.The wound part of the cordis supthe cylinder axis. posed to have its centreof gravity i.252.A uniform sphere of mass m and radius // rolls without \302\242o

i.59.

Fig.

i.60.

the ceilingof an elevatorcar.The carstartsgoingup with an acceleration w0. Find the acceleration w' of the cylinderrelativeto the carand the force F exertedby the cylinderon the ceiling(through the threads). A spool with a thread wound on it is placedon an inclined smooth plane set at an anglea 30 to the horizontal. The free end of the thread is attachedto the wall as shown in Fig. The mass of the spool is m ----200g, its moment of inertia relativeto its own axisI 0.45g.m2, the radius of the wound thread layer r 3.0cm. Find the acceleration of the spool axis. A uniform solid cylinder of mass m restson two horizontal planks.A thread is wound on the cylinder.The hangingend of the thread is pulledvertically down with a constant[orceF (Fig.1.62).

i.255.

\177

\302\260

i.6i.

\177

i.256.

o\177)

slippingdown an inclinedplane set at an anglea to the horizontal. Find: (a) the magnitudesof the friction coefficient at which slipping is absent; (b) the kineticenergy of the spheret secondsafter the beginning of motion. kg and radius // i.253.A uniform cylinder of mass rn a8.0 0 due to moment t i.3cm (Fig.i.60)starts descendingat find: gravity. Neglectingthe mass of the thread, \177

-\177

\177-

\177

Fig.

t.62.

Find the maximum magnitudeof the force F which stilldoesnot bring about any slidingof the cylinder,if the coefficient of friction between the cylinder and the planks is equal to k. What is the ac4* 5t)))

axisof the cylinder rollingdown the inclined plane? i.257.A spoolwith threadwound on it,of mass m, restson a rough horizontalsurface.Its moment of inertia relativeto its own axisis 2, where is a numericalfactor,and R is the outequal to I The radius of the wound thread layer is equal sideradiusof the spool. celeration

W,\177ax

\177

Of the

\177mTl

\177,

Find the acceleraR and 217.The mass of the threads is negligible. tion of the weight A after the system is set free. A uniform solid 1.260. cylinder Aof mass ml can freely rotateabout A confixed to a mount B of mass m2 (Fig. axis horizontal a stant horizontalforce F is appliedto the end K of a light thread tightly wound on the cylinder. The friction between the mount and the sup-

i.66).

porting horizontal plane is assumed

/111111111111111III1111111111IIIIit11111111

Fig.

to be absent.Find: of the point K; (a) the acceleration (b) the kineticenergy of this system t seconds after the beginningof

motion.

1.261.

A plank of mass rnl with a uniform sphereof mass m, placedon it restson a smooth horizontalplane.

t.63.

to r. The spool is pulled without slidingby the thread with a constant Find: force F directedat an anglea to the horizontal(Fig.t.63). axis the o f vector of the the acceleration spool on the (a) projection

x-axis; (b) the work performedby the ter the beginningof motion.

secondsafThe arrangementshown in Fig.i.64consistsof two identical 1.258.

uniform

force

F during the

solidcylinders,eachof mass m,

first

on which two light threads

constant horizontalforceF is appliedto the plank. With what will the plank and the centreof the spheremove proaccelerations

vided there is no sliding between the plank and the sphere? |.262.A uniform solidcylinder of mass m and radius is set in rotationabout itsaxiswith an angularvelocity coo,then loweredwith its lateralsurface onto a horizontalplane and released.The coefficient of friction between the cylinder and the plane is equal to k. /\177

Find:

(a) how long the cylinder will move with sliding; (b) the total work performed by the sliding friction force acting on the cylinder. A uniform ballof radius r rollswithout slippingdown from the top of a sphereof radius R. Find the angular velocity of the ball at the moment it breaksoff the sphere.The initial velocity of the

ball is negligible.

|.264.A uniform solidcylinder of radius/\177

horizontalplane passing into

i.64.

Fig.

15cm rollsover a

inclined plane forming

angle

i.65.

are wound symmetrically. Find the tensionof eachthread in the processo[lnotion.The friction in the axle of the upper cylinder is assumed to be absent. t.259.In the arrangementshown in Fig.t.65a weight A possesses mass M. Also known are the moment of mass m. a pulley B possesses iaei'tiaI of the pulley relativeto its axisand the radiiof the pulley 52

t.66.

1.263.

Fig.

i.68.

30 with the horizontal(Fig.i.67). Find the maximum value of the velocity v0 which stillpermitsthe cylinder to rollonto the inclined plane sectionwithout a jump.The slidingis assumedto be absent. \302\260

\302\242z

\177

53)))

i.265.A smallbody A is fixed to the insideof a thin hoop of radius R and mass equal to that of the body A. The hoop rigid rolls without slippingover a horizontal plane;at the moments when the body A getsinto the lowerposition, the centreof the hoop moves with velocity

Fig.

i.69.

Fig.

still.Explain the

\177 \177

rests on a smooth horizontal surface.One of the ends of the rod is struck with the impulse J = 3.0N.s in a horizontal direction perpendicular to N-s.Find the rod.As a result,the rod obtainsthe momentum p=3.0 the forcewith which one half of the rod will act on the other in

i.70.

At what values of v 0 will the (Fig.1.68). hoop move without bounc1.266. Determinethe kinetic of a tractorcrawlerbelt of mass m if the tractormoves with energy velocity v (Fig.1.69). 1.267. A uniform sphereof mass m and radius r rollswithout sliding over a horizontal plane,rotating about a horizontal axle OA In the process, (Fig.1.70). the centreof the spheremoves with velocity v along a circle of radiusR. Find the kineticenergy of the sphere. 1.268. Demonstrate that in the reference frame rotating with a constant angular velocity co about a stationary axis a body of mass m experiences the resultant = (a) centrifugal force of inertia = mcoZRc, where R c is the radius vector of the body's centreof inertia relative to the rotation axis; (b) Coriolisforce Fcor=2m where is the velocity of the body's centre of inertia in the rotating referenceframe. 1.269. A midpoint of a thin uniform rod AB of mass m and length l is rigidly fixed to a rotationaxleOO' as shown in 1.7i. he Fig. rod is set into rotation with a constantangular velocity co. FindTthe resultant moment of the centrifugalforces of inertiarelativeto the point C in the reference frame fixed to the axleOO' and to the rod. 1.270. A conicalpendulum, a thin nniform rod of length l and massm,, rotatesuniformly about a verticalaxiswith angular velocity co (the upper end of the rod is Find the angle0 between the hinged). rod and the vertical. 1.271. A uniform cube with edgea restson a horizontal whose friction coefficient equalsk. The cube is set in motion withplane an initial velocity, travels some distanceover the plane and comesto a standv0

processof motion. thin uniform square plate with side l and mass M can rotatefreely about a stationary verticalaxis coincidingwith one of A smallballof mass m flying with velocity v at right angles its sides. to the platestrikeselasticallythe centre of it. Find: (a) the velocity of the ball v' after the impact; (b) the horizontalcomponentof the resultantforce which the axis will exerton the plateafter the impact. t.275.A vertically orienteduniform rod of mass M and length l can rotateabout its upper end.A horizontally flying bullet of mass m strikesthe lowerend of the rod and gets stuck in it;as a result,the the

ing?

1.274.A

rod swings through an angle a. Assuming that m << M, find: (a) the velocity of the flying bullet; (b) the momentum incrementin the system \"bullet-rod\" during the impact;what causesthe changeof that momentum; (c) at what distancex from the upper end of the rod the bullet must strikefor the momentum of the system \"bullet-rod\"to remainconstant during the impact. A horizontally orienteduniform discof mass M and radius 1.276. .R rotatesfreely about a stationary verticalaxispassingthrough its centre.The dischas a radial guide along which can slidewithout frictiona smallbody of mass m.A light thread running down through the hollow axle of the disc is tied to the body. Initially the body was locatedat the edgeof the discand the whole system rotatedwith an angularvelocity coo.Then by meansof a force F appliedto the lower end of the thread the body was slowly pulled to the rotation

Fc\177

[vb\302\242o],

v\177

of the angular momentum of the disappearance in the plane at. right angles to the cube'smotion direction.Find the distancebetween the resultantsof gravitationalforces and the reactionforces exertedby the supporting plane. t.272.A smooth uniform rod AB of mass M and length l rotates freely with an angular velocity coo in a horizontalplane about a stationary vertical axis passingthrough its end A. A small sleeveof mass m startsslidingalongthe rod from the point A. Find the velocity v' of the sleeverelative to the rod at the moment it reachesits other end B. A uniform rod of mass m = 5.0kg and le\177gth l = 90 cm 1.273.

cuberelativeto the axislying

axis. Find:

! ..

(a) the angularvelocity of the system in its final state; (b) the work performed by the force F. 1.277.A man of mass ml stands on the edgeof a horizontaluniof rotating freely form discof mass m2 and radiusR which is capable about a stationary vertical axispassing through its centre.At a cer55)))

tain moment the man

starts moving along the edgeof

the

disc;he

shifts over an angleq)' relativeto the discand then stops. In the processof motion the velocity of the man varies with time as v\" (t). find: Assuming the dimensionsof the man to be negligible, (a) the anglethrough which the dischad turned by the moment the

man

stopped; man actedon

(b) the force moment (relative to the rotation axis) with which

disc in the processof motion. Two horizontaldiscsrotatefreely about a verticalaxispass1.278. ing through their centres.The moments of inertiaof the discsrelative to this axisare equal to and I2,and the angularvelocitiesto ah and When the upper discfell on the lower one, both discsbegan rotating,after sometime, as a singlewhole (due to friction). Find: the

the

I\177

\177)\177.

(a) the steady-state angular rotation velocity of the discs; (b) the work performed by the friction forces in this process. 1.279.A smalldiscand a thin uniform rod of lengthl,whosemass is timesgreaterthan the mass of the disc,lie on a smooth horizontal plane.The discis set in motion,in horizontaldirectionand perpendicularto the rod, with velocity v, after which it elastically collides with the end of the rod.Find the velocity of the discand the angular velocity of the rod after the collision. At what value of will the velocity of the disc after the collision be equal to zero?reverseits di\177l

\177l

rection?

1.280.A stationary platform which can rotate freely about a verticalaxis(Fig.1.72) supports a motor M and a balanceweight N. The moment

of i-nertia of

with the motor and the

the

platform

balanceweight

Fig.

1.72.

t80\302\260;

arrangementin the verticalpositionafter the motor turns the shaft 90 1.281. A horizontally oriented uniform rod AB of mass m = = 1.40kg and length l 0 = t00cm rotatesfreely about a vertical axisOO' passingthrough itsend A. The point A stationary is located at the middleof the axis OO' whose length is equal to l = 55cm. At what angularvelocity of the rod the horizontalcomponentof the force actingon the lower end of the axisOO' is equal to zero?What 56

\302\260.

1.283. \302\260

--=-

/\177

\177

\302\242)

\177o

BB' through

\177).

(a) the magnitudeand directionof the rod'sangularmomentum M relativeto the point C, as well as its angular momentum relativeto the rotation axis; (b) how much the modulus of the vector M relativeto the point C increases during a half-turn; (c) the moment of externalforces N acting on the axle OO' in the processof rotation. A top of mass m = 0.50 kg, whoseaxisis tiltedby an a\177gle due to gravity. The moment of 0 = 30 to the vertical,precesses inertia of the top relativeto its symmetry axis is equal to I = 2.0g.m2, the angularvelocity of rotation about that axisis equal to o) = 350rad/s,the distancefrom the point of rest to the centreof inertia of the top is l = 10cm.Find: (a) the angularvelocity of the top'sprecession; (b) the magnitudeand directionof the horizontalcomponent of the reactionforce actingon the top at the point of rest. = 5.0cm at the A gyroscope,a uniform disc of radius 1.284. is mounted on the floor end of a rod of length l = 10cm (Fig.1.73), of an elevatorcargoing up with a constantaccelrod erationw = 2.0m/s2. The other end of t is hingedat the point O. The gyroscope precesses with an angular velocity n = 0.5 rps. \177I Neglectingthe friction and the mass of the rod, find the properangularvelocity of the disc. t.285.A top of mass m = t.0kg and moment of inertia relativeto its own axis I = 4.0g-m Fig. i.73. o) = an angular velocity spins with = 310rad/s.Its point of rest is locatedon a blockwhich is shifted w = t.0m/s in a horizontaldirectionwith a constantacceleration The distancebetweenthe point of restand the centreof inertia of the top equals l = t0 cm.Find the magnitude and directionof the angular velocity of precession of mass m = 5.0kg and radius R = A uniform 1.286. = 6.0cm rotateswithsphere rad/s about an angular velocity o) = 1250 a horizontalaxlepassingthrough its centreand fixed on the mounting base by meansof bearings.The distancebetween the bearings equals l = 15cm. The base is set in rotation about a verticalaxis with an angula;r velocity o)' = 5.0rad/s.Find the modulus and directionof the gyroscopicforces. 1.287. A cylindricaldisc of a gyroscope of mass m = t5 kg and radius r = 5.0cm spins with an angular velocity o) = 330 rad/s. tl\177e

relativeto this axis is equal to I. A light frame is fixed to the motor'sshaft with a uniform sphereA rotating freely with an angular velocity 0 about a shaft BB' coinciding with the axisOO'.The moment of inertia of the sphererelative to the rotation axisis equal to I 0. Find: (a) the work performed by the motorin turnin'g the shaft BB' through through (b) the moment of externalforceswhich maintainsthe axisof the 90\302\260;

is in this casethe horizontalcomponent of the force acting on the upper end of the axis? The middleof a uniform rod of mass m and length l is rig1.282. fixed to a vertical axisOO' so that the angle between the rod idly The ends of the axisOO' are and the axisis equal to 0 (seeFig.1.71). provided with bearings.The system rotateswithout friction with an angularvelocity Find:

\177

\177.

\177)'.

57)))

distancebetween the bearingsin which the axle of the disc is mounted is equal to l t5 cm.The axle is forced to oscillateabout a horizontalaxiswith a periodT t.0s and amplitude 20 Find the maximum value of the gyroscopicforcesexertedby the axle on the The

-\177

bearings. A ship moves with i.288.

\177

36 km

velocity v

\302\260.

(p,\177

hour

per along arcof a circleof radius R 200 m. Find the moment of the gyroscopic forces exertedon the bearingsby the shaft with a flywheel whose moment of inertia relative to the rotation axis equals I a 3.8.10 kg.m2 and whose rotation velocity n----300 The rotation axis is orientedalong the length of the ship. rpm. i.289.A locomotiveis propelledby a turbine whoseaxle is parallel to the axesof wheels.The turbine'srotation directioncoincides with that of wheels.The moment of inertia of the turbine rotorrelative to its own axis is equal to I 240 kg-m2. Find the additional force exertedby the gyroscopicforces on the railswhen the locomotive moves along a circleof radius R 250 m with velocity v 50 km per hour. The gaugeis equal to l 1.5m. The angular 1500rpm. velocity of the turbine equals n \177-

\177

-----

--\177

1.6.ELASTICDEFORMATIONS

SOLID BODY

Relation between tensile (compressive) strain e and

\342\200\242

e= where

\177

stress

(t.6a)

\177/E,

h Young's modulus. Relation between lateral compressive(tensile) strain \177s

\342\200\242

nal tensile (compressive) strain

where

\177

\342\200\242

is Poisson'sratio. Relation between shear

e: e'

strain V

---=

e' and

(t.6b)

--\177e,

-f

longitudi-

and tangential

= \"de,

stress

\177:

(l.6c)

G is shear modulus.

where \342\200\242

Compressibility: i \177=

dV \"

(i.6d)

i.290.What pressurehas to be appliedto the ends of a steelcylinderto keepitslengthconstant on raisingits temperatureby t00 i.291.What internal pressure(in the absenceof an externalpressure) can be sustained (a) by a glasstube; (b) by a glass sphericalflask, if in both cases the wall thicknessis equal to hr t.0mm and the radius of the tube and the flask equals r -= 25 mm? i.292.A horizontally oriented copperrod of length l =-1.0m is rotatedabout a verticalaxis passingthrough its middle.What is the number of rps at which this rod ruptures? i.293.A ring of radius r -= 25 cm made of leadwire is rotated about a stationary verticalaxispassingthrough its centre and perpendicularto the plane of the ring. What is the number of rps at which the ring ruptures? i.294.A steelwire of diameterd t.0mm is stretchedhorizontally between two clampslocatedat the distance l 2.0m from eachother.A weight of mass m 0.25kg is suspendedfrom the midpoint O of the wire.What will the resultingdescentof the point O \302\260C?

-\177

\177

\177--

be in centimetres?

i.295.A uniform elasticplank moves over a smooth horizontal plane due to a constant force F0 distributeduniformly over the end face.The surface of the end face is equalto S,and Young's modulus of the materialto E. Find the compressive strain of the plank in the directionof the acting force. i.296.A thin uniform copperrod of length l and mass m rotates uniformly with an angularvelocity co in a horizontalplane about a vertical axispassingthrough one of its ends.Determine the tension in the rod as a function of the distancer from the rotation axis.Find the elongationof the rod. i.297.A solidcoppercylinder of length l 65 cm is placedon a horizontal surface and subjectedto a vertical compressiveforce F ----t000N directeddownward and distributeduniformly over the end face.What will be the resultingchange of the volume of the cylinder in cuhic millimetres? A copper 1.298. rod of lengthl is suspendedfrom the ceilingby one

of its ends.Find: (a) the elongationhl of the rod due to its own weight; (b) the relativeincrementof its volume AV/V. A bar made of materialwhose Young'smodulus is equal to E and Poisson's ratio to is subjectedto the hydrostatic pressure

i.299. p. Find: the

\177

fractionaldecrement of its volume; (a) (b) the relationshipbetween the compressibilityand the elastic constants E and Show that Poisson's ratio cannot exceedt/2. i.300.One end of a steelrectangulargirder is embedded into a wall (Fig.t.74).Due to gravity it sagsslightly. Find the radius of curvature of the neutral layer (see the dotted line in the figure) in I\177

\177.

\342\200\242

Volume

density

elastic strain energy: u

\177

Eee/2,

\177

G\177/2.

\177

(i.6e)

59)))

the vicinity of the point O if the length of the protruding sectionof

Fig.

the girderis equalto l = h

= t0 cm.

A steelplateof thicknessh has the shapeof a squarewhose 1.304. sideequalsl, with h << l.The plateis rigidly fixed to a verticalaxle

i.74. Fig.

6.0m and the thicknessof the girderequals

1.301.

The bendingof an elasticrod is describedby the elastic curve passingthrough centresof gravity of rod'scross-sections. At smallbendings the equationof this curve takesthe form d2y N (x)= E1 dx

\177

OOwhich is rotatedwith a constantangularacceleration(Fig.1.79). Find the deflection assuming the saggingto be small. Determinethe relationshipbetween the torque N and the torsion angle for (a) the tube whose wall thicknesshr is considerably lessthan the \177

1.305.

l\177Eer

q\177

Their length l, ra(b) for the solidrod of circularcross-section. dius r, and shear modulus G are supposed to be known.

Fig.

\177___\177

t.75.

\177,,

tube radius;

\177

where N (x) is the bendingmoment of the elasticforces in the crosssectioncorrespondingto the x coordinate,E is Young'smodulus, I is the moment o] inertia of the cross-section relativeto the axispassing through the neutral layer (I = Iz\177dS' Fig.t.75). withside Supposeone end of a steelrod of a squarecross-section into a wall, the protrudingsectionbeing of length l a is embedded

Fig.

t.77.

Fig.

t.78.

Fig.

Calculatethe 1.306.

t.76.

\177.79.

steeltube of

length torque twisting = 2.0about its axis,if the insideand = 3.0m through an angle = outside diametersof the tube are equal to = 30 mm and = 50 mm. \302\260

q\177

find the Assuming the mass of the rod to be negligible, (Fig.1.76). shapeof the elasticcurve and the deflectionof the rod if its end A experiences (a) the bendingmoment of the coupleNo; (b) a force F orientedalongthe y axis. 1.302. A steel girder of length l rests freely on two supports is equal to I (Fig.t.77).The moment of inertia of its cross-section (see the foregoingproblem).Neglectingthe mass of the girderand find the deflection due to the force assumingthe saggingto be slight, F appliedto the middleof the girder. 1.303. The thicknessof a rectangularsteelgirderequalsh. Using the equationof Problem1.301, find the deflection causedby the weight of the girderin two cases: into a wall with the length (a) one end of the girderis embedded of the protruding sectionbeing equal to l (Fig. t.78a); (b) the girderof length2lrestsfreely on two supports (Fig.t.78b). \177,,

\177,

d\177

Find the maximum power which can be transmitted by 1.307. meansof a steelshaft rotatingabout its axiswith an angularvelocity = 120rad/s, if its length l 200cm, radius r = 1.50cm, and the permissibletorsion angle = 2.5 is A uniform ring of mass m, with the outside radius 1.308. fitted tightly on a shaft of radius The shaft is rotatedabout its axis with a constant angular acceleration Find the moment of elasticforces in the ring as a function of the distancer from the rotation axis. \177

\177

\302\260.

\177p

r\177,

r\177.

\177.

Find the elasticdeformation of a steelrod of mass 1.309. = 3.1kg stretchedto a tensilestrainenergy e = 1.0.10 A steelcylindricalrod of length l and radius r is suspended 1.310.

by its end from the ceiling. (a) Find the elasticdeformation energy U of the (b) Define U in termsof tensilestrain hl/l of the

-\177.

rod. rod. 6t)))

1.31i.What

work has to be

to make a

out of a

performed hoop steel hand of length l = 2.0m, width h = 6.0cm, and thickness 6 = 2.0mm? The process is assumed to proceedwithin the elasticity range of the material. 1.312. Find the elasticdeformation energy of a steel rod whose one end is fixed and the other is twisted through an angleq) = 6.0 The length of the rod is equal to l ----t.0m, and the radius to r = ----t0 mm. 1.313. Find how the volume density of the elasticdeformation energy is distributedin a steelrod dependingon the distancer from its axis.The length of the rod is equal to l, the torsion angle to 1.314. Find the volume density of the elasticdeformation energy in fresh water at the depth of h = t000m.

the volume of water flowing acrossthe pipe'ssectionper unit time if the difference in water columnsis equal to Ah. is mounted alongthe axisof a gas A Pitottube (Fig.1.82) area is equal to cross-sectional whose Assumingthe vispipeline cosity to be negligible,find the volume of gas flowing acrossthe

1.317.

\302\260.

\177.

t.7.HYDRODYNAMICS The fundamental equation): \342\200\242

equation of dv

p where p is the fluid the case of gravity), \342\200\242

density,

Bernoulli'sequation. In 2

\342\200\242

any

steady

the

\177pgh-\177-p=

flow

= pgin

const

number

the flow

defining

(t.7b)

pvl]\177

of a viscous fluid:

pattern

(1.7c)

l is a characteristiclength, is the fluid viscosity. Poiseuille'slaw. The volume of liquid flowing through a circular

me/s):

r\177R* S\177

Pl--P\177 \177

p\177

(t.7d) p\177

i\177

\342\200\242

t.3t5.Idealfluid flows along a flat tube of constantcross-section, locatedin a horizontalplane and bent as shown in Fig.t.80(top of the fluid view).The flow is steady.Are the pressuresand velocities equal at points 1and 2? What is the shape of the streamlines? 1.316. Two manometric tubes are mounted on a horizontalpipe of varying cross-section at the sections and (Fig.t.8t).Find S\177

S\177

= 20 cm.

wide cylindricalv-essel 50 cm in is filled with water and restson a table. height find at Assuming the viscosity to be negligible, what heightfrom the bottom of the vessela small hole should be perforated for the water jet coming out of it to hit the surface of the table at the maximum distance

l,\177,x

from

the

vessel.

\177_\177

\177\177h Fig.

1.82.

Find l\177. tube

B and l are the tube'sradius and length, the pressure differencebetween the endsof the tube. Stokes'law. The friction forceon the sphereof radius r moving through a viscousfluid: F = 6\177lrv. (t.7e) where

i.3i8.

1.3t9.A

streamline.

Reynolds

sectionof the pipeperunit time,if the difference in the liquid colof the liquid and the gas are umns is equal to Ah, and the densities and p respectively.. P0 A wide vessel with a smallhole in the bottom is filled with water and kerosene. Neglectingthe viscosity, find the velothe thickness of the if water of the flow, city water layer is equal to h\177=-30 cm and that of the kerosenelayer to

ideal fluid

\177l

\342\200\242

(in

of mass forces(f

the volume density

Re = where

(t.7a)

pressuregradient.

pv u

along

= f--Vp,

\177-

Vp is the

of ideal fluid (Eulerian

hydrodynamics

A bent tube is lowered into a water stream as shown in 1.320. t .83. Fig. The velocity of the stream relativeto the tube is equalto v

\177

2.5m/s.The closedupper end of the

tube locatedat the height will the water let

t2 cm has a smallorifice.To what heighth

spurt?

The horizontalbottom of a wide vessel with an idealfluid 1.321. has a round orifice of radius/\177 over which a round closedcylinder is (Fig.t.84).The clearancebetween mounted,whose radius \177 the cylinder and the bottom of the vesselis very small,the fluid density is p. Find the staticpressureof the fluid in the clearanceas a function of the distancer from the axisof the orifice (and the cylinder), if the height of the fluid is equal to h. 1.322. What work should be done in orderto squeezeall water from a horizontally locatedcylinder (Fig.t.85)during the time t by meansof a constantforce actingon the piston?The volume of waareaof the oriter in the cylinder is equal to V, the cross-sectional B\177

B\177

63)))

fice to s, with s being considerably less than the piston area.The friction and viscosity are negligibly small. A cylindricalvessel of height h and base area S is filled with water. An orifice of areas << S is openedin the bottom of the vessel.Neglectingthe viscosity of water, determinehow soon all the water will pour out of the vessel.

|.323.

|.32\177. A horizontally oriented tube AB of length l rotateswith a constant angular velocity co about a stationary vertical axisOO' passingthrough the end ,4 (Fig.1.86). The tube is filled with an idealfluid. The end A of the tube is open, the closedend B has a very smallorifice. Find the velocity of the fluid relativeto the tube as a function of the column \"height\"

i.329.A sidewall of a wide open tank is provided with a narrowing tube (Fig.i.88) through which water flows out.Thecross-sectional area of the tube decreasesfrom S = 3.0cm to s = t.0cm The = \177

water level in the tank is h

4.6m higherthan

\177.

that in the

tube.

Fig..1.86. Neglectingthe viscosity of the water, find the horizontalcomponent of the forcetending to pull the tube out of the tank.

Demonstratethat 1.325.

in the case Fig. 1.83. idealflnid Eq. {t.7a)turns into Bernoulliequation. 1.326. On the oppositesidesof a wide vertical vessel filled with water two identicalholesare opened,eachhaving the cross-sectional

of a steady

of an

flow

Fig.

t.87.

Fig.

t.88.

vesselwith water is rotatedabout its verA cylindrical 1.330. tical axis with a constantangular velocity co. Find: Fig.

1.84.

Fig.

t.85.

area S = 0.50cm The height difference between them is equal to Ah ----5t cm. Find the resultant force of reactionof the water flowing out of the vessel. |.327.The sidewail of a wide verticalcylindricalvesselof height h = 75 cm has a narrow vertical slitrunning alltheway down to the bottom of the vessel.The lengthof the slitis l = 50 cm and the width b = 1.0mm. With the slitclosed,the vessel is filled with water. Find the resultantforce of reactionof the water flowing out of the vesselimmediately after the slitis opened. \177.

1.328. Water flows out of a big tank

(a) the shape of the free surface of the water; (b) the water pressuredistributionover the bottom of the vessel alongits radiusprovided the pressureat the centralpoint is equal to

Po.

discof radiusR = t0 cm is locatedwithi.33|.thin horizontal = 0.08P filled with oil whose A

in a

viscosity cylindrical cavity between the discand the horizontalplanes The clearance (Fig.i.89). \177l

alonga tube bent at right an-

gles:the insideradius of the tube is equalto r -- 0.50cm (Fig. 1.87). The lengthof the horizontalsectionof the tube is equalto l = 22 cm. The water flow rate is 0 ----0.50]itresper second.Find the moment of

Fig.

i.89.

i.0

to the point O.

of the cavity is equal to h ---- mm.Find the power developedby the viscousforces acting on the discwhen it rotateswith the angular velocity o) 60rad/s.The end effects are to be neglected.

(54

5--9451

reactionforcesof flowing water, actingon the tube'swalls,relative

\177

65)))

A long cylinder of radius R t 1.332.

with a

constantvelocity

is displacedalong its axis inside a stationary co-axialcylinder of

The spacebetweenthe cylinders is filled with viscous liqthe velocity of the liquid as a function of the distancer from the axisof the cylinders.The flow is laminar. A fluid with viscosity r1 fills the spacebetween two long co-axialcylinders of radii /71and R\177, with /71</7\177. The inner cylinder is stationary while the outer one is rotatedwith a constant The fluid flow is laminar.Taking into account angular velocity that the friction force acting on a unit areaof a cylindricalsurface of radius r is defined by the formula = fir (Oco/Or),find: (a) the angular velocity of the rotating fluid as a function of raradius/7\177.

uid. Find

1.333.

\302\242o\177.

\177r

dius r; (b) the moment of the frictionforces actingon a unit lengthof the outer cylinder. A tube of length l and radius /7 carriesa steady flow of 1.334. fluid whose density is p and viscosity r1.The fluid flow velocity depends on the distancer from the axisof the tube as v = v 0 (i Find: (a) the volume of the fluid flowing acrossthe sectionof the tube per unit time; tube'svolume; (b) the kineticenergy of the fluid within the the fluid; by (c) the friction force exertedon the tube the tube. of the ends at difference the (d) pressure In the arrangementashown in Fig.1.90a viscous liquid 1.335. whosedensity is p = i.0g/cm flows alonga tube out of a wide tank

= viscosities of glycerin and water are equal to rh = i3.9P and = 0.0iiP respectively. 1.338. A leadsphereis steadily sinkingin glycerin whoseviscosity is equalto = 13.9 P. What is the maximum diameterof the sphere \1771,,.

at which the flow around that sphere stillremainslaminar? It is known that the transitionto the turbulent flow corresponds to Reynolds number Re = (Herethe characteristic length is taken to be the sphere diameter.) \1771

0.5.

t.339.A steel ball of diameterd = 3.0mm starts sinkingwith

initialvelocity in olive oilwhoseviscosity is r1 ----0.90P. How soon after the beginningof motionwill the velocity of the ball differ from. the steady-state velocity by n = t.0%? zero

t.8.RELATIVISTIC \342\200\242

MECHANICS

Lorentz contraction of

of a moving at0

and slowing

length

V\177, at-]/t--

/=to

r\177/R\177).

\177

(\177lc)

where \342\200\242

is the proper length and Lorentz transformation*:

x'-- x--Vt V

\342\200\242

\177

At,

-\177

the

proper time

(t.8a)

of the moving

xV/c= t' t-t--

(V/c)\177

(Vl\177)

el\177ek:

clock.

(i.Sb)

Interval sz= is an invariant:

= --c S12-t12--l\177==inv,

(t.8c)

=\177

is the time interval between events I the points at which these events occurred. of velocity*: Transformation where tzs

and

is the distancebetween

lz\177,

\342\200\242

I Relativistic mass m

Fig.

i.90.

and rn

]/t-

= c\177

(t.8d)

v\177VIc\177

relativistic momentum:

o =

p = mv =

(\177,/0

rno V

(t.8e)

(\177,/\177)=

is the rest mass, or, simply, the mass. Relativisticequation of dynamics for a particle:

where m o

A. Find the velocity of the liquid flow, if h, = 10era, = 20era, and h 3 = 35 cm. All the distancesl are equal. radius of a pipelinedecreases The cross-sectional 1.336. gradually as r = r0e-\177, where a -- 0.50m -1,x is the distancefrom the pipeline inlet.Find the ratio of Reynolds numbers for two cross-sections

\342\200\242

h\177

separatedby Ax = 3.2 m. 1.337. When a sphere of radius r, = i.2mm moves in glycerin, the laminarflow is observed if the velocity of the sphere doesnot exceedv, = 23 cm/s.At what minimum velocity of a sphereof radius r,z = 5.5 cm will the flow in water becometurbulent?The v\177

v\177V/c=

d---\177P

= F,

the relativistic momentum of the particle. Total and kinetic energiesof a relativistic particle: E= mc== mec= q- T, T= (m -- mo) c=.

(t.8f)

p is

where \342\200\242

(t.8g)

tive the

Thereferenceframe K' is assumed to move with a velocity V in the posidirectionof the x axis of the frame K, with the x' and x axescoinciding and y' and y axes parallel. 67)))

\342\200\242

ticle

between

Relationship

E\"--p\177\"=

When considering the quantity: ing invariant \342\200\242

and

collision,and

1/r

(i.8h)

(r-t-2r,\177o\177).

collisionsof particlesit helps to use the followE\"

where E

p\177--

rn\177,

of a relativistic par-

and momentum

the energy

-- = \177'c\177

(i.8i)

m\177c',

prior to the ivare the total energy theand momentumtheof the system system) formed. ra o is the rest mass of particle (or

A rod moves lengthwisewith a constantvelocity v relative 1.340. frame K.At what value of v will the length to the inertialreference of the rod in this frame be q ----0.5%lessthan itsproperlength?

In a triangle the properlength of each sideequalsa. Find 1.341. relative frame in the reference

the perimeterof this triangle to it with a constantvelocity V along one of its (a)

bisectors;(b) sides.

Investigatethe resultsobtainedat

<< c and

--\177

moving

c, where c is the

velocity of light. Find the properlength of a rod if in the laboratory frame 1.342. m, and the itsvelocity is v = c/2,the length l ----i.00 of reference ----45 0 is motion of its direction and rod the between angle 45 a cone has A stationary upright taper angle 1.343. Find: m surface (a)its the lateral of area So----4.0 and the surface area, in the referenceframe taper angle; (b) its lateral the axis of the cone. v (\177/5)c along moving with a velocity frame K) did to what With 1.344. velocity (relative the reference measuredby s, the clockmove, if during the time interval t = 5.0 = 0.10 s? At slow became it by the clockof the frame K, A rod flies with constant velocity past a mark which is 1.345. frame K. In the frame K it takesAt ---stationary in the reference frame fixed ----20ns for the rod to fly past the mark.In the reference to the rod the mark movespast the rod for At' = 25 us.Find the prop-

0\302\260.

\302\260,

\177.

\177--

er length of the rod. The properlifetimeof an unstable particle is equal to 1.346. = ns. Find the distancethis particlewill traverse till its i0 Ate where itslifetimeis equal decay in the laboratory frame of reference, to

=20 ns.

the properdistancebetween the particlespriorto their hitting the target. A rod moves alonga rulerwith a constantvelocity.When 1.349. of both ends of the rod aremarkedsimultaneously in the positions frame fixed to the ruler,the difference of readingson the the reference = 4.0m. But when the positionsof the rod's ruler is equal to ends are marked simultaneously in the referenceframe fixed to the -rod, the difference of readingson the sameruler is equal to ----9.0m. Find the properlength of the rod and itsvelocity relative to the ruler. Two rodsof the sameproperlength lo move toward each 1.350. other parallelto a common horizontalaxis. In the referenceframe when fixed to one of the rods the timeintervalbetween the moments, is At. What to is the right and left ends of the rods coincide, equal the velocity of one rod relativeto the other? frame K i.351.Two unstable particlesmove in the reference ---0.990c. v a w ith velocity alonga straightlinein the samedirection frame is equal to The distancebetween them in this reference ----120m. At a certainmoment both particlesdecay simultaneously frame fixed to them.What timeintervalbetween the in the reference momentsof decay of the two particleswill be observed in the frame K? Which particledecays later in the frame K? frame K A rod AB oriented 1.352. alongthe axisof the reference moves in the positivedirectionof the x axiswith a constant velocity p. The pointA is the forward end of the rod,and the point B itsrear end. Find: (a) the properlength of the rod, if at the momentta the coordinate of the point A is equal to xa,and at the moment ts the coordihx\177

Ax\177

-\177

\177

\342\200\242

nate of the point B isequal to xs; (b) what timeintervalshould separatethe markingsof coordinates to of the rod'sends in the frame K for the difference of coordinates the rod. of to the becomeequal properlength 1.353. The rod A'B'moves with a constantvelocity p relativeto Both rods have the sameproperlength and the rod AB (Fig.i.91). \177e

a rouen moving with a velocity t.347.In the re[erenceframe K ---travelleda distancel 3.0km from itsbirthplaceto v = 0.990c the point where it decayed.Find:

(a) the (b) the

properlifetimeof this rouen; distancetravelledby the rouen

muon's standpoint\".

i.348.Two

particlesmoving

in a

the frame K

the

laboratory frame of reference

velocity v alongthe samestraight linewith the sameinterval At time the with againsta stationary target 68

\"[rom

= (3/4)cstrike = 50 ns. Find

O----\177u

Fig.

i.9i.

clocksare mounted,which are synchroB'.Supposethe moment when the clockB' gets opposite the clockA is taken for the beginningof the timecount in the reference frames fixed to eachof the rods.Determine: at the ends of each of them

nized pairwise:A with B and A' with

69)))

(a) the readingsof the clocksB and B' at the moment when they are oppositeeach other; (b) the samefor the clocksA and A'. t.354.There are two groups of mutually synchronized clocksK and K' moving relativeto eachother with a velocity v as shown in Fig.i.92.The moment when the clockA\" gets oppositethe clockA

(a) the approachvelocity of the particlesin the laboratory frame reference; (b) their relativevelocity. Two rods having the sameproperlength 10 move lengthwise !.360. toward eachother parallelto a common axiswith the samevelocity of

0 0 00 ,'-\",

,.._,,

,ooo@oo Fig.

i.92.

is taken for the positionof

beginningof the timecount. Draw the approximate hands of all the clocksat this moment \"in termsof the

K clocks\";\"in terms of the K' clocks\". frame K' moves in the positive directionof The reference the x axisof the frame K with a relativevelocity V. Supposethat O and O'coincide, the at the moment when the originsof coordinates clockreadingsat thesepoints areequal to zero in both frames.Find the displacement velocity x of the point (in the frame K) at which frames will be permanentthe readingsof the clocksof both reference Demonstratethat \177 V. ly identical. frame K two eventsoccurred At two points of the reference that if theseevents obey separatedby a timeinterval At. Demonstrate the cause-and-effect relationshipin the frame K (e.g.a shot fired and a bullet hitting a target},they obey that relationshipin any other inertialreferenceframe shows three eventsA, The space-time diagramof Fig. B, and C which occurredon the x axis of someinertialreference

t.355.

|.356.

\177

K'.

1.357.

1.93

frame.Find: (a) the timeinterval between the events ,4 and B in the reference frame where the two events occurredat the samepoint; (b) the distancebetween the points at which the events ,4 and C occurredin the referenceframe where thesetwo events aresimultaneous. |.358.The velocity componentsof a particlemoving in the xy plane of the referenceframe K are equalto v x and vy. Find the velocity v' of this particlein the frame K' which moves with the velocity V relativeto the frame K in the positive directionof its x axis. \177.359. Two particlesmove toward each other with velocities v1 0.50cand ----0.75crelativetoa laboratory frame of reference. --\177

Find: 70

v\177

.9 4 Fig.

relativeto eachrod in v

6\"

i.93.

What is the lengthof the laboratory frame of reference. frame fixed to the other rod? the reference

|.36|. Two relativisticparticlesmove at right anglesto eachother the velocity and the one laboratory frame of reference,

in a

with

other with the velocity

v\177.

Find their relative velocity.

1.362. An unstable particle moves in the referenceframe K' alongitsy' axiswith a velocity v'. In itsturn, the frame K' moves relativeto the frame K in the positivedirectionof itsx axiswith a frames coincide, the velocity V. The x' and x axesof the two reference

y' and y axesare parallel.Find the distancewhich the particletraverses in the frame K, if its properlifetimeis equal to |.363.A particlemoves in the frame K with a velocity v at an angle 0 to the x axis.Find the correspondingangle in the frame K' moving with a velocity V relativeto the frame K in the positivedirectionof itsx axis,if the x and x' axesof the two frames coincide. The rod AB orientedparallelto the x' axisof the referen\177 1.364. frame K' moves in this frame with a velocity v' alongitsy' axis.In itsturn, the frame K' moves with a velocity V relativeto the frame Find the angle {) between the rod and the K as shown in Fig.1.94.

x axisin the frame K.

The frame K' moves with 1.365.

the frame

a constantvelocity V relativeto the frame K',

w' of a particlein K. Find the acceleration

7i)))

if in the frame K

this particlemoves with a velocity v and accelera-

tion w along a straight line (a) in the directionof the vectorV; (b) perpendicularto the vector V.

An imaginary spacerocketlaunchedfrom the Earth moves 1.366. ----

w' i0g which isthe samein every instantaacceleration neous co-movinginertial referenceframe.The booststage lasted with an

I I

Fig.

i.94.

i.0year of terrestrialtime.Find how

much (in per cent) does the rocketvelocity dither from the velocity of light at the end of the boost stage.Whatdistancedoesthe rocketcoverby that moment? From the conditionsof the foregoingproblemdetermine frame fixed to the rocket.Remember the boost timeXo in the reference that this timeis defined by the formula x----

i.367.

2 (v/c) dt,

referenceframe. where dt is the time\"inthe geocentric |.368.How many timesdoesthe relativisticmass of a particle whose velocity dithers from the velocity of light by 0.0i0%exceed its rest mass? t.369.The density of a stationary body is equal to P0. Find the frame in which the velocity (relative to the body) of the reference = than of the is 25% Po. greater density body rl with a momentum p = i0.0GeV/c,where i.370.A proton moves c is the velocity of light.How much (in per cent) doesthe proton velocity dither from the velocity of light? |.371.Find the velocity at which the relativisticmomentum of a particleexceedsits Newtonian momentum rI = 2 times. i.372.What work has to be performed in orderto increasethe velocity of a particleof restmassm0 from 0.60c to 0.80c? Compare

resultobtainedwith the value calculatedfrom the classicalformula. i.373.Find the velocity at which the kineticenergy of a particle equalsits rest energy. i.374.At what values of the ratio of the kineticenergy to rest energy canthe velocity of a particlebe calculatedfrom the classical formula with the relativeerrorlessthan e = 0.0i0? the

momentum of a

particleof rest mass m o de-

itskineticenergy.Calculatethe

momentum of a proton MeV. 5C0 whose kineticenergy equals A beam of relativistic particleswith kineticenergy T strikes beam current equals the charge The an target. absorbing against Find and restmass of eachparticleareequalto e and m0 respectively. the pressuredevelopedby the beam on the target surface,and the

pends on

i.376.

power liberatedthere.

|.377.A spheremoves with a relativisticvelocity v through a gas whoseunit volume containsn slowly moving particles,eachof mass m. Find the pressurep exertedby the gas on a sphericalsurface element perpendicularto the velocity of the sphere,provided that the pressure is the sameboth particlesscatterelastically.Show that the and in the referenceframe frame fixed to the sphere in the reference

\177a

|.375.Find how the

gas. |.378.A particleof restmass m0 startsmoving at aofmoment t = 0

fixed to the

due to a constant force F. Find the timedependence the particle's velocity and of the distancecovered. i.379.A particleof rest mass me moves along the x axisof the

with the law x--\177/a2q 2, where a is frame K in accordance a constant, c is the velocity of light, and t is time. Find the force acting on the particle in this referenceframe. from the fundamental equation of relativistic Proceeding find: dynamics, of a particlecointhe acceleration (a) under what circumstances F on the force with cidesin direction acting it; (b) the proportionality factors relatingthe force F and the acceleration w in the caseswhen FA_v and F II v, where v is the velocity of the particle. A relativisticparticlewith momentum p and total energy E moves alongthe x axisof the frame K. Demonstratethat in the frame K' moving with a constantvelocity V relative to the frame K in the positive directionof its axisx the momentum and the total energy of the given particleare defined by the formulas: c\177t

i.380.

i.38i.

\342\200\242

where

Px--EVI

\302\242\177

E' E--

px\177___\177__V

V/c.

The photon energy in the frame K is equal to e.Making use 1.382. find I\177

of the transformation formulas citedin the foregoingproblem, the energy e' of this photon in the frame K' moving with a velocity motion direction.At what V relativeto the frame K in the photon's value of V is the energy of the photon equal to e' = e/27 for a particleis that the quantity E Demonstrate it has the samemagnitudein all inertialreference an invariant, frames.What is the magnitudeof this invariant? A neutron with kineticenergy T = 2m0cz, where me is its 1.384.

1.383.

i.e.

\177

p\177c

rest mass,strikesanother, stationary, neutron. Determine:

73)))

(a) the combinedkineticenergy of both neutrons in the frame of their centreof inertiaand the momentum of eachneutron in that \177

\177

frame; (b) the velocity of the centreof inertiaof this system of particles. Instruction.Make use of the invariant E remainingconstant on transition from one inertialreference frame to another (E is the total energy of the system, p is its composite momentum). 1.385. A particleof rest mass m o with kineticenergy T strikesa stationary particleof the samerestmass.Find the rest mass and the velocity of the compoundparticleformed as a result of the collision. 1.386. How high must be the kineticenergy of a proton striking another,stationary, proton for their combined kineticenergy in the frame of the centreof inertia to be equal to the totalkineticenergy of two protons moving toward each other with individual kinetic energiesT = 25.0GeV? t.387.A stationary particleof restmass rn o disintegratesinto three and m a. Find the maximum total particleswith restmassesml, energy that, for example,the particlern 1 may possess. t.388.A relativisticrocketemitsa gas jet with non-relativistic velocity u constantrelativeto the rocket.Find how the velocity v of the rocketdependson itsrest mass rn if the initialrest mass of the rocketequals rn o.

p\177c

rn\177,

\177

PART TWO

THERMODYNAMICS PHYSICS AND MOLECULAR

2.i.EQUATION \342\200\242

OF THE GAS STATE. PROCESSES

Ideal gas law: pv =

where \342\200\242

(2.ia)

\177r, -\177-

M is the molar mass. Barometric formula: p = Poe-Mgh/RT,

where \342\200\242

( p+-\177) where

(2.tb)

-- 0.

Po is the pressureat the height h Van der Weals equation of gas state (for a mole):

is the

mo\177ar

volume under given p and

2.i.

(2.ie)

(VM--b):RT, T.

30 1 containsidealgas at the temperaA vesselof volume V ture 0 After a portion of the gas has beenlet out, the pressurein remaining the vesseldecreased by Ap ----0.78atm (the temperature of the releasedgas.The gas density under mass Find the constant). the normal conditionsp = i.3g/1. by a tube with a valve 2.2.Two identicalvesselsare connected if the pressuredifferthe other into vessel one from the letting gas pass ence Ap i.10atm. Initially there was a vacuum in one vessel while the other containedideal gas at a temperaturetl----27 and pressurePl i.00atm.Then both vesselswere heatedto a temwill the pressurein the first perature ----107 Up to what valueincrease? vessel (which had vacuum initially) a mixtureof hydrogen 2.3.A vessel of volume V 20 1 contains and pressurep ----2.0atm. and helium at a temperaturet ----20 The mass of the mixtureis equal to ra 5.0g.Find the ratioof the massof hydrogen to that of helium in the given mixture. 2.4.A vessel contains a mixture of nitrogen (tnl = 7.0 g) and 290 K and presT a at dioxide carbon temperature g) sure P0 = 1.0atm. Find the density of this mixture,assuming the -----

\302\260C.

\177

\302\260C

-\177

\302\260C.

t\177

\177-

\302\260C

\177

(m\177

\177

-\177

gasesto be ideal. a mixtureof idealgases 2.5.A vesselof volume V = 7.51 contains 300K:v 1 0.i0moleof oxygen, = 0.20 at a temperatureT Assuming moleof nitrogen,and va 0.30moleof carbondioxide. the gasesto be ideal,find: (a) the pressureof the mixture; -----

-\177

v\177

-----

75)))

(b) the mean molarmass M of the given mixturewhich entersits equation of statepV = (reC).RT,where m is the mass of the mix-

ture.

2.6.A verticalcylinder closedfrom both ends is equippedwith an easily moving piston dividing the volume into two parts,eachcontaining one moleof air. In equilibriumat To ----300K the volume of the upper part is ----4.0timesgreaterthan that of the lower part. At what temperature will the ratio of these volumes be equal to \177l

= 3.0? ,l'2.7. A vessel

of volume V is evacuatedby meansof a piston air pump.One piston strokecapturesthe volume AV. How many strokes are neededto reducethe pressurein the vessel times?The process

is assumed to be isothermal,and the gas ideal. 2.8.Find the pressureof air in a vassal beingevacuatedas a function of evacuationtimet. Thevesselvolume is V, the initialpressure is/9o.The processis assumed to be isothermal,and the evacuation rata equal to C and independentof pressure. Note.The evacuationrata is the gas volume being evacuatedper unit time, with that volume being measuredunder the gas pressure attained by that moment. 2.9.A chamberof volume V = 87 1is evacuatedby pump whose evacuation rate (see Note to the foregoingproblem)equals C = l0 1/s.How soon will the pressurein the chamber decreaseby = 1000times? 2.10.A smooth verticaltube having two different sectionsis openfrom both ends and equippedwith \1771

\177t

\1771

2.i).

two pistons of different areas (Fig. Each One piston slideswithin a respectivetube section. moleof ideal gas is enclosedbetween the pistons tied with a non-stretchablethread.The crosssectionalarea of the upper piston is AS = i0em z greater than that of the lower one.The combined mass of the two pistonsis equal to m = 5.0kg. The outside air pressureis P0 = atm. By how many kelvins must the gas between the pistons be heatedto shift the pistons through l ----5.0cm? Find the maximum attainabletemperatureof each of the following processes:

i.0

2.tt.

(a) p =po--czV\177;(b) p = poe-\177v, where po,.czand arepositiveconstants,and

moleof gas.

I\177

2.t2.Find the ----

Fig.

2.t.

ideal gas in

isthe volume of one

attainablepressureof ideal gas in the where To and are positiveconstants,and processT To qV is the volume of one moleof gas.Draw the approximate p vs V plotof this process. 2.i3.A tall cylindricalvesselwith gaseousnitrogenislocatedin a uniform gravitational field in which the free-fallacceleration isequal to g.The temperatureof the nitrogenvariesalongthe height 76

minimum \302\242zV

\177,

\302\242z

so that its density is the samethroughoutthe volume.Find the

temperaturegradient dT/dh. 2.t4.Supposethe pressurep and the density p of air are related of height (n is a constanthare).Find the as p/p\" = constregardless gradient. temperature \177orresponding closeto 2.15.Letus assume that air is under standard conditions molar a nd the the that Earth'ssurface. the temperature Presuming massof air are independentof height, find the air pressureat the

height 5.0km over the surface and in a mine at the depth 5.0km below the surface. of air, as 2.t6.Assuming the temperatureand the molarmass to be independentof the height, well as the free-fall acceleration, at the tempefind the difference in heightsat which the air densities rature 0 differ (a) e times;(b) by rl = 1.0%. An idealgas of molarmass M is containedin a tall vortical 2.t7. cylindricalvesselwhosebasearea is S and heighth. The temperature of the gas is T, itspressureon the bottom baseis Po- Assumingthe g to be independentof the temperatureand the free-fallacceleration height, find the mass of gas in the vessel. An ideal gas of molarmass M is containedin a very tall verticalcylindricalvesselin the uniform gravitationalfield in which the free-fall acceleration equalsg. Assumingthe gas temperatureto to be the sameand equal T, find the height at which the centreof gravity of the gas is located. gravi2.19.An idealgasof molarmassM is locatedin the uniform is equal to g. Find tationalfield in which the free-fall acceleration the gas pressureas a function of height h, if p = P0 at h ----0, and the temperaturevaries with height as (b) T = To(lq- oh), (a) T = To(t--oh); where a is a positive constant. 2.20.A horizontalcylinder dosedfrom one end is rotatedwith constant a angularvelocity co about a verticalaxispassingthrough the open end of the cylinder.The outsideair pressureis equal to massof air to M. Find the Po,the temperatureto T, and the molar air pressureas a function of the distancer from the rotationaxis.The of r. molarmass is assumed to be indepeadent the density 2.21.Under what pressurewill= carbon dioxidehave calculations = out the 300 K? T at the 500 Carry temperature g/1 p both for an idealand for a Van derWaals gas. 2.22.One moleof nitrogenis containedin a vesselof volume V = \302\260C

2.|8.

= t.001. Find:

(a) the temperatureof the nitrogenat which=the pressurecan be calculatedfrom an idealgas law with an errorrI i0% (as compared from theVan derWaals equationof state); with the pressurecalculated (b) the gas pressureat this temperature. volume 2.23.One moleof a certaingas is= containedin a vesselof is 1.At a temperatureT1 300K the gaspressure Pl = V = 0.250 77)))

= 90atm, and at a = 350K the pressureis = = li0 atm. Find the temperature Van der Waals parameters this gas. 2.24.Find the isothermalcompressibilityoffora Van der Waals gas as a function of volume V at temperatureT. Note. By definition, = vt 0v Op 2.25.Making use of the resultobtained in the foregoingproblem, find at what temperature the isothermal u of a Van compressibility derWaals gas is greaterthan that of an ideal Examinethe case gas. when the molarvolume is much greaterthan the parameterb. T\177

p\177

\303\227

2.2.THE HEAT \342\200\242

FIRST

The first

law

\342\200\242

-]- A,

(2.2a)

energy

-----

the

system.

dV.

(2.2b)

\177

\342\200\242

Internal

energy

m U='--\177 \342\200\242

Molar heat

ideal gas:

RT :Cv'T:Mrn V--i ?--i p..\177__V

(2.2\302\242)

capacity in a polytropic process (pV n

(n--\177,)

= const):

i n---:-V (n--t) of one mole of a van der Waals

Internal

energy

gas:

U----Cv'T--

(2.2e)

V'-\177\"

2.26.Demonstrate that the interval energy U of the air in a room is independentof temperatureprovided the outsidepressurep is constant.CalculateU, if p is equal to the normalatmospheric pressure and the room's volume is equal to V = 40 m a,. 2.27.A thermally insulatedvesselcontaininga gas whose molar massisequal to M and the ratio of specific heats C,Cv- moves with a velocity v. Find the gas temperature incrementresultingfrom -----

\"\177

the sudden stoppageof the vessel. 2.28.Two thermally insulatedvesselsI and 2 are filled with air and connectedby a short tube equippedwith a valve. The volumes of the vessals, the pressuresand. temperatures of air in them are known (V1,Pl, TI and Find the air temperatureand T\177). pressure establishedafter the opening of the valve. V\177,

p\177,

2.29.Gaseoushydrogen contained under standard conditionsin a sealedvesselof volume V initially 5.01 was cooledby AT = 78

-----

2.35.

2.36.

(2.2d)

\177-

\342\200\242

2.3i.

\177

the increment of the internal Work performed by gas:

-\177

2.33.

of thermodynamics:

Q where

OF THERMODYNAMICS.

LAW CAPACITY

55 K.Find how much the internalenergy of the gaswill changeand what amount of heat will be lostby the gas. 2.30.What amount of heat is to be transferred to nitrogenin the isobaricheatingprocessfor that gas to perform the work A = 2.0:I? As a result of the isobaricheating by AT = 72 K one mole kJ. Find of a cert,ain idealgas obtains an amount of hea.t Q -- 1.60 the work performedby the gas,the incrementof its internal energy, and the value of 300K 2.32.Two molesof a certainidealgasat a temperatureTo =n = 2.0 so that the gas pressurereduced were cooledisochoricallsr times.Then, as a result of the isobaricprocess,the gasexpandedtill its temperature got backto the initialvalue.Find the total amount of heat absorbed by the gas in this process. Calculatethe value of 7 CffCv- for a gaseousmixtureconsisting of v I = 2.0molesof oxygen and v 2 3.0molesof carbon The gasesare assumed to be ideal. dioxide. gaseousmix2.34.Find the specificheat capacitiescv and c,foraThe of 20 and gasesare argon. ture consistingof 7.0g of nitrogen g assumed to be ideal. One moleof a certainidealgas iscontainedunder a weightlesspiston of a verticalcylinder at a temperatureT. The spaceover What work has to be performed the piston opensinto the atmosphere. in orderto increaseisothermally the gas volume under the piston n timesby slowly raisingthe piston?Thefrictionof the piston against the cylinder walls is negligiblysmall. A piston can freely move inside a horizontalcylinder closed from both ends.Initially,the piston separatesthe inside spaceof the cylinder into two equal parts eachof volume V0, in which an idealgas is containedunder the samepressureP0 and at the same temisotherperature.What work has to be performedin orderto increase of the t o that times of of one the volume compared gas part mally other by slowly moving the piston? at a temperature 2.37.Threemolesof an idealgasbeinginitially ---timesits initial n 5.0 = 273 K were isothermally expanded To heatedso that the pressurein the final volume and then isochorically of statebecameequal to that in the initial state.The total amount heat transferred to the gas during the processequalsQ = 80 k\177. Find the ratio 7 ----C\177,/Cv- for this gas. Draw the approximate plotsof isochoric,isobaric,isotherfor the caseof an idealgas, using the mal, and adiabaticprocesses \177l

2.38.

following variables: (a) p, T; (b) V, T. 2.39. One moleof oxygen being initially at a temperatureTo ---= 290K is adiabaticallycompressed to increaseits pressure

\177l

times.Find: the gas temperatureafter the compression; (a) (b) the work that has been performed on the gas. = 2.50.A certainmassof nitrogenwas compressed \1770.0

\177l

5.0times 79)))

(in

termsof volume),first adiabatically,and

then

In isothermally.

both casesthe initialstateof the gas was the same.Find the ratio of works expendedin eachcompression. the respective

2.41.A heat-conductingpiston can

freely move inside a

closed

thermally insulated cylinder with an ideal gas. In equilibriumthe pistondividesthe cylinder into two equal parts,the gas temperature Find the gas tembeingequal to To.The piston is slowly displaced.

peratureas a function of the ratio of the volumes of the greaterand smallersections. The adiabaticexponentof the gas is equal to 2.42.Find the rate v with which helium flows out of a thermally insulatedvesselinto vacuum through a smallhole.The flow rate of the gas insidethe vesselis assumed to be negligible undertheseconditions.The temperatureof helium in the vessel is T = i,000K. 2.43.The volume of one moleof idealgas with the. adiabatic to the law V a/T, where a is a conexponent is varied according stant. Find the amount of heat obtainedby the gas in this process if the gas temperature increasedby AT. 2.44.Demonstratethat the processin which the work performed to the corresponding incrementof its by an idealgas is proportional internal energy is describedby the equationpV n const,where n is a constant. 2.45.Find the molarheat capacity of an idealgas in a polytropic const if the adiabaticexponentof the gas is equal to processpV At what valuesof the polytropic constantn will the heat capacity of the gas be negative? 2.46.In a certainpolytropic processthe volume of argonwas increaseda-\177 4.0 times. Simultaneously, the pressure decreased 8.0times.Find the molarheat capacity of argon in this process, assumingthe gas to be ideal. 2.47.One moleof argon is expandedpolytropically, the polytropic constant being n----1.50. In the process,the gas temperature -----26 Find: AT K. changesby (a) the amount of heat obtainedby the gas; \1771

\177.

\177n

\177

'\177

\177

\177.

-\177

\177

(b) the work performedby the gas.

2.48.An idealgas whose adiabaticexponentequals isexpanded accordingto the law p aV, where a isa constant.The initialvol\177

\177

ume of the gas is equal to Vo. As a resultof expansion the volume in-

creasesB times. Find:

(a) the incrementof the internal energy of the gas; (b) the work performedby the gas; (c) the molarheat capacity of the gas in the process. 2.49.An idealgas whose adiabaticexponentequals isexpanded sothat the amount of heat transferredto the gas isequalto the decreaseof its internal energy. Find: (a) the molarheat capacity of the gas in this process; (b) the equation of the processin the variablesT, V; (c) the work performed by one moleof the gas when itsvolume increasesrI timesif the initial temperatureof the gas is To. \177

2.50.One moleof an idealgas whose adiabaticexponentequals undergoesa processin which the gaspressurerelatesto the tempera-

aT where a and a are constants.Find: ture as p the work performed by the gas if itstemperature (a) gets an increment AT; at what value (b) the molarheat capacity of the gas in this process; of a will the heat capacity be negative? An ideal gas with the adiabaticexponent ? undergoesa processin which its internalenergyrelatesto the volume as U = aV where a and a are constants.Find: (a) the work performedby the gas and the amount of heat to be transferred to this gas to increaseits internal energy by AU; in this (b) the molarheat capacity of the 2.52.An ideal gas has a molarheatgascapacity Cvprocess. at constant volume.Find the molarheat capacity of this gas as a function of its volume V, if the gas undergoesthe following process: (a) T-\177 Toe ; (b) p =poesy, where To, P0, and a areconstants. One moleof an idealgas whose adiabaticexponentequals? undergoesa process p Po + a/V, where P0 and a are positive con\177

\177-

\177,

2.5i.

\177,

\177v

2.53.

\177-

stants.Find:

(a) heat capacity of the gas as a function of its volume; (b) the internal energy incrementof the gas, the work performed by it, and the amount of heat transferred to the gas, if its volume increasedfrom V1 to 2.54.One moleof an ideal gas with heat capacity at constant pressureCp undergoesthe processT To -+- aV, where T and a are constants.Find: (a) heat capacity of the gas as a function of itsvolume; (b) the amount of heat transferred to the gas, if its volume increasedfrom V1 to Forthe caseof an idealgas find the equationof the process the variables T, V) in which the molarheat capacity varies as: (in (a) C Cv t- aT; (b) C Cv + (e) C = Cv + ap, where and a are constants. An idealgas has an adiabaticexponenty. In someprocess itsmolarheat capacity varies as C ----a/T, where a is a constant. V\177.

-\177

2.55.

V\177.

-\177

a, 2.56.

[\177,

\177V;

Find:

(a) the work performedby one moleof the gas during itsheating temperatureT to the temperature timeshigher; (b) the equationof the processin the variablesiv, V. 2.57.Find the work performedby one moleof a Van der Waals at gas during its isothermalexpansionfrom the volume V1 to a temperatureT. 2.58.One mole of oxygen is expandedfrom a volume V1 1.001 to 5.0l at a constanttemperatureT 280K. Calcul ate: (a) the increment of the internal energy of the gas: 6--9451 8 from the

\1771

V\177

\177

\177--

V\177

\177

t)))

(b) the amount of the absorbedheat. The gas is assumed to be a Van derWaals gas. 2.59.For a Van der Waals gas find: T, V; (a) the equationof the adiabaticcurve in the variables -as a funcheat molar of the the difference Cv capacities C p (b) tion of T and V. 2.60.Two thermally insulated vessels are interconnectedby a tube equippedwith a valve.One vesselof volume V1 i01 contains The other vesselof volume v =- 2.5molesof carbondioxide. 1001 is evacuated.The valve having beenopened,the gas adiabaticAssumingthe gasto obey the Van derWaals equation, ally expanded. find its temperature changeaccompanying the expansion. 3.0moles 2.61.What amount of heat has to be transferred to v while it exof carbon dioxideto keep its temperatureconstant ---The 17 t0 1 to 5.0 volume from the V2 V1 pands into vacuum der Waals Van be a to is assumed gas. gas \177

V\177

-\177

\177

2.3.KINETIC

THEORY OF GASES. BOLTZMANN'S LAW AND MAXWELL'S Number

\342\200\242

wall

collisionsexercisedby

surface per unit time:

-\177-

where n \342\200\242

Mean energy

\342\200\242

is their mean velocity.

= nkT.

(2.3b) (2.3c)

kT, -\177

is the sum of translational, rotational, tional degreesof freedom. where i

Maxwellian

\342\200\242

distribution dN (u)

where

\177-

v/v,, v, is the most

The m\177st molecules: \342\200\242

and the

double number of vibra-

distribution:

dN (v) = N

\177robable,

the

e 3/2

(

\177

where

the

potential

\177

(2.3h)

noe-(U-Uo)/\177T,

of a molecule.

energy

2.62.Modern vacuum

pumps permitthe pressuresdown to p ---be reachedat room temperatures. Assuming that the gas exhaustedis nitrogen,find the number of itsmolecules per t cm 3 and the mean distancebetween them at this pressure. \177

4.10 -\177

atm to

2.63.A vesselof volume V

at a

temperatureT

accountthat

----\177

----

-\177

5.01 containsm = 1.4g of nitrogen

t800K. Find

the gas pressure,taking into 30% of molecules are disassociated into atoms at

this temperature. 2.64.Under standard conditionsthe density of the helium and nitrogen mixtureequals p 0.60g/1.Find the concentrationof helium atoms in the given mixture. 2.65.A parallelbeam of nitrogenmoleculesmoving with velocity v 400 m/s impingeson a wall at an angle 30 to itsnormal. The concentrationof moleculesin the beam n----0.9.10 cm Find the pressureexertedby the beam on the wall assuming the moleculesto scatterin accordance with the perfectly elasticcollision

\177

\302\260

(\177

-\177

\177

-\177.

law.

2.66.How many degreesof freedom have the gas molecules,if under standard conditionsthe gas density is p t.3mg/cms and the velocity of sound propagationin it is v 330m/s. 2.67.Determinethe ratio of the sonicvelocity v in a gas to the root mean squarevelocity of molecules of this gas, if the molecules -\177

are

(a) monatomic;(b) rigid diatomic. has the temperature gas consistingof N-atomicmolecules

2.68.A

of molecules:

\342\200\242

\177

is the concentration of molecules,and iv) Equation of an ideal gas state: p

the

(2.3a)

n iv),

\177-

\177

DISTRIBUTION on a unit area of molecules gas

Boltzmann's formula:

\342\200\242

(2.3d)

avx,

e_mV\17712k

4y\177v B

dr.

(2.3e)

\177

us u du, \177

probablevelocity. mean,

and the

(translational, rotational,and vi-

gas consistsof

(a) diatomic;

a reduced form:

T at which all degrees of freedom

in such brational) are excited.Find the mean energy of molecules a gas.What fraction of this energy corresponds to that of translational motion? 2.69.Supposea gas is heated up to a temperatureat which all degreesof freedom (translational,rotational,and vibrational) of itsmoleculesareexcited.Find the molarheat capacity of such a gas in the isochoric process,as well as the adiabaticexponent-f, if the

(2.3[)

root mean square velocitiesof

(b) linear N-atomic; (c) network N-atomic

molecules. 2.70.An idealgas consistingof N-atomicmoleculesis expanded isobarically.Assuming that all degreesof freedom (translational, rotational,and vibrational)of the moleculesare excited,find what fraction of heat transferred to the gas in this processis spent to How high is perform the work of expansion. fraction in the case of a monatomic gas?))) tl\177is

2.7i.Find the molarmass and the number of degreesof freedom its heat capacitiesare known: Cg = of moleculesin a gas = 0.65J/(g.K)and cp = 0.9iJ/(g.K). if

2.72.Find the number of degreesof freedom of moleculesin

a gas

whose molarheat capacity 29 J/(mol.K); (a) at constantpressureis equal to = in the 29 is to C processpT const. J/(mol. K ) (b) equal 2.73.Find the adiabaticexponent7 for a mixtureconsistingof v 1 molesof a monatomic gas and v.o molesof gas of rigid diatomic \177

Cr\177

-\177

molecules.

2.74.A

thermally insulated vessel with gaseousnitrogen at a temperaturet = 27 moves with velbcity v = i00m/s.How much (in percent)and in what way will the gas pressurechangeon a sudden \302\260C

stoppageof the vessel? 2.75.Calculateat the temperaturet = i7 (a) the root mean squarevelocity and the mean kineticenergy of an oxygen moleculein the processof translationalmotion; (b) the root mean square velocity of a water dropletof diameter d = 0.t0 suspendedin the air. 2.76.A gas consistingof rigid diatomicmoleculesis expanded to reduce \177diabatically. How many timeshas the gas to be expanded the root mean square velocity of the molecules = t.50times? 2.77.The mass m = 15g of nitrogen is enclosedin a vessel at 300K.What amount of heat has to be transferred T a temperature to the gas to increasethe rootmean squarevelocity of itsmolecules \302\260C:

\177tm

\342\200\242

\177l

\177

= 2.0 times? 2.78.The temperatureof a gas consistingof rigid diatomicmoleculesis T ----300K.Calculatethe angularroot mean squarevelocity of a rotating moleculeif its moment of inertia is equal to I---= 2.i.10 g.cm% 2.79.A gas consistingof rigid diatomicmoleculeswas initially adiabaThen the gas was compressed under standard conditions.

\1771

-\177

= moleculein

5.0times.Find the the final state.

kineticenergy of a rotating of rigid diatomicmolecules 2.80.How will the rate of collisions if the the vessel's wall gas is expa,ndedadiabatically change, against times? 2.81.The volume of gas consistingof rigid diatomicmolecules was increased 2.0timesin a polytropic processwith the molar of heat capacity C R. How many timeswill the rate of collisions this result of as a be reduced wall a vessel's moleculesagainst process? 2.82.A gas consistingof rigid diatomicmoleculeswas expanded of the molecules in a polytropic process so that the rate of collisions heat capacity molar the F ind did not wall the vessel's change. against

tically

\177

mean

\177

of the gas in this process. 2.83.Calculatethe most probable,the

mean,and

the

rootmean

square velocitiesof a moleculeof a gas whose density under standard atmospheric pressureis equal to p 1.00g/1. \177

2.84.Find

whose velocitiesdiffer the fraction of gas molecules = 1.00%from the value of by less than (a) the most probablevelocity; (b) the root mean squarevelocity. 2.85.Determinethe gas temperatureat which exceeds (a) the root mean squarevelocity of hydrogen molecules 5\1771

their most probablevelocity by Av = 400 m/s; (b) the velocity distributionfunction F (v) for the oxygen moleculeswill have the maximum value at the velocity v 420 m/s. 2.86.In the caseof gaseousnitrogenfind: vl = (a) the temperatureat which the velocitiesof the molecules 600m/s are associatedwith equal values of 300m/s and the Maxwell distribution function F (v);

: : v\177

v at which the value of the Max(b) the velocity of the molecules well distributionfunction F (v) for the temperatureTo will be the sameas that for the temperature timeshigher. 2.87.At what temperatureof a nitrogenand oxygen mixture do differ of nitrogenand oxygen molecules the most probablevelocities by Av = 30 m/s? 2.88.The temperatureof a hydrogen and heliummixtureis T 300K.At what value of the molecularvelocity v will the Maxwell distributionfunction F (v) yield the samemagnitudefor both gases? 2.89.At what temperatureof a gas will the number of molecules, whose velocitiesfall within the given interval from v to v q-dr, be the greatest?The mass of eachmoleculeis equal to m. 2.90.Find the fraction of moleculeswhose velocity projectionson the x axis fall within the interval from vx to vx q- dye,, while the moduli of perpendicular velocity componentsfall within the interThe mass of each moleculeis m, and the to val from is T. temperature 2.91.Using the Maxwell distribution function, calculatethe mean velocity projection and the mean value of the modulus of this projection(I I) if the massof eachmoleculeis equal to m and the gas temperatureis T.. 2.92.From the Maxwelldistributionfunction find (v\177), the mean of the molecular value of the squaredvx projection velocity in a gas at a temperatureT. The mass of each moleculeis equal to m. . 2.93. Making use of the Maxwell distributionfunction, calculate the number of gas molecules reachinga unit areaof a wall perunit is equal to n, the temperature time,if the concentrationof molecules to T, and the mass of eachmoleculeis m. 2.94.Using the Maxwell distribution function, determinethe pressureexertedby gas on a wall,if the gas temperatureis T and the concentrationof moleculesis n. 2.95.Making use of the Maxwell distribution function,find (i/v), the mean value of the reciprocalof the velocity of molecules \177l

v\302\261q-

v\302\261

dr\302\261.

(v\177)

v\177

\177

85 84)))

idealgas at a temperatureT, if the mass of each moleculeis equal to m. Compare the value obtainedwith the reciprocalof the mean velocity. 2.96.A gasconsistsof moleculesof massm and is at a temperature in an

T. Making use of the Maxwell velocity distribution function, find the correspondingdistribution of the moleculesover the kinetic energiese. Determinethe most probablevalue of the kineticenergy ep. Doessp correspondto the most probablevelocity? of a gas in a thermal 2.97.What fraction of monatomicmolecules kineticenergiesdiffering from the mean value equilibriumpossesses % and less? by l = 2.98.What fraction of moleculesin a gas at a temperatureT has the kineticenergy of translationalmotionexceeding eo if So >>

1.0

6\177

2.99.The velocity distribution of moleculesin a beam coming out of a holein a vesselis described by the function F (v)=Av\177e-'\1772/2hT, where T is the temperatureof the gas in the vessel.Find the most probablevalues of in the beam;compare the result (a) the velocity of the molecules obtained with the most probablevelocity of the moleculesin the vessel;

in the beam. (b) the kineticenergy of the molecules An idealgasconsistingof molecules of mass rn with concen2.100. T. Usingthe Maxwell distributionfunctration n has a temperature tion, find the number of moleculesreachinga unit area of a wall to its normal per unit time. at the anglesbetween 0 and From the conditionsof the foregoingproblem find the number of molecules reachinga unit areaof a wall with the velocities in the interval from v to v @ dv per unit time. Find the force exertedou a particleby a uniform field if 2.102. the concentrationsof these particlesat two levels separatedby the distanceAh = 3.0cm (along the field) differ by = 2.0times. The temperatureof the system is equal to T = 280K.

2.|0|.

\177}

-\177

d\177}

\177l

2.|03.When examiningthe suspendedgamboge droplets undm their averagenumbersin the layers separatedby the microscope, distanceh-\177 40 were found to differ by = 2.0times.Theenvironmental temperatureis equal to T = 290K.'The diameterof and their density exceedsthat of the the dropletsis d = 0.40 a

\177tm

\177l

= 0.20g/cm3.

\177tm,

surroundingfluid by from

thesedata.

Find Avogadro'snumber

2.104. Supposethat

% is the ratio of the molecularconcentration of hydrogen to that of nitrogen at the Earth's surface,while is Find the ratio the correspondingratio at the height h = 3000m. at the temperatureT = 280K, assuming that the temperature are independentof the height. and the free fall acceleration a gas composed of two kinds 2.|05.A tall verticalvesselcontains The concentrations of molecules with of masses and of these molecules at the bottom of the vesselare equal to nx and \177l

\177i]\177lo

m\177

m\177,

m\177

\177

m\177.

n\177

n 1.Assuming the temperatureT and the respectively,with free-fall acceleration g to be independentof the height,find the height are equal. at which the concentrationsof these kinds of molecules A very tall vertical cylinder contains carbon dioxideat 2.106. T. Assumingthe gravitationalfield to be unia certain temperature form, find how the gas pressureon the bottom of the vessel will changewhen the gas temperatureincreases times. 2.i07.A very tall vertical cylinder containsa gas at a temperature T. Assumingthe gravitationalfield to be uniform, find the mean Doesthis value value of the potentialenergy of the gas molecules. or of depend on whether the gas consistsof one kind of molecules several kinds? 2.i08.A horizontaltube of length l = t00cm closedfrom both w. The tube ends is displaced lengthwisewith a constantacceleration containsargon at a temperature T = 330K. At what value of w will the argon concentrationsat the tube'sends differ by = t.0%? 2.109. Find the mass of a moleof colloid particlesif during their centrifugingwith an angularvelocity (o about a verticalaxisthe concentrationof the particlesat the distancer e from the rotation axisis timesgreaterthan that at the distance (in the samehorizontal plane). The densitiesof the particlesand the solvent are equal to p and to Po respectively. A horizontaltube with closedends is rotatedwith a cons2.110. tant angularvelocity (o about a verticalaxispassingthrough one of its ends.The tube containscarbon dioxideat a temperatureT 300K. The length of the tube is l t00cm.Find the value co concentrationsat the oppositeends at which the ratio of molecular n\177

\177

\177l

r\177

\177l

\177

2.0. 2.111. The potentialenergy of gas molecules in

of the tube is equal to

\177

a certaincentral field dependson the distancer from the field's centreas U (r) = ar is T, the concenwhere a is a positive constant.The gas temperature tration of moleculesat the centreof the field is n o Find: (a) the number of moleculeslocatedat the distancesbetween r and r dr from the centreof the field; from the (b) the most probabledistanceseparatingthe molecules centreof the field; locatedin the sphericallayer between (c) the fraction of molecules r and r -]- dr; in the centre (d) how many timesthe concentrationof molecules of the field will change if the temperaturedecreases times. From the conditionsof the foregoingproblemfind: (a) the number of moleculeswhose potentialenergy lieswithin the interval from U to U-]-dU; (b) the most probablevalue of the potentialenergy of a molecule; comparethis value with the potentialenergy of a moleculelocated at its most probabledistancefrom the centreof the field.))) \177l

\177,

\177

2.i12.

\177l

2.4.THE

Find the efficiency of a cycle consistingof two 2.118.

isobaricand if the pressurechangesn timeswithin the cycle. adiabaticlines, The working substanceis an idealgas whose adiabaticexponent is

SECONDLAW OF THERMODYNAMICS.

two

ENTROPY

Heat engine efficiency:

\342\200\242

(2.4a)

Q\177

=\177

n=-\177

O\177,

where Qa is the heat obtained by the working by the working substance. Efficiency of a Carnot cycle:

substance,

Q\177

the

heat released

\342\200\242

T1--T2 T1

\177

where \342\200\242

T1 and T2 are the temperatures of the Clausius inequality:

hot and

(2.4b) cold bodiesrespectively.

\177 \177

--\177Q-

(2.4c)

adiabaticexponentequals 2.|19. idealgofaswhose and two two isochoric \177.

goesthrough a cycle consisting isobariclines. Find the of the gas rises efficiency of such a cycle,if the absolutetemperature n timesboth in the isochoricheating and in P the isobaricexpansion. An ideal gas goes through a cycle 2.120. consistingof (a) isochoric,adiabatic, and isothermal An

\177

lines;

(b) isobaric, adiabatic, and isothermal

lines,

2.ii3.

the isothermal processproceedingat V of the whole cycle. the minimum temperature Fig. 2.2. Find the efficiency of each cycle if the absowithin the varies n-fold lute temperature cycle. The conditionsare the sameas in the foregoingproblem at the maxthat the isothermalprocessproceeds with the exception the o f whole imum temperature cycle. 2.t22.An idealgasgoesthrough a cycle consistingof isothermal, with the isothermalprocess proceedpolytropic,and adiabaticlines, ing at the maximum temperatureof the whole cycle.Find the efficvariesn-fold within iency of such a cycle if the absolutetemperature the cycle. An ideal gas with the adiabaticexponent? goes through a direct(clockwise) cycle consistingof adiabatic,isobaric,and isochoric lines.Find the efficiency of the cycle if in the adiabaticprocess the volume of the ideal gas

2.114.

2.124. Calculatethe efficiency of a cycle consistingof isothermal, if in the isothermal processthe volume isobaric,and isochoriclines,

where

\177iQ

is the

of heat

elementary amount

transferred

algebraic quantity). \342\200\242

Entropy

increment of a

\342\200\242

Fundamental

\342\200\242

the system

(\177Q

\177

(2.4d)

\" \177iQT

relation of thermodynamics:

T dS dynamic

system: AS

Relation between the

probability):

\177

entropy

-\177

and the

S = k In

f\177,

(2.4e)

dV.

statistical weight

\1772

(the thermo-

(2.4f)

where k is the Boltzmann constant.

In which casewill the efficiency of a Carnot cycle be higher: isincreasedby AT, or when the cold when the hot body temperature is decreased by the samemagnitude? body temperature Hydrogen is used in a Carnot cycle as a working substance. Find the efficiency of the cycle,if as a resultof an adiabaticexpansion (a) the gas volume increasesn = 2.0times; times. (b) the pressure decreasesn-----2.0 A heat engineemploying a Carnot cycle with an efficiency the thermalreservoirs of +l = 10%is used as a refrigeratingmachine, being the same. Find its refrigeratingefficiency e. An ideal gas goesthrough a cycle consistingof alternate isothermaland adiabaticcurves (Fig.2.2).The isothermalprocesses proceedat the temperaturesTl, T\177, and T3. Find the efficiency of such a cycle,if in eachisothermal expansionthe gas volume fncreases in the sameproportion. Find the efficiency of a cycle consistingof two isochoric and two adiabaticlines,if the volume of the ideal gas changes

2.t15.

2.116. 2.tt7.

equal to

= 10timeswithin the cycle.The working substanceis nitrogen.

with

2.121.

2.|23.

(a) increasesn-fold; (b) decreasesn-fold.

ideal gas with the adiabaticexponent (a) increasesn-fold; (b) decreasesn-fold. and Find the efficiency of a cycle consistingof two isochoric 2.125. two isothermallines if the volume varies \177-fold and the absolute temperature\177-fold within the cycle.The working substance is an ideal gas with the adiabaticexponent?. 2.|26.Find the efficiency of a cycle consistingof two isobaricand two isothermallines if the pressurevaries n-fold and the absolute temperature\177-fold within the cycle.The working substance is an ideal gas with the adiabaticexponent An ideal gas with the adiabaticexponent goes through 2.127. a cycle (Fig. 2.3)within which the absolute temperaturevaries Find the efficiency of this cycle. \177-fold. 2.128. Making use of the Clausiusinequality, demonstratethat

of the

\177.

\177

89)))

all cycles having the samemaximum temperatureTmax and the sameminimum temperatureT,\177 are lessefficient comparedto the Carnot cycle with the same and Train. 2.i29.Making use of the Carnot theorem, show that in the case of a physically uniform substancewhose stateis defined by the parameters T and V (OU/OV)T= T (Op/OT)vp, T\177a_\177

Find the entropy incrementof the gas in this processprovided its

increasesa = 2.0times. ideal gas with the adiabaticexponenty goesthrough processp = Po-aV, where Po and a are positive constants,

volume a

An 2.138.

V is the volume.At what volume will the gas entropy have the maximumvalue? 2.139. One moleof an idealgas goesthrough a processin which the entropy of the gas changeswith temperatureT as S aT + In T, where a is a positive constant, is the molarheat capacity of this gas at constantvolume.Find the volume dependence of the gas temperature in this processif T To at V = Vo. 2.140. Find the entropy incrementof one moleof a Van der Waals gas due to the isothermalvariation of volume from V1 to Vs. The

and

-----

C\177

where U (T, V) is the internal energy of the substance. Instruction.Considerthe infinitesimalCarnot cycle in the variables

p,2.130. Find the V.

entropy incrementof one when itsabsolutetemperature increasesn = of heating is

moleof carbondioxide 2.0timesif the process

(a) isochoric;(b) isobaric. The gas is to be regardedas ideal. The entropy of v ----4.0molesof an idealgas increasesby 2.131. AS = 23 J/K due to the isothermalexpansion.How many times should the volume v = 4.0 moles of the gas be increased?

2.132. Two molesof

ideal gas are cooled and then expandedisobaricallyto isochorically back to the initialvallower the gastemperature ue. Find the entropy incrementof the gas if in this processthe gas pressure changed n=3.3 times. 2.133. Helium of massrn= 1.7g is expanded adiabaticallyn = 3.0timesand then compressed isobarically down to the initial volume. an

\177-

der Waals corrections are assumed to be known. derWaals gas which had initially the volume V1 and the temperature was transferred to the statewith

Van

2.141. One moleof a Van

T\177

and the temperatureT\177. Find the corresponding of the gas, assuming its molarheat capacity increment entropy to be known. 2.142. At very low temperatures the heat capacity of crystals is = aT where a is a constant.Find the to C equal entropy of a crystal as a function of temperaturein this temperatureinterval. 2.t43.Find the entropy incrementof an aluminum bar of mass m = 3.0kg on its heating from the temperatureT1 = 300K up = 600K if in this temperature to interval the specificheat capacaluminum varies as c = a of bT, where a = 0.77 J/(g.K), ity the volume

V\177

C\177

\177,

T\177

\177

= 0.46mJ/(g.K\177). 2.144. In someprocessthe temperature of a substancedependson itsentropy S as T = aS\",where a and n are constants. Find the b

/ \177r

\177?

Find the entropy increment of the gas in this Fig. 2.3. process. Find the entropy incrementof v = 2.0 2.134. molesof an ideal gas whose adiabatic exponent y = 1.30if, as a result of a certain process,the gas volume increaseda = 2.0 timeswhile the pressuredropped = 3.0times., 2.135. Vessels1 and 2 containv = t.2molesof gaseoushelium. The ratio of the vessels'volumes V\177/V1 = a 2.0,and the ratio of the absolute temperaturesof helium in them Tx/T\177 = = t.5. Assuming the gas to be ideal,find the difference of gas entropiesin these vessels,S\177-Sx. 2.136. One moleof an idealgas with the adiabaticexponenty goes a through polytropic processas a resultof which the absolutetemperatureof the gas increasesT-fold.The polytropic constantequalsn. Find the entropy incrementof the gas in this process. The expansionprocessof v = 2.0molesof argon proceeds 2.137. in directproportionto itsvolume. so that the gas pressureincreases \177

-----

\177

C\177

correspondingheat capacity C of the substanceas a function of what conditionis C < 07

2.145.

Find the temperatureT as a function of the entropy S of a substancefor a polytropic processin which the heat capacity of the substanceequalsC.The entropy of the substanceisknown to be equal to So at the temperatureTo. Draw the approximateplots and T(S)for 2.146. One moleof an idealgaswith heat capacity Cv goesthrough a processin which itsentropy S dependson T as S = a/T, where a is a constant.The gas temperature to T\177. Find: varies from (a) the molarheat capacity of the gas as a function of itstempe-

C>0

C<0.

T\177

rature;

(b) the amount of heat transferredto the gas; (c) the work performed by the gas. 2.147. A working substancegoes through a cycle within which the absolutetemperaturevaries n-fold, and the shape of the cycle is shown in (a) Fig.2.4a;(b) Fig.2.4b,where T is the absolute

temperature,and S the entropy.Find the

efficiency of each cycle.)))

One of the 2.148.

two thermally insulatedvesselsinterconnected by a tube with a valve contains = 2.2molesof an idealgas.The other vessel is evacuated.The valve having been opened,the gas increaseditsvolume n ----3.0times.Find the entropy incrementof \177

the gas.

A weightlesspiston divides a thermally insulated cylinder 2.149.

into two equal parts. One part contains one moleof an ideal gas adiabaticexponent7, the other is evacuated.The initial gas temperatureis To. The piston is releasedand the gas fills the whole

with

(a) the probability of atoms gatheringin one half of the vessel; numericalvalue of N ensuringthe occurrence (b) the approximate of this event within the time interval t years (the age of the \177

t01\302\260

Universe).

distribution Find the statisticalweight of the most probable 2.155. =

l0 identicalmoleculesover two halves of the cylinder's volume.Find also the probability of such a distribution. of an ideal gas. Dividing A vesselcontainsN molecules 2.156. A and into two halves the vessel B, find the probability mentally Considerthe caseswhen N = 5 that the half A containsn molecules.

of N

= 0, t, 2, 3, 4, 5. A vessel of volume 2.157.

and n

of an ideal Vo containsN molecules gettinginto a certainseparatgas.Find the probability of n molecules ed part of the vessel of volume V. Examine,in particular,the case V Vo/2. Find the diameAn idealgas is under standard conditions. 2.158. ter of the spherewithin whose volume the relativefluctuation of the is equal to = t.0.t0-3.What is the average number of molecules inside such a sphere? number of molecules molecules One moleof an idealgasconsistingof monatomic 2.159. is enclosedin a vessel at a temperatureTo = 300K. How many times and in what way will the statisticalweight of this system by AT = t.0 K? (gas) vary if it is heated isochorically \177l

Fig. 2.4.

cylinder.Then the piston is slowly displacedback to the initial position. Find the incrementof the internal energy and the entropy of the gas resulting from these two processes. 2.|50.An ideal gas was expandedfrom the initial state to the volume V without any heat exchangewith the surroundingbodies. Will the final gas pressurebe the samein the caseof (a) a fast and in the caseof (b) a very slow expansionprocess? A thermally insulatedvesselis partitionedinto two parts so that the volume of one part is n 2.0timesgreaterthan that of 0.30moleof nitrogen,and the other.The smallerpart contains the greaterone 0.70moleof oxygen.The temperatureof the gasesis the same.A hole is puncturedin the partitionand the gases aremixed.Find the correspondingincrementof the system'sentropy, assumingthe gasesto be ideal. volume of the

2.|51.

2.5.LIQUIDS.CAPILLARY \342\200\242

Additional

(Laplace'sformula):

EFFECTS

(capillary)

pressurein a liquid

under

an arbitrary

surface

\177

hp=-

\177z

\177

\1771

v\177

2.|52.A ---pieceof copperof mass

= 300g with initial temin which the water 97 is placedinto a calorimeter perature of mass t00g is at a temperature 7 Find the entropy incrementof the system by the moment the temperatures equalize. The heat capacity of the calorimeteritself is negligibly small. 2.|53.Two identicalthermally insulated vessels interconnected by a tube with a valve containone moleof the sameidealgas each. The gas temperature in one vesselis equal to T1 and in the other, The molarheat capacity of the gas of constantvolume equals The valve having been opened,the gas comesto a new equilibrium state.Find the entropy incrementAS of the gas.Demonstratethat m\177

\342\200\242

--\177

t\177

\302\260C.

AS\1770.

2.|54.N atoms3 of gaseoushelium areenclosedin

at room

temperature.Find:

a cubicvessel

of a liquid:

= adS,

(2.5b)

dS is

where

\302\260C

-----

t.0 cm

-\177

where (z is the surface tension of a given liquid. Free energy increment of the surfacelayer

t\177

m\177

of volume

(2.5a)

, (--\177 .-\177)

\177

\342\200\242

during

the area increment of the surface layer. Amount of heat required to form a unit area of the liquid the isothermal increaseof its surface:

Find 2.160.

--T

d\177-\"

surface layer

(2.5c)

the capillary pressure (a) in mercury dropletsof diameterd (b) inside a soap bubble of diameterd == 3.0mm if the surface tension of the soap water solution is a = 45 mN/m. In the bottom of a vesselwith mercury there is a round hole of diameterd 70 \177tm. At what maximumthicknessof the mercury layer will the liquid stillnot flow out through this hole?

t.5

\177tm;

2.161.

\177-

93)))

2.i62.A vesselfilled with air under pressurePo containsa soap bubble of diameterd. The air pressurehaving beenreducedisotherFind the surface mally n-fold, the bubble diameterincreased \177l-fold. tensionof the soap water solution. 2.i63.Find the pressurein an air bubble of diameterd 4.0\177tm, locatedin water at a depth h 5.0m. The atmospheric pressure has the standard value Po. 2.i64.The diameterof a gasbubble formed at the bottom of a pond is d 5.0\177tm. When the bubble risesto the surface its diameter increasesn ----t.1 times.Find how deepis the pond at that spot. The atmospheric pressureis standard,the gas expansionis assumed to be isothermal. 2.i65.Find the difference in height of mercury columnsin two communicatingvertical capillarieswhose diametersare d1 \177

--\177

\177-

0.50mm

----

2.i66.A

t38.

1.00mm,

if the contactangle0 ---vertical capillarywith inside diameter

and

--\177

d\177

\177

\302\260

0.50mm

submergedinto water so that the length of its part protrudingover the water surface is equal to h 25 mm.Find the curvature radius --\177

of the

meniscus.

2.i67.A glasscapillary of

length l tt0 mm and inside diameterd----20\177tmis submergedvertically into water.The upper end of the capillary is sealed. The outsidepressureis standard.To what length x has the capillaryto be submorgedto makethe water levels inside and outside the capillarycoincide? 2.168. When a verticalcapillaryof length l with the sealedupper end was brought in contactwith the surface of a liquid,the level of this liquid roseto the height h. The liquid density is p, the inside diameterof the capillary is d, the contactangleis 0, the atmospheric pressureis Po.Find the surfacetensionof the liquid. 2.i69.A glass rod of diameterd1 ----1.5mm is inserted sym2.0mm. metricallyinto a glasscapillarywith insidediameter Then the whole arrangementis vertically orientedand brought in contactwith the surfaceof water.To what height will the water rise \177--

\177

d\177

in the

capillary?

2.t70.Two verticalplatessubmergedpartially in a wetting liquid form a wedge with a very smallangle The edgeof this wedge is vertical.The density of the liquid is p, itssurface tensionis a, the contactangle is 0.Find the heighth, to which the liquid rises,as a function of the \177i(p.

distancex the edge. 2.|7|. verticalwater jet flows out of a round hole.One of the horizontalsectionsof the has the diameterd --2.0 while from

the other sectionlocatedl

jet

= 20mm

lower has the diameterwhich is n ----t.5 timesless.Find the volume of the water flowing from

the hole e\177ch

second.

2.172. A water drop falls in air with a uniform velocity.Find the difference between the curvature radii of the drop'ssurface at the upper and lower points of the drop separatedby the distance

-----

2.3mm.

2.|73.

A mercury drop shaped as a round tablet of radius Assumand thicknessh is locatedbetweentwo horizontalglassplates. be of to the mass m a which has h find that , placed weight ing on the upperplateto diminishthe distancebetweenthe platesn-times. /\177

\177/\177

m The contactangleequals0. Calculate

if/\177

--\177

2.0cm,h

\177

0.38ram,

2.0,and 0 135. Find the attractionforce between two parallelglassplates, 2.174. separatedby a distanceh 0.10mm, after a water drop of mass 70 mg was introducedbetween them.The wetting is assumed m to be complete. 2.|75.Two glass discsof radius ----5.0 cm were wetted with water and put togetherso that the thicknessof the water layer be1.9\177tm. Assumingthe wetting to be complete, tween them was h find the force that has to he appliedat right anglesto the platesin orderto pull them apart. 2.|76.Two verticalparallelglassplatesare partially submerged 0.t0 mm, and in water. The distancebetween the plates is d their width is l \177-t2 cm. Assuming that the water between the platesdoesnot reachthe upper edgesof the platesand that the wetting is complete,find the force of their mutual attraction. 2.|77.Find the lifetimeof a soap bubble of radius R connected with the atmospherethrough a capillary of length l and inside radius r. The surface tension is a, the viscosity coefficient of the gas is 2.|78.A verticalcapillary is brought in contactwith the water surface.What amount of heat is liberatedwhile the water rises the suralongthe capillary?The wetting is assumed to be complete, n ----

\302\260

\177

--\177

\177

/\177

\177-

\177

\177i-

face tension equals 2.179.Find the free energy of the surface layer of mm; (a) a mercury dropletof diameterd (b) a soap bubble of diameterd 6.0mm if the surface tension of the soap water solution is equal to a =-45 mN/m. Find the incrementof the free energy of the surface layer when two identicalmercury droplets,each of diameterd mm,

'1.4

--\177

2.180.

\177

mergeisothermally.

1.5

2.|8i.

Find the work to be performed in orderto blow a soap bubble of radius if the outsideair pressureis equal to P0 and the surfacetension of the soap water solution is equal to a. A soap bubble of radius r is inflated with an ideal gas. 2.182. The atmospheric pressureis Po, the surface tensionof the so\177p water solution is a. Find the difference between the molar heat capacity of the gas during itsheating inside the bubble and the molarheat capacity of the gas under constant pressure, Cp. 2.183. Consideringthe Carnot cycle as appliedto a liquid film, show that in an isothermalprocessthe amount of heat requiredfor the formation of a unit area of the surface layer is equal to q----where da/dT is the temperaturederivative of the \177--T.da/dT, surface tension. /\177

C-

95))) \177)4

2.184. The A(r

at a

surface of a soap film was increasedisothermally by temperatureT. Knowing the surface tension of the soap

water solution a and the temperaturecoefficient

find the

d\177z/dT,

increment (a) of the entropy of the film'ssurface layer; (b) of the internal energy of the surface layer.

2.6.PHASE \342\200\242

Relations

\302\260C

2.191.

der Waals constants

between

Van

3b, Per =

2.190.

\177.

TRANSFORMATIONS

critical state of a substance:

Assumingthe saturatedvapour to be an idealgas find the increment of entropy and internal energy of the system. Water of mass m = 20 g is enclosed in a thermally insulated cylinder at the temperatureof 0 under a weightlesspiston whose area is S=4t0em The outsidepressure is equal to standard atmosphericpressure.To what height will the piston rise when the water absorbs Q = 20.0kJ of heat? One gram of saturatedwater vapour is enclosed in a thermThe outsidepresally insulatedcylinder under a weightlesspiston. sure beingstandard, m = g of water is introducedinto the cylinder at a temperatureto = 22 Neglectingthe heat capacity of the cylinder and the friction of the piston against the cylinder's walls, find the work performedby the force of the atmospheric pressure during the loweringof the piston. If an additionalpressure Ap of a saturated vapour over a convexsphericalsurface of a liquid is considerablyless than the vapour pressureover a planesurface,then Ap 2a/r, where and are the densitiesof the vapour and the liquid, a is.the surface tension, and r is the radius of curvature of the surface.Using this formula, find the diameterof water dropletsat which the saturated vapour pressureexceedsthe vapour pressureover the plane surface by = t.0% at a temperaturet 27 The vapour is assumed to be an ideal gas. Find the mass .of all molecules leavingone square centimetreof water surfacepersecondinto a saturatedwater vapour above it at a temperaturet 100 It is assumed that 3.6%of all water vapour molecules falling on the water surface are retained in the liquid phase. 2.194. Find the pressureof saturated tungsten vapour at a temK if a tungstenfilament is known to losea mass peratureT 2000 g/(s.cm from a unit area per unit time when evaporatinginto high vacuum at this temperature. By what magnitudewould the pressureexertedby water on the walls of the vesselhave increased if the intermolecular attraction forces had vanished? Find the internal pressure of a liquid if its density p and specificlatent heat of vaporizationq are known. The heat q is assumed to be equal to the work performed against the forces of the internal pressure,and the liquid obeys the Van der Waals

and the

parametersof

the

t.0

\302\260C.

\342\200\242

cr'\177

2762 ,

Tcr

\177

(2.6a)

27/\177'-----\177

Relation between the critical parametersfor a mole of substance: (2.6b) PcrV M cr (3/8)-FlTcr. \177-

\342\200\242

equation: Clausius-Clapeyron dp

2.192.

\177--

(p\177/p\177)

q12

T (V\177--

V\177)

where q12 is the specificheat absorbedin the transformation are the specificvolumes of phases t and 2.

(2.6c) t

-+ 2, V[

and

V\177

p\177

\177

\1771

A saturated water vapour is contained in a cylindrical 2.185. vessel under a weightlesspiston at a temperaturet = t00 As a result of a slow introductionof the piston a smallfraction of the 0.70g gets condensed.What amount of work was vapour Am performed over the gas? The vapour is assumed to be ideal, the volume of the liquid is to be neglected. 6.01 containswater togetherwith A vesselof volume V 2.186. itssaturatedvapour under a pressureof 40 arm and at a temperature 501/kg volume of the vapour is equal to of 250 The specific The total massof the system water-vapour under these conditions. equalsm = 5.0kg. Find the mass and the volume of the vapour. Thesaturatedwater vapour is enclosedin a cylinder under 2.187. \302\260C.

\177

--\177

V\177

\302\260C.

occupiesa volume Vo -= 5.01 at the temperaturet ---Find the mass of the liquid phase formed after the volume under the piston decreased isothermally V =- 1.61.The saturated to be ideal. is assumed vapour A volume occupiedby a saturatedvapour is reducediso2.188. thermally n-fold.Find what fraction of the final volume is occupied by the liquid phase if the specificvolumes of the saturated vapour and the liquid phase differ by N times (N > n). Solve the same problem under the conditionthat the final volume of the substance to the midpointof a horizontalportionof the isothermal corresponds a piston and

= t00

\302\260C.

I\177o

\302\260C.

2.193.

\177

\302\260C.

\177l

---=

t.2.t0

-\177a

\177t--\177

\177)

2.195. 2.196.

p\177

1.00kg, boilingat staninto saturated vapour. t urns dard atmospheric completely pressure,

equation.Calculate in water. 2.197.Demonstratethat Eqs.(2.6a)and (2.6b) are valid for a substance, obeying the Van der Waals equation, in critical state. Instruction.Makeuse of the fact that the criticalstatecorresponds to the point of inflection in the isothermalcurve p (V). 2.t98.Calculatethe Van der Waals constantsfor carbondioxide if itscriticaltemperature = 304 K and criticalpressure = 73 atm.

7--9451

\177l

line in the diagramp, An 2.189.

amount of water of mass rn

\177

p\177

T\177,

p\177,

-\177

97)))

2.|99.Find the specificvolume of benzene(C6H6)in

criticalstate

itscriticaltemperatureTcr 562K and criticalpressurePer 47 atm. Write the Van der Waals equationvia the reducedpara2.200.

\177--

---\177

-----

and T, having taken the correspondingcriticalvalues for the units of pressure, volume,and temperature. Using the equaexceedsits tion obtained,find how many timesthe gas temperature criticaltemperatureif the gaspressureis t2 timesas high as critical pressure,and the volume of gas is equal to half the criticalvolume. 2.201. Knowing the Van der Waals constants,find: 1.00kg can the maximum volume which water of mass m (a)

meters

Assumethat: the specific latent heat of vaporizationq is independent of T, the specific volume of liquid is negligible as compared to that of vapour, saturatedvapour obeys the equationof state for an ideal gas.Investigateunder what conditionstheseassumptionsare permis-

sible.

into the equilibrium state? At what Fig. 2.5. water temperatureof the supercooled does it turn into ice completely? 2.206. Find the incrementof the icemelting temperaturein the of 0 when the pressureis increasedby Ap= 1.00atm. vicinity 0.09tcm3/g. that of water by\177AV volume of iceexceeds The specific under vapour 2.207.Find the specificvolume of saturated water of pressureby Ap '--3.2kPa is known standard pressureif a decrease to decreasethe water boiling temperatureby AT = 0.9K. 2.208. Assuming the saturated water vapour to be ideal, find its pressureat the temperaturet0t.t 2.209.A smallamount of water and its saturated vapour are enclosedin a vesselat a temperaturet 100 Howmuch (in per cent) of will the mass of the saturatedvapour increaseif the temperature K? Assume that the vapour is an the system goesup by AT idealgas and the specificvolume of water is negligibleas compared to that of vapour. Find the pressure of saturated vapour as a function of temperaturep (T) if at a temperatureTo its pressureequals

2.21t.An ice which was initially under standard conditionswas compressed up to the pressurep ----640 atm. Assuming the lowering of the ice melting temperatureto be a linearfunction of press4re under the given conditions, find what fraction of the icemelted.The specificvolume of water is lessthan that of iceby AV' 0.09cma/g. 2.212. In the vicinity of the triple point the saturated vapour pressure p of carbondioxidedependson temperatureT as logp = -= a -- b/T, where a and b are constants.If p is expressed in atmospheres,then for the sublimationprocessa 9.05and b t.80kK, and for the vaporizationprocessa 6.78and b t.31kK.Find: (a) temperatureand pressure at the triple point; (b) the values of the specificlatent heats of sublimation, vaporization, and melting in the vicinity of the triple point. 2.213. Water of mass rn t.00kg is heatedfrom the 100 at which it evaporatestemperature t,--\177 t0 up to completely. Find the entropy increment of the system. 2.214. The ice with the initial temperature -----0 was first melted,then heated to the temperature = 100 and evaporated. Find the incrementof the system'sspecificentropy. 2.215. A pieceof copper of mass m 90g at a temperaturetx = 90 was placedin a calorimeterin which iceof mass 50 g was at a temperature--3 Find the entropy incrementof the piece of copperby the moment the thermal equilibrium is reached. A chunk of ice of mass ml 2.216. t00g at a temperaturetx 0 was placedin a calorimeterin which water of mass 100g was at a temperature Assuming the heat capacity of the calorimeterto be negligible,find the entropy incrementof the system by the moment the thermalequilibriumisreached.Consider two cases:(a) te 60 (b) t\177-----94 2.217. Molten leadof mass m 5.0g at a temperature = 327 of was (themeltingtemperature lead) poured into a calorimeter packed with a large amount of iceat a temperature tI = 0 Find the entropy incrementof the system lead-iceby the moment the thermal equilibriumisreached.The specificlatent heat of meltingof leadis equal to q 22.5J/g and its specificheat capacity is equal to c = 0.125 J/(g.K). 2.2i8. A water vapour filling the spaceunder the piston of a cylinder, is compressed (or expanded) so that it remainssaturatedall the time, beingjust on the verge of condensation. Find the molar heat capacity C of the vapour in this processas a function of temperature T, assuming the vapour to be an idealgas and neglecting the specificvolume of water in comparison with that of vapour. CalculateC at a temperaturet t00

\177,

\177

occupy in liquid state; (b) the maximum pressure of the saturated water vapour. Calculatethe temperatureand density of carbon dioxide 2.202. in criticalstate, assuming the gas to be a Van der Waals one. 2.203. What fraction of the volume of s vesselmust liquid ether in orderto pass into criticalstatewhen at occupy room temperature criticaltemperatureis reached?Ether has Tcr = 467 K, p\177 = 35.5atm, P M = 74 g/mol.

Demonstratethat 2.204. line 1-5correspondingto

the straight the isothermal-isobaric phase transition cuts the Van der Waals isotherm so that are equal (Fig.2.5). areas and 2.205.What fraction of water supercooleddown to the temperature t

=--20

\302\260C

\302\260C.

2.210.

\302\260C

t\177

-----

\177

\302\260C

t.5

--\177

\177

\302\260C

turns into icewhen the system passes

\177--

\177

t\177

under standard pressure

\177--

\177

t\177

\177

\302\260C.

-----

\177

-----

--\177

\177

\302\260C

m\177

-\177

t\177.

\177

\302\260C.

\302\260C;

\177

\302\260C

t\177

\302\260C.

\177

\302\260C.

99)))

One moleof water being in 2.219.

equilibriumwith a negligible amount of itssaturated vapour at a temperatureT1 was completely convertedinto saturated vapour at a temperature Find the entropy incrementof the system. The vapour is assumed to be an ideal in comparison with gas,the specificvolume of the liquid is negligibIe that of the vapour. T\177.

2.7.TRANSPORT

PHENOMENA

Relativenumber of gas moleculestraversing

\342\200\242

lisions:

N/No

-\177

distances without col-

the

e sp,

(2.7a)

free path. Mean free path of a gas molecule:

where X \342\200\242

the mean

\177

where d

the

effective diameter

cules per unit volume.

t__

a molecule,and

D, viscosity

Coefficientsof diffusion

\342\200\242

\177l,

(2.7b)

and

heat

of mole-

the number

conductivity

\303\227

gases:

where p is the gas density, and cv is its specificheat capacityat constant volume. Friction forceacting on a unit area of plates during their motion parallel to each other in a highly rarefied gas: \342\200\242

F= 1

-\177-<v)

where

u\177

\342\200\242

rarefied

and u2

are the

plul-

(2.7d)

u\177l,

flux

transferred

gas:

=-g-<v) pcv

between

-- T2 I,

two

wails

by highly

(2.7e)

are the temperatures of the walls. Calculatewhat fract'ion of gas molecules 2.220. the mean the distancesexceeding (a) traverseswithout collisions free path (b) has the free path values lying within the interval from to A, narrow molecularbeam makesits way into a vessel Find the mean free path of molefilled with gas under low pressure. cules if the beam intensity decreases \177l-fold over the distanceAl. 2.222. Let adt be the probability of a gas moleculeexperiencing a collision during the time interval dt; a is a constant.Find: no collisions the during experiencing probability of a molecule (a) where T1and

T\177

\177,;

\177

2\177.

2.221.

the time interval t;

(b) the mean time interval between successivecollisions. the mean free path and the mean time interval beof gaseousnitrogen molecules tween successivecollisions (a) under standard conditions; Find 2.223.

I0o

\302\260C

-\177

\302\260C

tion). Find: (a) the gas pressurebelow which the mean free path of the moleculesk > l; (b) the correspondingmolecularconcentrationand the mean

distancebetween the molecules. 2.228. For the caseof nitrogen under standard conditionsfind: mean number of collisions the by eachmolecule experienced (a) second; per occurringbetween the molecules (b) the total number of collisions within tcma of nitrogen per second. How doesthe mean free path k and the number of collisions 2.229. of eachmolecule per unit time depend on the absolutetemperature of an ideal gas undergoing (a) an isochoricprocess; (b) an isobaricprocess? As a result of someprocessthe pressureof an ideal gas 2.230. increasesn-fold. How many timeshave the mean free path k and each moleculeper unit time v changed the number of collisions'of and how, if the processis (a) isochoric;(b) isothermal? An ideal gas consistingof rigid diatomicmolecules goes 2.231. How do the mean free path k and the adiabaticprocess. an through of eachmoleculeper secondv depend in this nunlber of collisions processon T? (a) the volume V; (b) the pressurep; (c) the temperatureexAn ideal gas goes through a polytropic processwith 2.232. \177

velocitiesof the plates.

Density of a thermal

t = 0 and pressurep = 1.0 nPa'(sucha pres(b) at temperature vacuum of means be reached can sure pumps). contemporary by How many timesdoesthe mean free path of nitrogen mole2.224. culesexceedthe mean distancebetween the moleculesunder standard conditions? under standard Find the mean free path of gas molecules 2.225. conditionsif the Van der Waals constantof this gas is equalto b = 40 ml/mol. An acousticwe've propagates 2.226. through nitrogenunder stanAt what frequency will the wavelength be equal dard conditions. to the mean free path of the gas molecules? 2.227.Oxygen is enclosedat the temperature0 in a vessel dimensionl = t0 mm (this is the linear with the characteristic the dimensiondetermining characterof a physical processin ques-

of ponent n. Find the mean free path k and the number of collisions each moleculeper second as a function of (a) the volume V; (b) the pressurep; (c) the temperatureT. Determinethe molarheat capacity of a polytropic process 2.233. through which an ideal gas consistingof rigid diatomicmolecules between the molecules goes and in which the number of collisions \177

remains constant (a) in a unit volume;(b)

in the

total volume

of the

gas.)))

2.234. An ideal gas of molarmass M

is enclosedin a vessel of

volume V whose thin walls are kept at a constant temperatureT. a moment t 0 a smallhole of area S is opened,and the gas startsescapinginto vacuum.Find the gas concentration n as a function of time t if at the initial moment n(0) = n o 2.235. A vessel filled with gas is divided into two equal parts I and 2 by a thin heat-insulatingpartition with two holes.One hole has a smalldiameter,and the other has a very largediameter (in comparisonwith the mean free path of molecules). In part 2 the gas is kept at a temperature timeshigher than that of part How will the concentrationof molecules in part 2 changeand how many times after the large hole is closed? 2.236. As a result of a certainprocessthe viscosity coefficient of an ideal gas increasesa = 2.0times and its diffusion coefficient = 4.0times. How doesthe gas pressurechange and how many times? 2.237.How will a diffusion coefficient D and the viscosity coefficient of an idealgas changeif its volume increases n times: (a) isothermally;(b) isobarically? 2.238. An idealgas consists of rigid diatomicmolecules, tIow will a diffusion coefficient D and viscosity coefficient changeand how many timesif the gasvolume is decreased adiabaticallyn l0times? 2.239. An ideal gas goes through a polytropic Find the process. polytropic exponentn if in this processt'he coefficient (a) of diffusion; (b) of viscosity; (c) of heat conductivity remains At

-----

\177l

\177

\177l

\1771

\177

constant.

2.240. Knowing the viscosity coefficient of heliumunderstandard effective diameterof the helium atom. 2.241. The heat conductivity of helium is 8.7timesthat of argon

conditions,calculatethe (under standard

Find the ratio of conditions).

of argon and helium

atoms.

effective

diameters

2.242.Understandardconditionsheliumfills up the spacebetween two longcoaxialcylinders.The mean radius of the cylindersis to R, the gap between them is equal to AT/, with AT/<\177 R.equal The outer cylinder rotateswith a fairly low angular velocity co about the stationary inner cylinder.Find the moment of friction forces actingon a unit lengthof the innercylinder.Downtb what magnitude should the heliumpressurebe lowered(keepingthe temperature constant) to decreasethe sought moment of friction forcesn i0 times if AR = 6 mm? 2.243.A gas fills up the spacebetweentwo long coaxialcylinders of radii R1 and R2, with R1 R2. The outer cylinder rotates with a fairly low angular velocity co about the stationary inner cylinder. The moment of friction forces acting on a unit length of the inner cylinder is equal to N Find the viscosity coefficient of the gas taking into accountthat the frictionforce actingon a unit areaof the cylindrical surface of radius r is determinedby the formula \177---\177

<\177

\177r

102

(Oo\177/Or).

\177l

a common axis and are radius of eachdiscis The each other. h from a distance at located a low angularvelocwith is disc rotated h. One with a to >> a, equal ity co relativeto the other,stationary, disc.Find the moment of friction forcesacting on the stationary disc if the viscosity coefficient of the gas between the discsis equal to Solve the foregoingproblem, assuming that the discs 2.245. arelocatedin an ultra-rarefiedgasof molarmass M, at temperatureT

2.244.Two identicalparalleldiscshave

and under pressurep. 2.246. Making use of

find Poiseuille's equation (1.7d),

the mass

timethrough the pipeof lengthI and radiusa are maintained at its ends. if constant pressures Pl and 2.247.Oneend of a rod,enclosedin a thermally insulatingsheath, The rod is comis kept at a temperatureT1 while the other,at heat conductivand and are w hose two sections of 11 lengths posed Find the temperatureof the interface. and ity coefficients 2.248. Two rods whose lengthsare and and heat conductivity coefficients and are placedend to end.Find the heat conductivity coefficient of a uniform rod of length l\177-la whose conductivity The lateralsurfaces is the sameas that of the system of thesetwo rods. of the rods are assumed to be thermally insulated. 2.249.A rod of length l with thermally insulated lateralsurface consistsof materialwhose heat conductivity coefficient varies with a/T, where a is a constant.The ends of the rod temperatureas Find the function T (x), where are kept at temperatures T1 and end whose the from x is the distance temperatureis T1, and the heat flow density. are 2.250. Two chunks of metal with heat capacitiesC1 and S and area cross-sectional l and a rod of interconnected length by fairly low heat conductivity The whole system is thermally insulated from the environment.At a moment t = 0 the temperature differencebetween the two chunks of metalequals(AT)0. Assuming the heat capacity of the rod to be negligible,find the temperature difference between the chunks as a function of time. 2.251. Find the temperaturedistribution in a substanceplaced The between two parallelplateskept at temperaturesT1 and of coefficient plateseparationis equal to l, the heat conductivity t\177

of gasflowing

perunit

p\177

T\177.

l\177

\303\227\177.

\303\2271

l\177

\303\2271

l\177

\303\227\177

\177

\303\227

T\177.

C\177

\303\227.

T\177.

the substance

\177-V

The spacebetween two large horizontal plates is filled 2.252. l = 50 mm. The lower \303\227

helium.The plate separationequals plate is kept at a temperatureT1 = 290K, the upper, atis close = 330K. Find the heat flow density if the gas pressure to standard. 2.253. The spacebetween two largeparallelplatesseparatedby a distancel 5.0mm is filled with helium under a pressurep = t.0 Pa. One plate is kept at a temperaturetl = 17 and the other,at a temperature = 37 Find the mean free path of helium

with

\177

\302\260C

\302\260C.

t\177

atoms and the heat flow density.

t03)))

2.254. Find the temperaturedistribution in coaxial

the space between filled with a uniform heat conductingsubstanceif the o[the cylinders areconstant and are equal to T1 and temperatures T 2 respectively. 2.255.Solvethe foregoingproblemfor the caseof two concentric spheresof radii/\1771 and and temperatures and T2. 2.256. A constant electric current flows along a uniform wire with cross-sectional radius and heat conductivity coefficient A unit volume of the wire generatesa thermal power w. Find the temperaturedistribution acrossthe wire provided the steady-state temperatureat the wire surface is equal to T o 2.257.The thermalpower of w is generateduniformly insidea uniform sphereof radius/\177density and heat conductivity coefficient Find the temperaturedistribution in the sphere provided the steady-statetemperatureat its surface is equal to T o two

cylinders of

radii/\1771

/\1772

and

THREE

PART

/\1772

ELECTRODYNAMICS

T\177

/\177

\303\227.

3.i.CONSTANT

ELECTRICFIELD and

Strength

\342\200\242

potential of t E

VACUUM

of a

the field q

4aCe ra

charge q:

point

q\177

4\177e

(3Aa)

\303\227.

Relation between

\342\200\242

field strength

i.e.field Gauss's strength

\177E \342\200\242

Potential and

where \342\200\242

(3.1b)

dS=q/eo, of the

strength

4ue0

of a

point

(3.1c)

dipole with electric mo-

]/tQ-3

0 is the angle between the vectorsr and p. Energy W of the dipolep in an external electric field,

and the moment

on the

E--\177

''W r

4ue---\177

dipole: W

\342\200\242

field

\177Edr=

(3Ad)

of forcesacting

=- --pE, N

cos\177

= [pE].

(3.1e)

Force F acting on a dipole, and its projectionFx: F=p

where VE x

potential:

--Vq),

(p\177

and

is equal to the antigradient of the potential theorem and circulation of the vector E:

\342\200\242

ment

\177--

OEIOlis the derivative gradient of the

is the

8E --\177-[,

Fx = p .VEx,

of the vector function E x.

E with respectto

(3.1f) the

dipoledirection,

3.1.Calculatethe ratio of the electrostaticto gravitationalinter-

actionforcesbetween two electrons,between two protons.At value of the

specificcharge q/m of a

what

particlewould theseforcesbe-

comeequal (in their absolutevalues)in the caseof interactionof identicalparticles? 3.2.What would be the interactionforce between two copper spheres,eachof massi g, separatedby the distancet m, if the total electronicchargein them differed from the total chargeof the nuclei by one per cent? 3.3.Two small equally charged spheres,each of mass m, are suspended from the samepoint by silk threads of length l. The distancebetween the spheresx << l. Find the rate dq/dt with which \17705)))

the chargeleaksoff each sphere if their approachvelocity varies as v a/]/x,where a is a constant. are locatedat the points with Two positivechargesql and radiusvectorsr 1 and r 2. Find a negativechargeq3 and a radiusvector r3 of the point at which it has to be placedfor the forceacting on each of the three chargesto be equal to zero. A thin wire ring of radius r has an electriccharge q. What will be the incrementof the force stretchingthe wire if a point charge q0 is placedat the ring'scentre? is locatedin the plane xy A positive point charge 50 at the point with radius vector r0----2i \177-3j, where i and j are the unit vectors of the x and y axes.Find the vector of the electricfield strength E and its magnitude at the point with radius vector r 8i Here r0 and r are expressedin \177

3.4.

q\177

3.5.

3.6.

t\177C

5j.

metres,

3.7.Point chargesq and --qare locatedat the verticesof a square with diagonals2l as shown -0 in Fig.3.1.Find the magnitude of the electric

\177

/\177

\177

3.9.

3.I0.

--q.

f\177.

3.11.

)\177.

3.12.

)\177---=

3.I3.

106

f\177

)\1770

\177.0

the

\342\200\242

rod).

Investigate the obtained expressionsat r>>a. A very long straight uniformly charged thread carries 3.I4. a charge per unit length. Find the magnitudeand directionof the electricfield strength at a point which is at a distancey from the thread and lieson the perpendicular passingthrough one of the )\177

thread'sends.

3.I5.A thread carrying a uniform charge per unit length has the configurationsshown in Fig.3.2a and b. Assuming a curvature )\177

..... /

\177-\302\242

field strength at a point locatedsymmetrically Fig. with respectto the verticesof the square at a distancex from its centre. 20 cm is uniformly charged A thin half-ringof radius with a total chargeq ----0.70nC. Find the magnitudeof the electric field strength at the curvature centreof this half-ring. A thin wire ring of radiusr carriesa chargeq.Find the magnitude of the electricfield strengthon the axisof the ring as a function of distancel from its centre.Investigatethe obtained function at l )> r. Find the maximumstrengthmagnitudeand the correspondplot of the function E(l). ing distance Draw the approximate A point charge q is locatedat the centreof a thin ring of Find the magnitude radius/\177 with uniformly distributed charge of the electricfield strength vector at the point lying on the axis of the ring at a distancex from its centre,if x )) A system consistsof a thin chargedwire ring of radius and a very long uniformly chargedthread orientedalong the axis of the ring, with one of its ends coincidingwith the centreof the of the ring. The total chargeof the ring is equal to q. The charge thread (per unit length) is equal to Find the interactionforcebetween the ring and the thread. A thin nonconducting ring of radius has a linearcharge cosT, where is a constant, q) is the azimuthal density angld. Find the magnitudeof the electricfield strength (a) at the centreof the ring; (b) on the axisof the ring as a function of the distancex from its centre. Investigate the obtainedfunction at x >)f\177. A thin straight rod of length 2a carrying a uniformly distributed charge q is locatedin vacuum. Find the magnitude of the

3.8.

electricfield strength as a function of the distancer from the rod's centrealongthe straight line (a) perpendicularto the rod and passing through its centre; (b) coincidingwith the rod'sdirection(at the points lying outside

3.t.

3.2.

Fig.3.3. radius to be considerablylessthan the length of the thread,find the magnitude of the electricfield strength at the point O. 3.I6.A sphereof radius r carriesa surface chargeof density = at, where a is a constant vector,and r is the radius vector of a point of the sphererelativeto its centre.Find the electricfield strength vector at the centreof the sphere. 3.I7.Supposethe surface chargedensity over a sphereof radius/\177 dependson a polarangle 0 as (r = (r0 cos0, where (r0 is a positive constant.Showthat such a chargedistributioncan be represented as a result of a smallrelativeshift of two uniformly charged balls of radius // whose chargesare equal in magnitudeand opposite in find the electricfield strength sign.Resortingto this ropresentation, vector inside the given sphere. 3.18.Find the olectricfield strongth vector at the centreof a ball of radius with volume chargedensity p = ar, where a is a constant vector,and r is a radiusvector drawn from the ball'scentre. 3.I9.A very long uniformly charged thread oriented the axisof a circleof radius restson itscentrewith one of along the ends. The chargeof the thread per unit length is equal to Find the flux of the vector E acrossthe circlearea. 3.20.Two point chargesq and --q are separatedby the distanco 2l(Fig.3.3). Find the flux of the electricfield strengthvectoracross a circleof radius 3.2I.A ball of radius is uniformly chargedwith the volume density p. Find the flux of the electricfield strength vector across Fig.

f\177

(\177

\177--

f\177

)\177.

/\177.

f\177

\17707)))

sectionformed by the plane locatedat a distancer0 R the ball's from the centreof the ball. threads carriesa uniform 3.22.Each of the two long parallel are The threads unit separatedby a distancel. length. charge per Find the maximummagnitudeof the electricfield strength in the symmetry plane of this system locatedbetween the threads. 3.23.An infinitely long cylindrical surface of circularcrosssectionis uniformly chargedlengthwisewith the surface density a a0 cosqo, where qo is the polarangleof the cylindricalcoordinate \177

\177,

\177

system whose z axis coincideswith the axis of the given surface. Find the magnitudeand directionof the electricfield strengthvector

on the z axis. 3.24.The electricfield strengthdependsonly on the x and g coordinates accordingto the law E a (xi gj)/(x2 g2), where a is a constant, i and j are the unit vectorsof the x and g axes.Find the flux of the vector E through a sphereof radius/\177 with itscentre at the origin of coordinates. carriesa positivechargewhose volume A ball of radius centreas density dependsonly on a separationr from the the is a constant. where Assuming permittivities P0 r/Tl), P\177 P0 (1 of the ball and the environmentto be equal to unity, find: (a) the magnitudeof the electricfield strengthas a function of the distancer both inside and outside the ball; distance (b) the maximumintensity Emax and the corresponding of a ball of radius/\177 carrying a spherically A system consists symmetric chargeand the surroundingspacefilled with a chargeof volume density p a/r, where a is a constant, r is the distance Find the from the centreof the chargeat which the magis vector field electric of the nitude independentof r outside strength How high is this strength?The permittivitiesof theball the and the surroundingspaceare assumed to be equal to unity. 3.27.A spaceis filled up with a charge with volume density p p0e-\177r3, where P0 and a are positive constants,r is the distance from the centreof this system.Find the magnitudeof the electric field strengthvectoras a function of r. Investigatethe obtainedexpresat ara \177 t and ar \177 sion for the smalland largevaluesof r, Insidea ball charged uniformly with 'volume density p with thereis a sphericalcavity. The centreof the cavity is displaced field Findthe a distance of ball the the centre t o strength by respect E insidethe cavity, assumingthe permittivity equal to unity. Insidean infinitely long circularcylinder chargeduniformly with volume density p there is a circularcylindricalcavity. The distancebetweenthe axesof the cylinder and the cavity is equal to Find the electricfield strength E insidethe cavity. The permittivity is assumedto be equal to unity. There are two thin wire rings,eachof radius/\177, whose axes Find the potential The chargesof the rings are q and coincide. differencebetweenthe centresof the rings separatedby a distance \177

3.25.

\177-

ball's

/\177

r\177n.

3.26.

\177

ball.

ball's

ball.

\177

3.28.

i.e.

\177

3.29.

a. 3.30.

t08

--q.

3.31.There is an infinitely longstraight thread carrying a charge Calculatethe potentialdifference lineardensity = 0.40 between points 1 and 2 if point 2 is removed = 2.0timesfarther from the thread than point 1. 3.32.Find the electricfield potentialand strength at the centre of radius of a hemisphere charged uniformly with the surface density a. 3.33.A very thin round plateof radius R carrying a uniform surwith

t\177C/m.

\177,

\177l

/\177

a is locatedin vacuum. Find the electricfield axisas a function of a dispotentialand strength along the plate's at l--\177 0 tance l from itscentre.Investigatethe obtainedexpression and l >> R. 3.34.Find the potentialqo at the edgeof a thin discof radius carrying the uniformly distributed charge with surface densi-

face chargedensity

a. 3.35.Find the

electricfield strength vector if the potential of this field has the form qo at, where a is a constantvector,and r is the radius vectorof a point of the field. 3.36.Determinethe electricfield strength vectorif the potential of this field depends on x, y coordinatesas a(x yZ); (b) q)----axy, a) (p where a is a constant.Draw the approximate shape of these fields .usinglinesof force(in the x, y plane). 3.37.The potentialof a certainelectrostaticfield has the form bz where a and b areconstants.Find the maga (x --b nitude and directionof the electricfield strengthvector.What shape have the equipotentialsurfaces in the following cases: (a) a>0, b>0;(b) a>0, b<\1770? 3.38. A charge q is uniformly distributedover the volume of a sphereof radius/\177. Assumingthe permittivity to bo oqual to unity throughout, find the potential (a) at the centroof the sphere; (b) insidethe spheroas a function of the distancer from itscentre. Demonstratethat the potentialof the field generatedby 3.39. a dipole with the electricmomentp (Fig. 3.4)may be representedas ---r is the radius vector. where (p pr/4\177eo Using this expression,find the magnitudeof the electricfield strength vector as a function of r and 0. 3.40.A point dipolewith an electricmomentp \177.\177/ ' orientedin the positive directionof the z axisis located at the origin of coordinates.Find the projectionsE, and of the electricfield strength to the z axisat vector(on the plane perpendicular Fig. 3.4. At which points is E the point S (seeFig.3.4)). to p? perpendicular 3.41.A point electricdipolewith a momentp is placedin the external uniform electricfield whose strength equals Eo, with -\177

-\177

c\177

\177

Y\177)

\177

\177,

r\177,

E.\177

|09)))

surfacesenclosingthe Eo. In this caseone of the equipotential of this radius Find the a f orms sphere. sphere. dipole 3.42.Two thin parallelthreadscarry a uniform chargewith linear densitiesX and --X.The distancebetween the threads is equal to l. Find the potentialof the electricfield and the magnitudeof its strength vectorat the distancer >> l at the angle0 to the vector l (Fig.3.5). 3.43.Two coaxialrings, each of radius R, madeof thin wire are separatedby a smalldistancel (l << R) and carry the chargesq and

\177'

--q.Find the electricfield potentialand

strength at the axisof the

the potential (x, y) of an electrostaticfield E = a (x2 --y2) j, where a is a constant,i and j are the unit vectors of the x and y axes. 3.50.Determinethe potential (x, y, z) of an electrostaticfield E = ayi (ax -4- bz) j byk, where a and b are constants,i, j, k are the unit vectors of the axesx, y, z. 3.51.The field potentialin a certain regionof spacedependsonly on the x coordinate as = --ax3 b, where a and b areconstants. Find the distribution of the spacecharge p (x). 3.52.A uniformly distributed spacechargefills up the spacebetween two largeparallelplatesseparatedby a distanced.The potential difference between the platesis equal to At what value of chargedensity p is the field strength in the vicinity of one of the plates equal to zero? What will then be the field strength near the other plate? 3.53.The field potentialinside a chargedball depends only on the distancefrom its centreas = ar2 b, where a and b areconstants. Find the spacecharge distribution p (r) inside the ball.

3.49.Find 2axyi

\177

-\177-

q\177

\177

-\177

q\177

Aq\177.

-\177

q\177

Fig.

3.5.

3.6.

Fig.

3.7.

Show in the same system as a function of the z ceordinate(Fig.3.6). obtained. functions Investigate plotsof the drawing the approximate thesefunctions at x >> R. 3.44.Two infinite planesseparatedby a distancel carry a uniform surface charge of densities(r and --\177 (Fig. 3.7).The planes have round coaxialholesof radius R, with l << R. Taking the origin axisas shown in the figure, find the potential O and the x coordinate of itsstrengthvectorExon the of the electricfield and the projection Draw the approxaxesof the system as functions of the x coordinate. imate plot (x). 3.45.An electriccapacitorconsistsof thin round parallelplates, eachof radius R, separatedby a distancel (l << R) and uniformly potentialof the chargedwith surface densities and --\177. Find the at the axes vector its of strength electricfield and the magnitude of the capacitoras functions of a distancex from the platesif x >> l. I

3.2.CONDUCTORSAND

]

\302\242

\177

at x >> R. Investigate the obtainedexpressions at a distance 3.46.A dipolewith an electricmoment p islocated r from a long thread chargeduniformly with a linear density Find the force F acting on the dipoleif the vector p is oriented (a) along the thread; (b) along the radius vector r; vector r. (c) at right angles to the thread and the radius molecules two water between 3.47.Find the interactionforce are moments ---electric their nm if l i0 distance a separatedby orientedalongthe samestraight line.The moment of eachmolecule -29 C.m. equalsp = 0.62.10 field E = 3.48.Find the potential (x, y) of an electrostatic vectors unit are the = a (yi + xj), where a is a constant, i and j of the x and y axes. \302\242

DIELECTRICS ELECTRICFIELD Electric field strength near the surface of a conductor En O]eO. Flux of polarization P acrossa closedsurface:

IN AN \342\200\242

\342\200\242

P dS = where \342\200\242

(3.2b)

it:

-\177

(3.2a)

--q',

q' is the algebraicsum of bound chargesenclosedby Vector D and Gauss'stheorem for D = %E

in vacuum:

dS=

.\177D

this

surface. (3.2c)

is the algebraicsum of eztraneous chargesinside a closedsurface. Relationsat the boundary between two dielectrics: P\177n--P\177n= --o'\177 E2\"\177=Er\177, D\177n--D\177n=o, (3.2d) where o' and are the surface densitiesof bound and extraneouscharges,and the unit vector n of the normal is directedfrom medium t to medium 2. In isotropic dielectrics: P= xe0E, D= ee0E, e = x. (3.2e) In the case of an isotropic uniform dielectric filling up all the space between the equipotential surfaces: E = Eo]e. (3.2f) 3.54.A smallball is suspended over an infinite horizontalconductingplane by meansof an insulatingelasticthread of stiffness k. As soon as the ball was charged,it descended by x cm and its separation f\177om the plane becameequal to Find the chargeof the where q \342\200\242

\177

\342\200\242

1-\177-

\342\200\242

ball.

111)))

3.55.A point chargeq is locatedat

distancel from

the infinite

conductingplane. What amount of work has to be performed in order to slowly remove this charge very far from the plane. 3.56.Two point charges,q and --q,areseparatedby a distancel, both being locatedat a distancel/2 from the infinite conducting plane. Find: (a) the modulus of the vector of the electricforce acting on each charge;

(b) the magnitudeof the electricfield strengthvector at the midpoint between these charges. 3.57.A point chargeq is locatedbetween two mutually perpendicular conducting half-planes. Its distance from each half-plane is equal to l. Find the modulus of the vector of the force acting

3.64.A point chargeq is locatedat a distancer from the centreO of an unchargedconductingsphericallayer whose inside and outside and /\177 respectively.Find the potentialat radii are equal to the point O if r\177 A system consistsof two concentricconducting spheres, with the inside sphere of radius a carrying a positive charge ql. What chargeq2 has to be deposited on the outsidesphereof radius b to reducethe potentialof the inside sphere to zero?How doesthe potential depend in this caseon a distancer from the centreof the system? Draw the approximate plotof this dependence. Four largemetal plates are locatedat a smalldistanced from oneanotheras shown in Fig. The extremeplatesareinter/\1771

3.65.

/\1771.

q\177

3.66.

3.8.

3.58.charge. A point dipolewith

on the

an electricmoment p is locatedat a distancel from an infinite conductingplane. Find the modulus if the vector p is of the vector of the forceacting on the dipole to the plane. perpendicular A point chargeq is locatedat a distancel from an infinite conductingplane.Determinethe surface density of chargesinduced r from the baseof the perpenon the planeas a function of separation dicular drawn to the plane from the charge. 3.60.A thin infiniteIy long thread carrying a charge per unit length is orientedparallelto the infinite conductingplahe. The distancebetween the thread and the plane is equal to Find: (a) the modulus of the vectorof the force acting on a unit length of the thread; (b) the distributionof surface chargedensity (r (x) over the plane, to the conducting where x is the distance\177rom the planeperpendicular surface and passing through the thread. A very long straight thread is orientedat right angles to an infinite conductingplane;its end is separatedfrom the plane by a distance The thread carriesa uniform charge of linear density 4. Supposethe point O is the traceof the thread on the plane. Find the surface density of the induced charge on the plane (a) at the point O; (b) as a function of a distancer from the poirit O. 3.62.A thin wire ring of radius R carriesa chargeq. The ring is orientedparallelto an infinite conductingplaneand is separated Find: by a distancel from the surface charge density at the point of the plane symmetrical (a) with respectto the ring; (b) the strengthand the potentialof the electricfield at the centre of the ring. Find the potential of an unchargedconductingsphereoutside of which a point chargeq is locatedat a distancel from the

Fig.

3.59.

\177,

3.61.

3.8.

connectedby means of a conductor while a potential difference Find: Aq0 is appliedto internal plates. (a) the values of the electricfield strength between neighbouring plates; (b) the total chargeper unit areaof eachplate. Two infinite conductingplates 1 and 2 are separatedby 3.67. a distance 1.A point chargeq is locatedbetween the platesat a distance x from plate 1. Find the chargesinduced on each plate. 3.68.Find the electricforce experiencedby a charge reduced

to a unit area of an arbitrary conductorif the surface density of the .

chargeequals(r.

3.69.A metalball of radius/\177 = 1.5cm has a chargeq = 10 Find the modulus of the vectorof the resultantforceactingon a charge \177tC.

locatedon one half

ball.

sphere'scentre.

of the an unchargedconductingball of radius is placed in an externaluniform electricfield, a surface charge density a = = (r 0 cos0 is inducedon the surface (here a0 is a constant, 0 is a polarangle).Find the magnitudeof the resultant electricforce acting on an induced charge of the samesign. An electricfield of strength E = kV/em produces polariin zation water equivalentto the correctorientationof only one out of N molecules. Find N. The electricmomentof a water molecule -29 C.m. equals p = 3.72.A non-polarmoleculewith polarizability is locatedat a great distancel from a polar moleculewith electricmoment p. Find the magnitudeof the interactionforce between the molecules if the vectorp is orientedalonga straight line passingthroughboth

1t2

8--9451

it.

3.63.

q\177

3.70.When

/\177

ball's

1.0

3.71.

0.62,i0

\177

molecules.

t13)))

of a thin 3.73.A non-polarmoleculeis locatedat thexaxis from the

uniformly centre what distance ring's Tt. At radius of chargedring molecule the on F given is the.magnitudeof the force acting (a) equal to zero; (b) maximum? Draw the approximate plot Fx (x). of a ball made of 3.74.A point chargeq is locatedat the centre e. Find the polariwith dielectric permittivity uniform isotropic zation P as a function of the radius vector r relativeto the centre of the system, as well as the charge q' insidea spherewhoso radius is lessthan the radiusof the interface the Demonstratethat at a dielectric-conductor --\177 bound charge (e i)/e, surface density of the dielectric's is the surface density of the charge where e is the permittivity, on the conductor. shape, carrying a charge q, is 3.76.A conductorof arbitrary of permittivity e (Fig. 3.9). dielectric uniform surrounded with

ball.

3.75.

\177'

\177

(b) the circulation of the vector D around the closedpath F to thesurface of length l (seeFig. whose plane is perpendicular of the dielectricand parallelto the vector Eo.

3.ii)

\177

Fig.

3.t0.

Find the total bound chargesat the inner and outer surfaces of tho dielectric. layer 3.77.A uniform isotropicdielectricis shaped asof athespherical electricfield plots with radii a and b. Draw the approximate the distancer from the centreof strength E and the potential avscertain has dielectric the positive extraneouscharge if the layer distributeduniformly: over the volume of (a) over the internal surface of the layer; (b) \302\242p

the

layer.

lying on the boundary between point A (Fig.3.i0) field strength in Vacuum is equal to glass and vacuum the electric the vector E0 and the normal between = the E 0 i0.0V/m, angle n of the boundary linebeingequalto a0 = 30 Find the field strength E and n, E in glassnearthe point A, the angle between the vector the at bound the point A. of charges surface the density as well as dielectric 3.79.Near the plane surface of a uniform isotropic with permittivity e the electricfield strength in vacuum is equal to E0, the vectorEo forming an angle0 with the normalof the dielectric'ssurface (Fig.3.ii).Assumingthe field to be uniform both inside and outside the dielectric,find: the vector E through a sphere of radius R with (a) the flux of centrelocatedat the surface of the dielectric;

3.78.Near the

\302\260.

\177z

3.80.An infinite plane of uniform dielectricwith permittivity e is uniformly charged with extraneouscharge of spacedensity p. The thicknessof the plate is equal to 2d.Find: (a) the magnitudeof the electricfield strength and the potential as functions of distancel from the middlepoint of the plane (where the tiotential is assumed to be equal to zero); having chosenthe to the plate, draw the approximate x coordinateaxisperpendicular plotsof the projectionEx (x)of the vectorE and the potential(p (x); (b) the surface and spacedensitiesof the bound charge. 3.81.Extraneouschargesare uniformly

distributed with space ball of radius Tt made of uniform isotropic dielectricwith permittivity e. Find: (a) the magnitudeof the electricfield strength as a function of distancer from the centreof the ball;draw the approximate plots

density p

over a

E (r) and (r); (b) the spaceand surface densitiesof the bound charges. 3.82. A round dielectricdiscof radius R and thicknessd is statically polarizedso that it gains the uniform polarizationP, with the vector P lying in the plane of the disc.Find the strength E of the electricfield at the centreof the disc if d (< R. 3.83.Under certain conditions the polarizationof an infinite unchargeddielectricplatetakesthe form P ----P0 (1 x\177/d2), where to the plate,x is the distancefrom the P0 is a vector perpendicular middleof the plate, d is its half-thickness.Find the strength E of the electricfield insidethe plate and the potentialdifference between itssur\302\242p

faces.

3.84.Initially the spacebetween the platesof the capacitoris filled with air,

and the field strengthin the gap is equal Fig. 3.t2. to E 0. Then half the gap is filled with uniform isotropicdielectric with permittivity e as shown in Fig.3.t2. Find the moduli of the vectors E and D in both parts of the gap (1and 2) if the introductionof the dielectric 8-

i4)))

\1770

(a) does not changethe voltage acrossthe plates; (b) leavesthe chargesat the platesconstant. Solvethe foregoingproblemfor the casewhen half the gap 3.85.

is filled with the dielectricin the way shown in

Fig.

3.t3.

Fig.3.13.

--\177

/\177

charge of one sign. 3.89.A point chargeq is locatedin vacuum at a distancel from the plane surface, of a uniform isotropicdielectricfilling up all the The permittivity of the dielectric equalse.Find: half-space. the surface of the bound density chargesas a function of distance (a) r from the point charge q; analyse the obtainedresult at (b) the total bound charge on the surface of the dielectric. 3.90. Making use of the formulation and the solutionof the forefind the magnitudeof the force exertedby the charges problem, going bound on the surface of the dielectricon the point charge q. 3.91.A point chargeq is locatedon the plane dividing vacuum

l-,-0;

and infinite uniform isotropicdielectricwith permittivity e. Find the moduliof the vectorsD and E as well as the potential as functions of distancer from the charge q. 3.92.A small conductingball carrying a chargeq is locatedin a uniform isotropicdielectricwith permittivity e at a distance l from an infinite boundary plane between the dielectricand vacuum. Find the surface density of the bound chargeson the boundary plane as a function of distancer from the ball. Analyse the obtainedresult (\177

--\177

half-spacefilled with uniform isotropicdielectricwith permittivity e has the conductingboundary plane.Insidethe dielectric, at a distancel from this plane,there is a smallmetalball possessinga chargeq. Find the surface density of the bound chargesat the boundary plane as a function of distancer from the ball. 3.94.A plate of thicknessh made of uniform statically polarized dielectricis placedinsidea capacitorwhoseparallelplatesare interof the dielectricis equal connectedby a conductor. The polarization

Fig. 3A4.

3.86.Half the spacebetween two concentricelectrodesof a spherwith uniform isotropic ical capacitoris filled, as shown in Fig.3.14, dielectricwith permittivity e. The charge of the capacitoris q. Find the magnitudeof the electricfield strengthbetween the electrodesas a function of distancer from the curvature centreof the electrodes. 3.87.Two smallidenticalballscarrying the chargesof the same sign are suspendedfrom the samepoint by insulating threads of equal length. When the surroundingspacewas filled with kerosene the divergenceangle between the threads remainedconstant.What is the density of the materialof which the ballsare made? 3.88.A uniform electricfield of strength E 100V/m is generated inside a ball made of uniform isotropicdielectricwith permitcm. Find the The radius of the ball is ----3.0 tivity e----5.00. maximumsurface density of the bound chargesand the total bound

for l

3.93.A

Fig.

3.t5.

The separationbetween the capacitorplatesis d. (Fig.3.15). Find the strength and inductionvectors for the electricfield both

to P

inside and outside the plates. 3.95.A long round dielectriccylinder is polarizedso that the vector P ----(zr, where (z is a positiveconstantand r is the distance from the axis.Find the spacedensity p' of bound chargesas a function of distance r from the axis.

3.96.A

dielectricball is polarizeduniformly and statically. Its equalsP.Taking into account that a ball polarizedin polarization this way may be represented as a result of a smallshift of all positive chargesof the dielectricrelativeto all negativecharges, (a) find the electricfield strength E inside the ball; (b) demonstratethat the field outside the ball is that of a dipolo locatedat the centreof the ball,the potentialof that field being equal to (p ----p0r/dae0,where P0 is the electricmoment of the ball, and r is the distancefrom its centre. 3.9\"/.Utilizing the solution of the foregoing problem,find the elecEo in a sphericalcavity in an infinite statically polarized uniform dielectricif the dielectrics polarization is P, and far from tric field strength

the cavity the field

3.98.A

strength

dielectmcball is placedin a uniform electric field of strength E0. Under theseconditionsthe dielectricbecomos polarizeduniformly. Find the electricfield strengthE insidethe ball and the polarization P of the dielectricwhose permittivity equals e. Make use of the result obtainedin Problem3.96. 3.99.An infinitely long round dielectriccylinder is polarized umformly and statically, the polarizationP being perpendicular to the axisof the cylinder.Find the electricfield strengthE insidethe uniform

dielectric.

3.i00.A long round cylinder madeof uniform

dielectricis placed electric field of strength E0. The axis of the to vector E0. Under these conditions cylinder is perpendicular the dielectricbecomes polarizeduniformly. Making use of the result in a

uniform

tt7)))

obtained in the foregoing problem,find the electricfield strength

E in the cylinder and the polarization P of the dielectricwhose permittivity is equal to e.

3.3.ELECTRICCAPACITANCE. ENERGY \342\200\242

\342\200\242

energy

of a system of W=-\177-

\342\200\242

\177

Total electric energy of a system Total electric energy of

(3.3a) charges:

point

(3.3b)

with continuous

90 dr. charged bodies I

charge

w= T two

where Wx aad W2 are the self-energies of the energy. Energy of a charged capacitor:

distribution:

(3.3e) and

bodies,and

2: (3.3d) Wxz

the interaction

Volume density

qV= 2

= 2C q\177'

\177

(3.3e)

of electricfield energy: ED eeoc w -\177-----2

3.101.

3.102.

(3.3f)

3.103.

3.104.

(e\177

t18

B\177

3.106.

>B1.

e\177.

3.107.

E\177

this capacitorif e,RiE\177< 3.108. Two long straight wires with equal radiia are locatedparallelto each cross-sectional Rz other in air. The distance between their axes of the equals b. Find the mutual capacitance wires per unit length under the conditionb a. A longstraight wire is located 3.109. parallelto an infinite conductingplate.The wire cross-sec- Fig. 3.16. tional radius is equal to a, the distancebetween the axisof the wire and the plane equals b. Find the mutual capacitanceof this system per unit lengthof the wire under the condi>\177

\177

Find the capacitance of an isolatedball-shaped conductor of radius B1surrounded by an adjacentconcentric layer of dielectric with permittivity e and outsideradiusBv Two parallel-plate air capacitors,each of capacitanceC, were connectedin seriesto a battery with emf Then one of the capacitorswas filled up with uniform dielectricwith permittivity e. How many times did the electricfield strength in that capacitor decrease?What amount of charge flows through the battery? The spacebetween the platesof a parallel-plate capacitor is filled consecutively with two dielectriclayers and 2 having the thicknessesdI and d2 and the permittivities el and respectively. The area of each plate is equal to Find: of the capacitor; (a) the capacitance (b) the density of the bound chargeson the boundary plane if the voltage acrossthe capacitorequals V aud the electricfield is directed from layer to layer 2. The gap between the plates of a parallel-platecapacitor is filled with isotropic dielectricwhosepermittivity e varies linearly from ex to e2 to the plates. ex) in the directionperpendicular The area of each plate equals S, the separationbetween the plates is equal to d. Find: (a) the capacitanceof the capacitor;

e\177B\177E\177?

\342\200\242

B\177

q\177q)\177.

W = Wx+ W2+W12,

\342\200\242

3.105.

B\177

ELECTRICFIELD

Capacitanceof a parallel-platecapacitor: C = eeoS/d. Interaction

(b) the spacedensity of the bound chargesas a function of toward is q and the field E in it is directed the chargeof the capacitor the growing e values. of a sphericalcapacitorwhose elecFind the capacitance and which is filled with isotropic trodeshave radii and dielectricwhosepermittivity varies as e -- a/r,where a is a constant, and r is the distancefrom the centreof the capacitor. is filled with two cylindricallayers A cylindricalcapacitor The insideradii of the of dielectricwith permittivitiesel and maximumpermissible The to and are equal B1 layers values of electricfield strength are equal to Elm and Ezra for these At what relationshipbetween e, B, and Em will the dielectrics. increase result in the field strengthreachingthe breakdown voltage value for both dielectricssimultaneously? There is a double-layer cylindricalcapacitorwhose parametersareshowninFig.3.\1776. The breakdown field strength values reand for these dielectricsare equal to spectively.What is the breakdown voltage of \177\177-\177-.\177. if

e\177

tion a << b.

Find the capacitance 3.110. of a system of two identicalmetal if the distancebetween their centresis equal to b, with b a. The system is locatedin a uniform dielectricwith permittivity e. Determine of a system consistingof a metal 3.111. the capacitance ball of radius a and an infinite conductingplane separatedfrom the centreof the ball by the distancel if l >) a. Find the capacitanceof a system of identicalcapacitors 3.112. between points A and B shown in (a) Fig.3.17a;(b)Fig.3.i7b.

ballsof radius a >\177

Fig.

3.t7. li\177)))

3.1i3.Fouridenticalmetal plates are locatedin air at equal distancesd from one another. The area of each plate is equal to S. Find the capacitance of the system between points A and B if the platesare interconnectedas shown (a) in

Fig.3.22.

Fig.

3.i8. \177F

2.0

the maximum voltage V\177----4.0 kV. What will the system of these two capacitorswithstand if they \177F,

nectedin series? Find 3.115.

voltage

are con-

potentialdifference between points A =and B if the emf is equal to 110V of the system shown in Fig.3.19 and the capacitanceratio C2/C\177 =\1771 = 2.0. the

\177

3.22.

3.i9.

Find the capacitance of 3.116.

an infinite circuitformed by the link of the same consistingof two identicalcapacitors, repetition each with capacitanceC (Fig. 3.20).

Fig.

3.20.

-\177

C\177

--q\177B

between the left and right platesof each capacitor.

\177

t\177F

3.122.

3,24. Fig, 3,25. In the circuitshown in Fig.3.25the emf of each battery 3.123. are equal is equal to $ ----60 V, and the capacitorcapacitances and C2 3.0\177F. Find the chargeswhich will to Ca 2.0 flow after the shortingof the switch Sw through section\177 1,2 and 3 in the directionsindicatedby the arrows. between points Find the potential difference qoa 3.124. A and B of the circuit shown in Fig.3.26. \177

\177

\177F

--q\177t\177

I, T

\177-

t\177F

\177

110V is connected parallel a voltage V of a circuit consistingof two unchargedcapacitorsconnectedin seriesand possessingthe capacitances C2 ----2.0 and C3 3.09F. What charge will flow through the connectingwires? What chargeswill flow after the shorting of the switch Sw in the circuitillustratedin Fig.3.24through sections and 2 in the directionsindicatedby the arrows?

Fig. 3.2t.

The emf A circuithas a sectionAB shown in Fig.3.21. 3.117. are equal of the sourceequals $ ----10V, the capacitorcapacitances to C1 1.0 and 2.0vF, and the potentialdifference A -\177-5.0 V. Find the voltage acrosseach capacitor. In a circuitshown in Fig.3.22find the potentialdifference 3.118.

3.23.

q\177A

l Fig.

Fig.

-- B betweenpoints the potentialdifference Determine 3.120. Under what conditionis and B of the circuitshown in Fig.3.23. it equal to zero? 1.0vF charged up to A capacitorof capacitance 3.121. to the terminals in

3.1i4.A capacitorof capacitanceCt = 1.0 withstands the kV while a capacitorof capacitance maximumvoltage Va----6.0 =

120

Fig.3.t8a;(b) in Fig.3.18b.

Fig.

chargeof each capacitorin the circuitshown

Find the 3.119.

\1770

3.125.

Fig.

c\177

3.26.

141 Fig.

3.27.

Determine the potentialat point of the circuitshown in Fig.3.27,assuming the potentialat the point O to be equal to zero. \1772i)))

Using the symmetry of the formula obtained,write the expressions potentials,at points 2 and 3. Find the capacitanceof the circuit shown in Fig.3.28 3.i26. between points A and B. for the

What 3.132.

in the circuitshown amount of heat will be generated 2?. the switchSwis shifted from position I to position

after Fig.3.30

Fig. Fig.

3.28.

3.30.

Fig.

3.3i.

3.133. What amount of heat will be generatedin the circuitshown 3.31 after the switch Sw is shifted from position 1 to posiFig. tion 2? A system consistsof two thin concentric metal shellsof 3.134. radiiR1 and R2 with correspondingchargesql and q:.Find the selfof of each the interaction values and in

3.t27.Determinethe interactionenergy of the point chargeslocatedat the cornersof a squarewith the sidea in the circuitsshown in Fig. 3.29.

shell, energy and the total electricenergy of the system. 3.t35.A charge q is distributed uniformly over the volume of a ball of radius R. Assumingthe permittivity to be equal to unity,

energy the

shells

W\177

WI\177,

find: (a) the (b) the

electrostaticself-energy of the ball; ratio of the energy W1 storedin the ball to the energy pervadingthe surroundingspace. 3.t36.A point chargeq ----3.0 is locatedat the centreof a spherical layer of uniform isotropicdielectricwith permittivity e = 3.0. The insideradius of the layer is equal to a = 250ram, the outside ram. Find the electrostaticenergy inside the radius is b----500 dielectriclayer. 3.t37.A sphericalshell of radius with uniform charge q is Find the work performed by the electric expandedto a radius forces in this process. A spherical shellof radius R 1 with a uniform chargeq has 3.138. a point chargeq0 at its centre.Find the work performedby the electric forces during the shell expansionfrom W\177

\177xC

Fig.

3.29.

There is an infinite straight chain of alternating charges 3.128. The distancebetween the neighbouringchargesis equal --q. to a. Find the interaction energy of each charge with all the

and

others. Instruction.Make use of seriesin

the expansionof In (l

\177-

\177z)

in a

power

\177z.

3.t29.A point chargeq is Jocatedat a distancel from an infinite tonductingplane.Find the interactionenergy 6f that chargewith choseinducedon the plane. 3.t30.Calculatethe interactionenergy of two ballswhosecharges The distancebetween the ql and q2 are sphericallysymmetrical. centresof the balls is equal to l. Instruction.Start with finding the interactionenergy of a ball and

sphericallayer. A capacitorof capacitance 3.131. C1 = i.0 carrying initially a voltage V = 300V is connectedin parallelwith an uncharged capacitorof capacitanceC:= 2.0 Find the incrementof the electricenergy of this system by the moment equilibriumis reached. Explainthe result obtained. a thin

\177xF

\177tF.

t22

R\177

R\177.

radius

to radius

3.t39.A /\1771

spherical shell is

uniformly

chargedwith the surface deffsity Usingthe energy conservationlaw, find the magnitude of the electricforce acting on a unit area of

--\177-

\177.

the

shell.

A point 3.140.

centreO of

charge q is locatedat the sphericalunchargedconducting

Fig.

3.32.

The insideand outside layer provided with a smallorifice (Fig.3.32). What amount of the layer are equal to a and b respectively. work has to be performed to slowly transfer the chargeq from the point O through the orifice and into infinity?

radiiof

t23)))

S. 3.14t.Each plateof a parallel-plateair capacitorhas an areathe

amount of work has to be performed to slowly increase if distance between the plates from xl to which is equal to q, or (b) the of the capacitor, (a) the capacitance voltageacrossthe capacitor,which is equal to V, is kept constant in the process? 3.t42.Insidea parallel-platecapacitorthere is a plate parallel to the outerplates,whose thicknessis equal to = 0.60of the gap width. When the plate is absent the capacitorcapacitanceequals c 20nF. First, the capacitorwas connectedin parallelto a constant voltage sourceproducingV = 200V, then it was disconnected from it, after which the plate was slowly removed from the gap. Find the work performed during the removal,if the plate is (a) made of metal; (b) made of glass. A parallel-plate capacitorwas lowered into water in a hor-

3.4.ELECTRICCURRENT

What

\342\200\242

Ohm's law

for an inhomogeneous segment

x\177

\177l

-----

3.143.

with water filling up the gap between the plates izontal position, d -= t.0mm wide.Then a constantvoltage V -- 500V was applied to the capacitor. Find the water pressure increment in the

gap.

3.t44.A parallel-platecapacitoris locatedhorizontally so that

one of its platesis submergedinto liquid while the other is over its The permittivity of the liquid is equal to e, surface (Fig.3.33). its density is equal to p. To what height will the level of the liquid in the capacitor riseafter its platesget a chargeof surface density \177?

where

is the voltage Differential form

V1\177

\342\200\242

. . . , .

is the

(3.4a)

strength

Power P of current

=0,

and thermal

power Q:

I, Q\177BI\177. P=VI=(\177--\177+\177) and

density

a metal:

]

u is the Number

velocity of

avera\177

\177

(3.4e)

Qsp=pj*

(3.4f)

cnu,

callers.

of ions recombining per unit volume nr

r is

(3.4d)

specificthermal power Qsp:

P,p=j{E+E*),

Current

(3.4b)

electric circuit):

Specificpower Psp of current

where

+\17712

\177

VI___\177

of a circuit:

drop acrossthe segment. of Ohm's law: j = o (E E*), a of field producedby extraneousforces.

Kirchhoff's laws (for

where

--q)2 qh

\177

gas per unit time:

(3.4g)

\177,

coefficient.

the re\177mbination

3.i47.A long cylinder with uniformly chargedsurface and crosssectionalradius a 1.0cm moves with a constant velocity = 10m/s along its axis. An electricfield strength at the surface of the cylinder is equalto E = 0.9kV/cm.Find the resultingconvection current, that is, the current caused by mechanical transfer of a charge. 3.i48.An air cylindricalcapacitorwith adcvoltage V = 200V appliedacrossit is beingsubmergedvertically into a vessel filled with water at a velocity v 5.0mm/s.Theelectrodes of the capacitor are separatedby a distanced = 2.0ram, the mean curvature radi\177 of the electrodesis equal to r = 50 ram. Find the currentflowing in this case along lead wires, if d \177 r. 5 7 3.i49.At the temperature0 the electric resistanceof conductor 2 is times that of conductor1.Their temperaturecoefficientsof resistanceare equal to and respectively. Find the temperaturecoefficient of resistance of a circuit segmentconsisting of these two conductorswhen they are connected (a) in series; (b) in parallel. Fig. 3.35. 3.t50. Find the resistanceof a wire frame shaped as a cube(Fig.3.35)when measuredbetween points (a) 1-7;(b) 1-2;(c) 1-3. The resistanceof each edge of the frame is \177

Fig.

3.33

\177

A cylindrical layer of dielectricwith permittivity e is 3.145. insertedinto a cylindricalcapacitorto fill up all the spacebetween The meanradius of the electrodea the electrodes. equalsR, the gap between them is equal to d, with d << R. The constant voltage l/ of the capacitor.Find the magnitude is appliedacrossthe electrodes of the electricforce pulling the dielectricinto the capacitor.

3.t46.A capacitorconsistsof two stationary plates shaped as made of dielectric R and a movable semi-circle of radius

plate

permittivity e and capableof rotating about an axisO between The thicknessof the movable plate the stationary plates(Fig.3.34). is equal to d which is practicallythe separationbetween the stationary plates.A potentialdifference V is appliedto the capacitor. Find the magnitudeof the moment of forces relativeto the axis O acting on the movable plate in the position shown in the with

figure.

\302\260C

\177

a\177

125)))

3.i5i.

what value of the resistance in the circuitshown in Fig.3.36will the total resistancebetween points A and B be independentof the number of cells? At

/\177x

Fig.

3.36.

3.152. Fig.3.37shows an infinite

circuitformed by the repetition samelink, consistingof resistanceR1 4.0 and = 3.0 Find the resistanceof this circuitbetween points A and B.

of the

\177

\177',

Fig.

\177,

.... \177

R\177

\177.

a.a7.

wire grid with 3.i53.Thereis an infinite wire between

squarecells(Fig.3.38). neighbouringjoint connections

The resistanceof each R of the is equal to R0. Find the resistance whole grid between points A and B. _ Instruction.Make use of principlesof

-symmetry and superposition. A homogeneous 3.154. poorly conducting _ medium of resistivity p fills up the space between two thin coaxialideally conduct- _ ing cylinders.The radii of the cylinders are equal to a and b, with a \177 b, the length of each cylinder is l. Neglectingthe edge Fig. 3.38. effects, find the resistanceof the medium between the cylinders. A metalball of radius a is surrounded by a thin concentric 3.155. metalshellof radius b. The spacebetween these electrodesis filled medium of resistivity p. up with a poorly conductinghomogeneous interelectrode the o f Analyse the obtained resistance g ap. Find the

solution at b--\177 3.i56.The spacebetween twowithconductingconcentricspheresof poorly conducting homogeneous radiia and b (a \177 b) is filled up of sucha system equalsC.Find the resistivmedium.The capacitance ity of the medium if the potentialdifference between the spheres, from an external voltage, decreases when they are disconnected At. interval time the during \177l-fold

3.i57.Two metalballsof the sameradiusa arelocatedin a homo-

geneouspoorly conductingmedium

with

resistivity

p. Find

the

resistanceof the medium between the ballsprovided that the separation between them is much greaterthan the radius of the ball. 3.t58.A metal ball of radius a is locatedat a distancel from an infinite ideally conductingplane.The spacearound the ball is filled with a homogeneous poorly conductingmedium with resistivity p. In the caseof a << l find: (a) the current density at the conductingplane as a function of distancer from the ball if the potentialdifference between the ball and the plane is equal to V; (b) the electricresistanceof the medium between the ball and

plane. Two long parallelwires are locatedin a poorly conducting 3.i59. medium with resistivity p. The distancebetween the axes of the radius of eachwire equals a. wires is equal to l, the cross-section In the casea<<lfind: from the axes the

(a) the currentdensity at the point equally removed of the wires by a distancer if the potentialdifference between the wires is equal to V; (b) the electricresistanceof the medinm per unit length of the

wires.

The gap between the plates of a parallel-platecapacitor 3.160. 100G\177.m. The capacitance is filled with glass of resisti\177ily p 4.0nF. Find the leakagecurrent of the of the capacitorequalsC capacitorwhen a voltage V 2.0kV is appliedto it. 3.i61.Two conductorsof arbitrary shape are embeddedinto an infinite homogeneouspoorly conductingmedium with resistivity p and permittivity e.Find the value of a product/\177G for this system, where is the resistanceof the medium between the condnctors, of the of the wires in the presence and C is the mutual capacitance \177-

/\177

medium.

A conductorwith 3.162.

permittivity

resistivity p bounds on a dielectricwith surface the at the conductor's

e. At a certainpoint A

electricdisplacement equals D, the vector D being direcledaway from the conductorand forming an angle a with the normal of the surface.Find the surface density of chargeson the conductorat the point A and the current density in the conductorin the vicinity of the samepoint. The gap between the plates of a parallel-platecapacitor is filled up with an inhomogeneous poorly conductingmedium whose to the conductivity varies linearly in the directionperpendicular = = has an 2.0 Each to t.0 pS/m plate pS/m. plates from cm2, and the separationbetween the plates is d = area S-=--230 = 2.0mm. Find the current flowing through the capacitordue to a voltage V = 300V. Demonstratethat the law of refraction of direct cnrrent linesat the boundary between two conductingmediahas the form

3.t63.

\177z

\177re

3.|64.

i27)))

tan

\177z

\177z\177/tan

the media,\177 and

where cr1 and are the conductivitiesof are the anglesbetween the currentlinesand the

cr\177/(\177l,

\1771

cr\177

normal of the boundary surface. Two cylindricalconductorswith equal cross-sections and different resistivities and areput end to end. Find the charge at the boundary of the conductorsif a currentI flows from conductor I to conductor2. The gap between the platesof a parallel-plate capacitoris filled up with two dielectriclayers I and 2 with thicknessesd1 and and resistivities Adc voltage permittivities and Pl and V is appliedto the capacitor, with electricfield directedfrom layer to layer 2. Find or, the surface density of extraneouschargesat the boundary between the dielectriclayers, and the condition under which = O. An inhomogeneous poorly conducting medium fills up the spacebetween plates and 2 of a parallel-plate capacitor.Its permittivity and resistivity vary from values el,Pl at plate to values e 2, at plate 2. A de voltage is appliedto the capacitor through which a steady currentI flows from plate to plate2.Find the total extraneouscharge in the given medium. The spacebetween the platesof a parallel-plate capacitor is filled up with inhomogeneous poorly conductingmedium whose to the plates. resistivity varies linearly in the directionperpendicular The ratio of the maximumvalue of resistivity to the minimum one is equal to rI The gap width equalsd. Find the volume density of the chargein the gap if a voltage V is appliedto the capacitor. e is assumedto be 1everywhere. A long round conductorot cross-sectional area S is made of materialwhose resistivity depends only on a distancer from the axis of the conductoras p = 2, where is a constant.Find: (a) the resistanceper unit length of such a conductor; (b) the electricfield strengthin the conductordue to which a current I flows through

3.165.

p\177

3.t66.

d\177,

e\177,

e\177

3.167.

p\177.

\177r

of the latter decrease parallelwith the voltmeter, the readings 2.0times,whereas the readings of the ammeterincreasethe samenumber of times.Find the voltmeter readingsafter the connectionof the resistance. 3.t74.Find a potentialdifference -- between=points 1and 2 20 of the circuitshown in Fig.3.39if R\177=\1770 $1---= 2.0V. The internalresist----5.0V, and ances of the current sourcesare negligible. 3.\17775. Two sourcesof current of equal emf are connectedin series and have different internal resistances//1 and \177//\177). Find the externalresistance// at which the potentialdifference acrossthe terminalsof one Fig. 3.39. of the so\177trces (which one in particular?)becomesequal to zero. 3.t76.N sourcesof current with different emf's are connected to as shown in Fig.3.40.The emf's of the sourcesareproportional

\177

\1771

q\177t

q\177

f\177,

R\177

f\177,

\177,\177

//\177.

(//\177

p\177

3.t68.

3.t69.

\177z/r

\177

it.

A capacitorwith capacitance 3.170. C 400 pF R of \177

via a

resistance

value

\177

650 f2

to a source

is connected constant voltage V0.

How soon will the voltage developed acrossthe capacitorreacha \177

0.90V0?

3.t71.A capacitorfilled with

loseshalf

the Assuming the

dielectricof permittivity chargeacquiredduring a time interval x ----3.0rain. chargeto leak only through the dielectricfiller,cale ----

2.\177

culateits resistivity. 3.172. A circuitconsists of a sourceof a constantemf $ and a resist anee R and a capacitorwith capacitance C connectedin series. The internal resistanceof the sourceis negligible.At a moment t 0 the capacitance of the capacitoris abruptly decreased xl-fold. Find the current flowing through the circuitas a function of time t. 3.173. An ammeter and a voltmeterareconnectedin seriesto a batis connected tery with an emf $ = 6.0V. When a certainresistance -----

128

3.40.

Fig.

their internal resistances, i.e.$

Fig.

where

3,41. is an assignedcon-

stant.The leadwire resistanceis negligible.Find: the current in the \177z//,

\177z

(a) circuit; (b) the potential difference between points A and B dividing the circuit in n and n links. 3.t77.In the circuitshown in Fig.3.41the sourceshave emf's

N-

$1=----1.0V and //1 l0 and

= 2.5V

and the resistances have the values 20 The internal resistances of the sources are negligible.Find a potentialdifference q)A-q) B between the pIatesA and B of the capacitorC. In the circuitshown in Fig. 3.42the emf of the sourceis equal to $ = 5.0V and the resistancesare equal to R 1 ----4.0 and /\177 6.0f2. The internal resistanceof the sourceequals -------0.\1770 Find the currents flowing through lhe resistances f\177

$\177

R\177

-----

f\177.

3.178.

f\177

\177

/\177

f\177.

and Rz.

R\177

3.\17779. Fig.3.43illustrates a potentiometric circuitby means of which we can vary a voltage V appliedto a certaindevicepossessing a resistanceR. The potentiometer has a length l and a resistance

9--9451

t29)))

R0, and voltage V0 is appliedto itsterminals.Find the voltage V fed to the deviceas a function of distancex. Analyse separatelythe case B >> B 0.

3.t85.Find the current flowing through the resistanceR=1 of10the are equal to [2, circuitshown in Fig.3.48if the resistances ----20 and R 3 = 30 g2, and the potentialsof points 1,2, and 3 6 V, and (Pa -- 5 V are equal to qh = t0 V, R\177

\177,

R\177

\177

q)\177

Fig.

3.42.

Fig.

3.43.

3.i80.Find the emf and the internal resistanceof a sourcewhich is equivalent to two batteriesconnectedin parallelwhose emf's and $2and internal resistances are equal to to and Find the magnitudeand directionof the current flowing 3.181. \1771

through the

resistance

in the

/\177

circuitshown

Fig.3.44if

the

Fig. Fig.

Fig.3.45.

3.44.

= 3.7 V and emf's of the sourcesare equal to $1=1.5V and = i0 [2, = 20[2, = 5.0g2. the resistancesare equal to The internal resistancesof the sourcesare negligible. In the circuitshown in Fig.3.45the sourceshave emf's 3.182. = 2.0V, $3= 2.5V, and the resistancesare St = 1.5V, to equal R1 10[2,/\177 20 B 3 = 30\177. The internal resistances of the sources are negligible.Find: $\177

/\177

/\1771

$\177

-----

\177

Fig.

3.48.

V = 25 V is maintained between Find the magnitude and and B of the circuit(Fig.3.49).

A 3.186.

points

3.47. constant voltage Fig.

/\177a.

R\177

3.49.

Fig.

3.50.

directionof the currentflowing through the segmentCD if the resist= i.0[2, -- 2.0g2, Ra 3.0[2, and = ancesare equal to \177

=4.0 [2.

R\177

3.t87.Find the resistancebetween points A Fig.3.50. 3.|88.Find how the

shown in

after timet (Fig.3.51)

R\177

and B of the circuit

voltage acrossthe capacitorC varies with

the shortingof the switch Sw at the moment

\177,

(a) the current flowing through the

resistanceR1; (b) a potential difference

q)A-

between the points A and B. 3.183. Find the currentflowing through q\177

the resistanceR in the circuitshown in Fig.3.46.Theinternal resistancesof the

Fig.

3.46.

batteriesare negligible. Find a potentialdifference between the plates 3.184. of a capacitorC in the circuitshown in Fig.3.47if the sourceshave ----i.0V and the resistances emf's $1= 4.0V and are equal ----i0\177, R\177---- 20 [2, and R 3-30\177. to The internal resistances of the sourcesare negligible. \302\242PA

$\177

R\177

t30

--\302\242P\177

Fig.

3.5i.

Fig.

3.52.

3.t89.What

amount of heat will be generatedin a coilof resistdue to a chargeq passingthrough it if the currentin the coil down to zero uniformly during a time interval At; (a) decreases down to zero halving its value every At seconds? decreases (b) A dc source with internal resistanceR 0 is loadedwith 3.|90. R interconnectedas shown in Fig. 3.52. three identicalresistances ance/\177

131)))

what value of R

be the highest?

will the thermal power generatedin this circuit

3.i91.Makesure that

Rl and

R\177

the currentdistributionover two resistances to the minimum thermal connectedin parallelcorresponds

power generatedin this circuit. A storage battery with emf 3.192.

$ 2.6V loadedwith an externalresistanceproducesa current I 1.0A. In this casethe \177

\177

potentialdifference between the terminals of the storage battery equals V 2.0V. Find the thermalpower generatedin the battery and the power developedin it by electricforces. A voltage V is appliedto adcelectricmotor.The armature is equal to R. At what value of current flowing resistance winding through the winding will the useful powerof the motor be the highest? is the motor efficiency in this case? What is it equal to? What Itow much (in per cent) has a filament diameterdecreased due to evaporation if the maintenanceof the previous temperature required an increaseof the voltage by = 1.0%?The amount of heat transferred from the filament into surroundingspaceis assumed to the filament surface area. to be proportional resistanceR A conductorhas a temperature-independent 0 it is connected and a total heat capacity C. At the moment t to adcvoltage V. Find the timedependenceof a conductor's temperature T assuming the thermal power dissipatedinto surrounding To), where k is a constant, To is the spaceto vary as q environmental temperature(equal to the conductor'stemperature at the initial moment). A circuit shown in Fig.3.53has resistances R1----20f\177 and 30 At what value of the resistance will the thermal

3.i93.

0.3

\177,

the number

is loadedwith an externalresistanceR = 10 Find n of parallelgroups consistingof an equal number of f\177.

which the externalresistance generates the highest thermal power. C = 5.00 is connectedto A capacitorof capacitance 200V (Fig.3.55).Then the switch a sourceof constant emf Sw was thrown over from contact to contact 2. Find the amount 330\177. of heat generatedin a resistance = 500 if Betweenthe plates of a parallel-platecapacitorthere is 3.200. a metallicplate whose thicknesstakesup 11 0.60of the capacitor

cellsconnectedin series,at

3.i99.

\177tF

\177--

\177,

\177

//\177

-----

R\177

--\177

3.194.

\177l

3.195.

Fig.

-\177

:k(T--

3.196. -----

R\177

f\177.

R:\177

3.55.

Fig.

3.56.

When that plate is absent the capacitorhas a capacity C = gap. ----20 nF. The capacitoris connectedto a dc voltage source V = ----t00V. The metallicplateis slowly extractedfrom the gap.Find: (a) the energy increment of the capacitor; work performedin the processof plate extrac(b) the mechanical tion. A glassplatetotally fills up the gap between the electrodes 3.201. of a parallel-platecapacitorwhose capacitancein the absenceof that glassplate is equal to C ----20 nF. The capacitoris connected to a de voltagesourceV 100V. The plateis slowly, and without friction,extractedfrom the gap.Find the capacitorenergy increment and the mechanical work performed in the processof plate extrac\177

tion.

Fig.

3.53.

Fig.

3.54.

power generatedin it be practicallyindependentof Asmallvariations The voltage between the points and B is supof that resistance? in this case. constant to be posed In a circuit shown in Fig.3.54 resistances//, and 3.197. The internal resistances are known, as well as emf's and R\177

\177,

$\177.

of the sourcesare negligible.At what value of the resistance// will the thermal power generatedin it be the highest?What is it equal to? combinationbattery consistingof a large A series-parallel 3.198, = identical of number N 300 cells,eachwith an internal resistance

t32

to adcvoltagesourceV A cylindricalcapacitorconnected 3.202. The separation touchesthe surface of water with its end (Fig.3.56). d betweenthe capacitorelectrodes is substantially less than their in the gap mean radius.Find a height h to which the water will rise.The capillaryeffects are to be neglected. 3.203. The radii of sphericalcapacitorelectrodesare equal to a and b, with a \177 b. The interelectrode spaceis filled with homogeneous substanceof permittivity e and resistivity p. Initially the capacitoris not charged.At the momentt = 0 the internalelectrode gets a charge q0. Find: (a) the time variation of the charge on the internal electrode; (b) the amount of heat generatedduring the spreading of the le\177rel

charge.

t33)))

C ----2.00 of capacitance 3.204.The electrodesof a capacitor are interThen the electrodes \177F

1.00inC.=

carry oppositechargesq0 5.0M\177. Find: connectedthrouffh a resistance resistance that during a timeinterthe through flowing charge (a) 2.00s; val the (b) the amount of heat generatedin the resistanceduring same interval. In a circuitshown in Fig.3.57the capacitanceof each 3.205. to R. One of the capacitors i s capacitor equalto C and the resistance, then at the and was connectedto a voltage Vo switch the of ---means shorted 0 was moment t by Sw. Find: (a) a current I in the circuit as a function of time t; (b) the amount of generated heat provided a dependenceI (t) is known. 25 cm wound of a thin Fig. 3.57. A coilof radiusr 3.206. 500m rotateswith an copperwire of length l rad/s about its axis.The coilis connectangular velocity o) 300 sliding contacts.The by ed to a ballistic galvanometer means of 2i f2. Find the specific totalresistanceof the circuitis equal to R sudden if a in stoppage of the charge of current carriers copper the galvanoflow nC l0 through a coil makes charge q -----

-----

\177

-----

-\177

--\177

meter.

\177

of electronsin a straight wire 3.207.Find the total momentum current A. I a m 1000 l carrying of length 1.0A/mm

\177--70

A copperwire carriesa current of density ] 3.208. to eachcopperatom, -\177

\177.

Assuming that one free electroncorresponds an electronduring evaluate the distancewhich will be coveredby i0 mm along the wire. its displacementl ----1000 m and crossA straight copperwire of length l 3.209. sectionalarea S 1.0mm carriesa current I 4.5A. Assuming -\177

\177

\177--

to each copperatom, firid: that one free electroncorresponds to electron an it takes displacefrom one end of the (a) the time wire to the other; in the (b) the sum of electricforces acting on all free electrons given wire. A homogeneous proton beam acceleratedby a potential of radius r difference V--\177 600kV has a round cross-section of the beam surface the on field the electric strength Find 5.0ram. axis of the and surface and the potentialdifference between the mA. 50 I to is the beam if the beam current equal Two largeparallelplates are locatedin vacuum. One of whose initialvelocity them servesas a cathode,a sourceof electrons the t oward directed f low oppositeplateprodAn electron is neg]igible. the in the gap between the potential a spacecharge causing is a ax where a positive constant, and x is plates to vary as Find: cathode. (he distance from the

3.210.

--\177

\177

3.21I.

\177aces

\177

\177/\177,

\302\242p

a function

the current density.

(b) The air between two parallelplatesseparatedby a distance 3.212.

/\177

\342\200\242

space charge as

(a) the volume density of the

d ----20 mm is ionizedby X-ray radiation.Each plate has an area S 500cm Find the concentrationof positiveions if at a voltage 3.0\177A flows betweenthe plates,which V ----i00V a current I are + is well below the saturationcurrent.The air ion mobilities 1.9i and ----i.37 c m\177/(V-s). uT-\177 cm\177/(V.s) A gas is ionizedin the immediate 3.213. vicinity of the surface 1(Fig. 3.58)separatedfrom electrode2 by a disof plane electrode tance l. An alternatingvoltage varying with timet as V Vo sin \177

\177.

\177

u\177

\177-

is appliedto

the electrodes.On decreasing frequency (o it was observed that the galvano- \17711 \177, g meter G indicatesa current only at (o \177 where (o0 is a certain cut-off frequency. Find V--\177I the mobility of ions reachingelectrode2 under theseconditions. 3.2t4.The air between two closelylocated plates is uniformly ionizedby ultravioletradiation.The air volume between the platesis equal Fig. 3.58. 500cma, the observedsaturationcurrent to V Find: is equal to I,\177 ----0.48 ion of the number pairsproducedin a unit volume per unit (a) time; (b) the equilibriumconcentrationof ion pairsif the recombination cma/s. coefficient for air ions is equal to r----1.67.10 3.215. Having been operatedlong enough,the ionizerproducing 3.5.i0cm-a.s of ion pairsper unit volume of air per unit time was switched off. Assuming that the only processtending to of ions in air is their recombinationwith coefficreducethe number -a cma/s, find how soon after the ionizer'sswitching ient r 1.67.i0 2.0times. off the ion concentrationdecreases A parallel-plate air capacitorwhose plates are separated 3.216. by a distanced ----5.0mm is first-chargedto a potentialdifference 90V and then disconnectedfrom a dc voltage source.Find V the timeinterval during which the voltageacrossthe capacitordecreasesby 1.0%,taking into accountthat the averagenumber of ion pairsformed in air under standard conditionsper unit volume 5.0cm -a.s and that the given voltper unit timeis equal to current. the to saturation age corresponds The gap between two plane platesof a capacitorequal to 3.217. d is filled with a gas.One of the platesemitsv0 electronsper second this way which, moving in an electricfield, ionize gas molecules; eachelectronproduces new electrons(and ions)alonga unit length currentat the oppositeplate,neglectof itspath. Find the electronic ing the ionization of gas moleculesby formed ions. the

\177o0,

\177.__\177Vo

\177

\177A.

-\177

\177

n\177

\177-

-----

\177l

\177

\177l

\177

-\177

n\177

\177z

t35)))

134

The 3.218.

gas between the capacitorplatesseparatedby a distanced is uniformly ionizedby ultra:\177iolet radiation so that ni electrons per unit volume per secondareformed.Theseelectronsmoving eachelectron ionizegas molecules, in the electricfield of the capacitor of its path.. unit electrons length new (and ions) per producing find the electroniccurrent denionization the ions, by Neglecting sity at the plate possessinga higher potential. \177z

3.5.

MAGNETICS

field

Magnetic

locity v:

Circulation

\342\200\242

\1770

\177

[jr]

Lorentz force:

\342\200\242

Ampere

force:

Force and

\342\200\242

(3.5c) (3.5d) (3.5e)

a magnetic dipole Pm

N=[pmB],

pm \177 ,

ISn:

the

(3.5f) dipole direction.

= I',

(3.5g)

9o currents. of sum macroscopic I is the algebraic Relations at the boundary between two magnetics:

-\177-

B\177n,

case of magnetics in

\177o

100 radius (a) at the centreof the /\177

Hx.\302\242

which

Fig.

3.60.

Find the magneticinduction of the field at the point O 3.223. whose shape is illustrated of a loopwith current a and b, as well as the angle are the radii in 3.60a, Fig. (a)

known; (b) in

Fig.3.60b,the radius a

and the side b are known.

A currentI flows alonga lengthy thin-walledtube of radius 3.224. R with longitudinalslitof width h. Find the inductionof the magnetic field inside the tube under the conditionh << R. A currentI flows in a longstraight wire with cross-section 3.225.

\177=

-t-l-

(3.5j)

round thin-wire loop

magnetic induction

Find 3.61).

(3.5h)

(a)

(3.5i)

H\177z.

A circulatesin a

loop;

3.59.

having the form of a thin half-ringof radius R (Fig. the induction of the magnetic field at the point O.

I'Vector is the total molecular current. tt and its circulation:

1.00 A current I 3.219. = mm. Find the

t36

-----

Fig.

q\177

to OB/On is the derivative of a vector B with respect Circulation of magnetization

For the

it:

+ q [vB]. dF = I [dl, B].

[jB]dV, moment of forces acting = 0B

Bxn

\302\260

(3.5b)

H=--B-B --J, \177Hdr=l,

where

I=5.0

\177)B

J where

I [dl, r]

and

\177--

where

\177to

4\"--\177

F ----qE dF

30\302\260;

(3.5a)

Gauss'stheorem for = [aoI, dS= 0.

\177B \342\200\242

dB=

3.59.

2(\177

\177 dV,

a vector B

non-relativistic

q[vr]

\177t0

4\177

Biot-Savart law: dB =

\342\200\242

q moving with

charge

point

B= \342\200\242

3.221.

q\177

FIELD.

MAGNETIC

CONSTANT

\342\200\242

at the point lying on the axis of the loop at a distancex from its centre. A current I flows along a thin wire shaped as a regular 3.220. polygon with n sideswhich can be inscribedinto a circleof radius R. Find the magneticinductionat the centreof the polygon. Analyse at n-\177 the obtained expression Find the magneticinductionat the centreof a rectangular wire frame whose diagonal is equal to d \177-t6 cm and the angle the current flowing in between the diagonalsis equal to = = 5.0A. the frame equals I A flows alonga thin wire shaped as shown A current 3.222. The radiusof a curved part of the wire is equal to/\177 = in Fig. = i20mm, the angle = 90 Find the magneticinduction of the field at the point O.

= (b) 100mm

Fig.

3.6i.

(b) Fig.

(c)

3.62.

3.226.Find

the magnetic induction of the field at the point O if a current-carryingwire has the shape shown in Fig.3.62a, b, c. The radius of the curved part of the wire is TI, the linearparts are assumed to be very long. t37)))

very long wire carrying a current I = 5.0A is bent at right angles.Find the magneticinductionat a point lying on a perpendicularto=the wire, drawn through the point of bending, at 35 cm from a distance l Find the magneticinductionat the point O if the wire car3.228. rying a current I ----8.0A has the shape shown in Fig.3.63a, b, c.

3.227.A

it.

3.234,Insidea longstraight uniform wire of round cross-section there is a long round cylindricalcavity whose axis is parallelto the axis of the wire and displacedfrom the latter by a distanceI. A directcurrentof density j flows alongthe wire.Find the magnetic inductioninsidethe cavity. Consider,in particular,the casel 0. Find the current density as a function of distancer from 3.235. -\177

axisof a radially symmetrical parallelstream of electronsif the magnetic induction inside the stream varies as B br% where b and are positive constants. has length l and cross-section A single-layercoil(solenoid) 3.236. radius R, A number of turns per unit length is equal to n. Find the magneticinductionat the centreof the coilwhen a current I flows the

\177

\177z

through

3.63.

Theradius of the curved part of the wire is R

parts of the wire are very long.

i00mm,

----

the

linear

Find the magnitudeand directionof the magneticinduction 3.229.

vector B (a) of an infinite plane carrying a current of linear density the vectori is the sameat all points of the plane; (b) of two parallelinfinite planes carrying currentsof lineardenare constant at all points of the vectorsi and sitiesi and the correspondingplanes. flows inside an infinite A uniform current of density 3.230. plateof thickness2d parallelto its surface.Find the magneticinduction induced by this current as a function of the distance x from the medianplane of the plate.The magneticpermeabilityis assumed to be equal to unity both inside and outside the plate. A direct current I flows along a lengthy straight wire. From the point O < g (Fig. 3.64)the current spreadsradially all over an infinite conductingplane perpendicuFig. 3.64. lar to the wire. Find the magnetic induction at all points of space. A current I flows along a round loop.Find the integral 3.232. B dr alongthe axisof the loopwithin the rangefrom --\177 to -\177. f Explainthe result obtained. A directcurrent of density j flows along a round uniform radiusR. Find the magneticinduction with cross-section wire straight vector of this currentat the point whose positionrelative to the axis of the wire is defined by a radiusvectorr. The magneticpermeability is assumed to be equal to unity throughout all the space.

--i;

--i

]

3.231.

\177

3.233.

it.

radius 3.237.A very long straight solenoidhas a cross-section R and n turns per unit length. A directcurrent I flows through the solenoid. Supposethat x is the distancefrom the end of the solenoid, measuredalong its axis.Find: (a) the magneticinductionB on the axisas a function of x; dcaw an approximate plot of B vs the ratio x/R',

(b) the distancex0 to the point on the axisat which the value of B differs by 1%from that in the middlesectionof the sole\177

noid.

\177l

conductingstrip of width h ----2.0cm is tightly radiusR --= wound in the shape of a very longcoilwith cross-section A directcurrent 2.5cm to make a single-layerstraight solenoid. A flows through the strip.Find the magnetic induction I----5.0 insideand outsidethe solenoidas a function of the distancer from A thin 3.238.

\177--

its axis. 3.239. N

= 2.5-t03 wire turns are uniformly

wound on a wooden A current I flows through very smallcross-section. the wire. Find the ratio of the magneticinductioninsidethe core to that at the centre of the toroid. i0 A flows in a long straight round A directcurrent I 3.240. conductor.Find the magneticflux through a half of wire'scrosssectionper one metreof its length. The A very long straight solenoidcarriesa current area of the solenoidis equal to S, the number of cross-sectional turns per unit length is equal to n. Find the flux o[ the vector B through the end plane of the solenoid. 3.242. Fig.3.65shows a toroidalsol-

toroidalcoreof

\177l

\177--

3.241.

is rectangular. enoid whose cross-section Find the magnetic flux through this if the current through the cross-section winding equals I ----1.7 A, the total number o[ turns is N i000,the ratio of the outside diameterto the inside one is 5.0cm. height is equal to h

Fig.

3.65.

\177

\177l

\177--1.6, and

the

\177

139)))

t38

round loopwith curmagneticmomentof a thin = mm and the mag100 R to is the of rent if the radius loop equal netic induction at its centreis equal to B \177--6.0 \177T. with a cur3.244.Calculatethe magnetic moment of a thin wireThe diameter 3.66). rent I ----0.8A, wound tightly on half a tore(Fig. of the tore is equal to d = 5.0cm, the number of the cross-section of turns is N = 500. Find the 3.243.

3.251.

Find the magnitudeand directionof a force vector acting on a unit length of a thin wire, carrying a current I ----8.0A, at a point O, if the wire is bent as shown in (a) Fig.3.68a,with curvature radius /\177-----l0 cm; the distancebetween the long parallelsegments (b) Fig.3.68b, of the wire being equal to 1 = 20 cm. 3.252.A coilcarrying a currentI = i0mA is placedin a uniform magneticfield so that its axis coincideswith the field direction. The single-layerwinding of the coilis made of copperwire with

Fig.

3.66.

Fig.

3.67.

insulated wire forms a plane spiral of N = i00 turns carrying a current I = 8 mA. The radii of inside and tight outsideturns (Fig.3.67)are equal to a = 50mm and b 100ram. A thin 3.245.

\177--

Find:

centreof the spiral; spiral with a given current. t hin disc of radius chargeduniformly A non-conducting 3.246. rotates about its axis with over one side with surface density an angular velocity (o. Find: (a) the magneticinduction at the centreof the disc; (b) the magnetic moment of the disc. charged 3.247.A non-conductingsphereof radius = 50 mm with an uniformly with surface density \177-----i0.0\177.C/m rotates its = the axis about rad/s (o 70 through passing velocity angular centre.Find the magnetic induction at the centreof the sphere. A charge q is uniformly distributedover the volume of 3.248. a uniform ball of massm and radius R which rotateswith an angular velocity o) about the axispassingthroughitscentre.Find the respective magneticmoment and its ratio to the mechanicalmoment. 3.249.A longdielectriccylinderof radius R isstaticallypolarized so that at all itspoints the polarizationis equal to P = at, where is a positive constant, and r is the distancefrom the axis. The cylinder is set into rotationabout itsaxiswith an angularvelocity ca. Find the magneticinduction B at the centreof the cylinder. Two protons move parallelto each other with an equal 3.250. km/s. Find the ratio of forces of magneticand v-----300 velocity electricalinteractionof the protons. (a) the magneticinduction at the (b) the magnetic moment of the

/\177

(\177

/\177

(b)

(a) Fig.

3.68,

Fig.

3.69.

--30mm. diameterd----0.i0 mm, radius of turns is equal to what value of the inductionof the externalmagneticfield can the coilwinding be ruptured? area S = 2.5mm a A copperwire with cross-sectional 3.253. bent to make three sidesof a square can turn about a horizontal The wire is locatedin uniform verticalmagnetic axisOO' (Fig.3.69). field.Find the magneticinductionif on passing a currentI = 16A 20 through the wire the latter deflects by an angle 0 A small coilC with N = 200 turns is mounted on one 3.254. end of a balancebeamand introducedbetween the po]esof an electroareaof the coil magnet as shown in Fig.3.70.The cross-sectional /\177

\302\260.

\177

g'.////////\177

Fig.

3.70.

\302\242\302\242

i40

= 1.0em the length of the arm OA of the balancebeam is 30cm. When there is no. current=in the coilthe balanceis in equilibrium.On passinga current I 22 mA through the coilthe is S

l ----

\177,

equilibriumis restoredby putting the additional,counterweight

of)))

mass hm = 60mg on the balancepan. Find the magneticinduction at the spot where the coilis.located. 0.90A is located A square frame carrying a current I 3.255. in the sameplane as a long straight wire carrying a current I0 = 5.0A. The frame sidehas a length a ----8.0cm. The axis of the -----

-----

frame passingthrough the midpointsof oppositesidesis parallelto the wire and is separatedfrom it by the distancewhich is timesgreaterthan the sideof the frame.Find: (a) Ampere force acting on the frame; (b) the mechanicalwork to be performed in orderto turn the frame through i80 about its axis, with the currents maintained \302\260

constant. 3.256. Two long parallelwires of negligibleresistanceare connectedat one end to a resistance/\177 and at the otherend to adcvoltagesource.The distancebetweenthe axesof the wiresis = 20times radius of each wire. At what value greaterthan the cross-sectional of resistance doesthe resultant force of interactionbetween the \177l

/\177

wires turn

into zero?

3.257.A directcurrent I flows in a longstraightconductorwhose cross-section has the form of a thin half-ringof radius The same currentflows in the opposite directionalonga thin conductorlocated /\177.

3.6i).Find the

on the \"axis\" of the first conductor(point O in Fig. magnetic interactionforce between the given conductors reducedto a unit of their length. Two long thin parallelconductorsof the 3.258.

shape shown in

Fig.3.7icarry direct currents I1

The separationbetween the conductorsis a, the width of the right-handconductoris equal to b. 1 With both conductorslying in one plane, find the magnetic interaction force between them reduced to a unit of their length. 3.259.A system consistsof two parallelplanes Fig. 3.7i. carrying currents producing a uniform magnetic field of induction B between the planes.Outside this spacethere is no magneticfield.Find the magneticforce acting per unit area of each plane. A conductingGurrent-carryingplane is plac\177d in an external 3.26{). uniform magneticfield.As a result,the magneticinductionbecomes

and

I\177.

equal to BI on one sideof the plane and to Be, on the other.Find the magneticforce acting per unit area of the plane in the cases illustratedin Fig. 3.72.Determinethe directionof the current in the plane in each case. In an electromagnetic 3.261. pump designedfor transferring molten metalsa pipesectionwith metalis locatedin a uniform magnetic field of inductionB (Fig. 3.73).A current I is made to flow acrossthis pipe section in the directionperpendicularboth to the vector B and to the axis of the pipe.Find the gauge pressureproducedby the pump if B = 0.t0T, I----t00 A, and a = 2.0cm. 3.262.A current I flows in a long thin\177alled cylinder of radius R. What pressure do the walls of the cylinder experience? Fig. 3.73. 3.263.What pressure does the lateral surface of a long straight solenoidwith n when a currentI flows through it? turns per unit length experience 3.264.A currentI flows in a longsingle-layersolenoidwith crosssectionalradius/\177. The number of turns per unit length of the solenoid equals n. Find the limiting current at which the winding may rupture if the tensilestrength of the wire is equal to A parallel-plate 3.265. capacitorwith areaof each plateequal to S and the separationbetween them to d is put into a streamof conductingliquid with resistivity p. The liquid moves parallelto the plateswith a constant velocity v. The whole system is locatedin a uniform magneticfield of inductionB, vector B being parallelto to the streamdirection.The capacitor the platesand perpendicular plates are interconnectedby means of an externalresistanceR. What amount of power is generated in that resistance?At what value of/\177 is the generatedpower the highest?What is this highest power equal to? 3.266. A straight round copperconductorof radius R = 5.0mm carriesa current I = 50 A. Find the potentialdifference between the axis of the conductorand its surface.The concentrationof the conductionelectronsin copperis equal to n = 0.9.\1770 cm -z. 3.267.In Hall effect measurementsin a sodium conductorthe strengthof a transversefield was found to be equalto E 5.0\177tV/om= A/cm and magneticinductionB with a current density ] = = i.00T. Find the concentrationof the conductionelectronsand Ft\177rn.

\177z

-----

\177

9\17700

its ratio to the total number of atoms in the given conductor. 3.268. Find the mobility of the conductionelectronsin a copper conductorif in Hall effect measurementsperformed in the magnetic field of inductionB = i00mT the transverseelectricfield strength timeslessthan of the given conductorturned out to be = that of the longitudinal electricfield. r A small current-carryingloopis locatedat a distance 3.269. from a longstraight conductorwith current The magneticmoment \177l

Fig.

142

3.72.

3.i't0s

143)))

to Pr,. Find the magnitudeand directionof the loopis equal to forcevectorapplied the loopif the vectorPr, of the

(a) is parallelto the straight conductor; (b) is orientedalong the radius vector r; in directionwith the magneticfield producedby the (c) coincides current I at the point where the loopis located. 3.270.A smallcurrent-carryingcoilhaving a magneticmomentI Pr, is locatedat the axisof a round loopof radius// with current flowing through it. Find the magnitudeof the vector force applied to the coilif its distancefrom the centreof the loopis equal to x and the vector Pm coincidesin directionwith the axis of the loop. Find the interactionforce of two coilswith magneticmo3.271. = 6.0mA.m and collinearaxesif ments Plr, ----4.0mA. m e and i s the coils between the separation equal to 1 ----20cm which exceeds linear dimensions. their considerably thin 3.272.A permanentmagnet has the shape of a sufficiently ---cm. i.0 is// disc of the radius The its axis. discmagnetizedalong Evaluate the magnitudeof a molecularcurrent I' flowing along the rim of the discif the magneticinductionat the point on the axisof the disc,lying at a distancex ----t0 cm from its centre,is equal to \177

P\177r,

-----

\177xT.

vacuum at a plane surface of a uniform isotropicmagnetic equal to B, the vectorB forming an of the magnetThe permeability anglea with the normalof the surface. ic is equal to Find the magnitudeof the magneticinductionB' in the magneticin the vicinity of its surface. 3.274.The magneticinduction in vacuum at a plane surface of a magneticis equal to B and the vectorB forms an angle0 with the

3.273.The magneticinduction in is \177t.

(b) the volume molecularcurrent How are these currents directedtoward each other? 3.276.Half of an infinitely longstraightcurrent-carryingsolenoid is filled with magneticsubstanceas shown in Fig.3.75.Draw the I\177.

Fig.

3.75.

apprcximate plots of magneticinductionB, strengthH, and magnetization on the axisas functions of x. 3.277.An infinitely long wire with a current I flowing in it is locatedin the boundary plane between two non-conductingmedia with permeabilities and Find the modulus of the magnetic inductionvector throughout the spaceas a function of the distance r from the wire.It shouldbe borne in mind that the linesof the vec-

]

\177tl

tor B are

\177t\177.

circleswhose centreslieon the axisof the wire.

round current-carryingloopliesin the plane boundary between magneticand vacuum.The permeabilityof the magnetic is equal to Find the magneticinductionB at an arbitrary point on

3.278.A

\177.

axis of the loopif in the absenceof the magneticthe magnetic induction at the samepoint becomesequal t 9 B0. Generalizethe obtainedresult to all points of the field. 3.279.When a ball made of uniform magneticis introducedinto an externaluniform magneticfield with inductionB0, it getsuniformFind the magneticinductionB inside the ball with ly magnetized. permeability recallthat the magneticfield insidea uniformly mag netized ball is uniform and its strength is equal to H' --$/3, where J is the magnetization. 3.280. N = 300turns of thin wire are uniformly wound on a permanent magnetshaped as a cylinder whose length is equal to l -------t5 cm.When a current I ----3.0A was passedthrough the wiring the field outsidethe magnet disappeared. Find the coerciveforce Hc of the materialfrom which the magnetwas manufactured. the

\177;

-----

3.281. A permanentmagnetis shaped as a ring with a narrow gap between the poles. The mean diameterof the ring equalsd 20 cm. The width of the gap is equal to b = 2.0mm and the magneticinduction in the gap is equal to B.----40 mT.Assuming that the scattering of the magneticflux at the gap edgesis negligible, find the modulus -----

Fig.

3.74.

of the magnetio The permeability normaln of the surface (Fig.3.75). Find: is equal to surface S of (a) the flux of the vector H through the spherical the of surface the magnetic; lies on centre whose radius R, F with (b) the circulationof the vector B around the square path \177.

side l locatedas shown in the figure. 3.275.A directcurrentI flows in a longround uniform cylindrical wire made of paramagneticwith susceptibility Find: current (a) the surface molecular \177.

I\177;

144

of the magneticfield strength vector inside the magnet. 3.282.An iron coreshaped as a torewith mean radius// 250mm supports a winding with the total number of turns N = t000.The corehas a cross-cutof width b ----t.00ram. With a current = A flowing through the winding, the magneticinduction in ----0.85 the gap is equalto B ----0.75T.Assumingthe scatteringof the magnetic flux at the gap edgesto be negligible,find the permeability of iron -----

under these conditions. 1(I---9451

145)))

magnetizationcurve 3.283. Fig.3.76illustratesa basic draw the this

plot,

purity grade). Using (commercial

l\177llllill ll\177llllli\177i

permeability

III III IIIIIII IIIIIII IIIIIII IIIIIII IIIIIII

1111111 II

\177.5

_K.\177

t111111111\177

..... iiii .... ::::

1111 IIIII III Itlllllllltltllllt IIIII iii \177iiiiiIII iii ii iii II O.5 111111111 I

III III

SlillIIIII I

]

\177

regionwhere the magneticfield is practicallyabsent.What amount of work was performed during this process?

3.6.ELECTROMAGNETIC MAXWELL'S

of the paramagneticif the maxim(b) the magneticsusceptibility

attractionforce equals Frnax -- t60 3.287.A smallball of volume V made of paramagnetic with susceptibility was slowly displacedalongthe axisof a current-carrying coilfrom the point where the magneticinductionequalsB out to the

iii

of iron

\342\200\242

INDUCTION.

EQUATIONS

Faraday's law of electromagneticinduction:

IIIIIIIII

\177l

. \342\200\242

''''

\177'

'''' II li

vase of a solenoidand

the

(3.6a)

coil:

doughnut

q) ----Nq)l,

is the

(3.6b)

of turns, q)l is the magnetic flux Inductanceof a solenoid: L= o nsV. a Intrinsic of current and interaction energy

where N

number

through

each turn.

(3.6e)

\177t\177t

\342\200\242

Fig.

3.76.

LI =T

energy

two

currents:

\177

as a function of the magneticfield strengthH. At what value of equal to? H is the permeability the greatest?What is ----50 cm supports A thin iron ring with mean diameterd 3.284. 800turns carrying currentI = 3.0A. a winding consistingof N b = 2.0mm. Neglectingthe scatterof width The ring has a cross-cut the flux at edges,and using the plot shown of the gap magnetic ing of iron under these conditions. in Fig.3.76,find the permeability rod made of paramagneticwith thin A long 3.285. cylindrical area S is a cross-sectional and having % magneticsusceptibility

\342\200\242

Volume density

I\177rnax

$ Displacementcurrent

Maxwell'sequations V X

.\177

a current-carryingcoil.

locatedalong the axis of One end of the rod is locatedat the coilcentrewhere

the magnetic induction is equal to B whereas the other end is locatedin the regionwhere the magnetic field is practicallyabsent.What is the force that the coilexertson the rod? In the arrangementshown in Fig.3.77 it 3.286. to measure(by means of a balance)the is possible force with which a paramagneticball of volume V--4i mm 3 isattrabtedto a poleof the electromag- Fig. 3.77. net M. The magneticinduction at the axis of the depends on the height x as B B0 exp(--axe), where poleshoe Find: B o = t.50T, a -- t00m the maximum attracthe ball experiences (a) at what height

\303\227

BH = T\"

(3.6e)

density: idls

\342\200\242

(3.6d)

L\177sI\177Is.

energy:

of magnetic field

\177

-\177

Wls--\177

\177

----

in differential

form:

V.B=0,

V.D=p,

E= ----\177-, H

(3.6f)

\"\177-\"

=J-[--\177-,

(3.6g)

\177 div rot (the rotor) and (the divergence). Field transformation formulas for transition from a referenceframe K referenceframe K' moving with the velocity v0 relative to it. In the case vo << c z E -t- [voB], B [voE]/c {3.6h) In the general case

where

\303\227

\177

B'= --

E' =

Eli

=Eli,

E'I __

\177

[voBI t

]/\177\302\261-[-

B\177I-----B

B,t =

B_L

\177

(3.6i)

--[voE]/\302\242

\177/t--

--(vo/c)\177

-\177.

tion; 146

x,\177

where the symbols II and and perpendicularto the to\"

\302\261

denote the

vector v 0.

field

components, respectivelyparallel t47)))

A wire bent as a parabolay = ax is locatedin 3.288. the vector B \177

a uniform

to being perpendicular magneticfield of inductionB,t ----0 a connector transs tarts sliding moment the At the planex, y. w lationwisefrom the parabolaapexwith a constant acceleration in the induction loop of emf Find the electromagnetic (Fig.3.78).

3.8i.

Fig. The connectorof length l and resistanceR slides to the right with a constantvelocity v. Find the current induced in shown in

thus formed as a function of y.

,9 Fig.

,z\"

3.78.

Fig.

3.79.

of length l 3.289.A rectangularloopwith a sliding connector the to field perpendicular loop plane is locatedin a uniform magnetic has to B.The connector is induction The (Fig.3.79). magneticthe sidesABequal resistances have CD //1 and an electricresistanceR, the of loop, self-inductance the and respectively.Neglecting during its motion with a find the current flowing in the connector constant velocity v. A metaldisc of radius a-\177 25 cm rotateswith a constant 3.290. about its axis. Find the potential angular velocity (o ----i30rad/s difference between the centreand the rim of the disc if (a) the externalmagneticfield is absent; B 5.0mT (b) the externaluniform magneticfield of induction to the disc. is directedperpendicular of diameterd 3.29|.A thin wire AC constant shaped as a semi-circle angular velocity (o i00 rad/s 20cm rotateswith a with in a uniform magnetic field of induction B----5.0mT, wire and the of A end the axis ea B. The rotation passesthrough to the diameterAC. Find the value of a lineintegral is perpendicular

r between the connectorand the the loop as a function of separation wire. the The o f resistance straight H-shaped conductorand the selfinductanceof the loopare assumed to be negligible. 3.294.A squareframe with sidea and a longstraightwire carrying a current arelocatedin the sameplane as shown in Fig.3.82. The frame translatesto the right with a constantvelocity v. Find the emf inducedin the frame as a function of distancex.

R\177

-\177

\177

-\177

\177t

I E dr

along the wire from point

to point

the obC.Generalize

Fig.

3.83.

3.295. A metalrod of mass m can rotate about a horizontalaxis The O, sliding along a circularconductorof radius a (Fig. 3.83). arrangementis locatedin a uniform magneticfield of inductionB

to the ring plane.The axis and the ring are directedperpendicular connectedto an emf sourceto form a circuitof resistance//. Neglectfind the law ing the friction, circuitinductance,and ring resistance, to make the rod rotate accordingto which the sourceemf must with a constant angular velocity 3.296. A copperconnectorof mass m slidesdown two smooth cops perbars, etat an anglea to the horizontal,due to gravity (Fig.3.84). //. The At the top the bars are interconnectedthrough a resistance separationbetween the bars is equal to 1.The system is locatedin to the plane a uniform magneticfield of inductionB, perpendicular of the bars,the connectThe resistances in which the connectorslides. of the loop, or and the slidingcontacts,as well as the self-inductance are assumed to be negligible.Find the steady-statevelocity of the connector. \"\177ary

tained result.

3.292.A wire loopenclosinga semi-circleof radius a is located on the boundary of a uniform magnetic field of induction B 0 the loopis set into rotation with At the moment t (Fig.3.80). about an axisO coincidingwith a acceleration a constantangular the loop line of vector B on the boundary. Find the emf inducedin function. this of plot as a function of timet. Draw the approximate positive. The arrow in the figure shows the emf directiontaken to be a current and a H-shaped 3.293.A long straight wire carryinglocated in the sameplane as conductorwith sliding connectorare \177-

\177

t48

Fig. 3.82,

\17749)))

3.297.The system differs from the one examinedin the foregoing problem (Fig. 3.84) by a capacitorof capacitance C replacingthe resistanceR. Find the acceleration of the connector. R

slideswithout friction along the

wires with a constant velocity v. Assuming the resistancesof the wires, the connector,the sliding find: contacts,and the self-inductanceof the freme to be negligible, (a) the magnitude and the directionof the current induced in the connector; (b) the force required to maintain the connector'svelocity con-

stant.

3.298. A wire shaped as a semi-circle of radius a rotatesabout an an angular velocity (o in a uniform magneticfield of induction B (Fig.3.85). The rotationaxis is perpendicular to the field The axis 00'with

direction. totalresistanceof the circuitis equal to R. Neglectingthe magneticfield of the induced current, find the mean amount of thermal power being generated in the loopduring a rotation period. 3.299.A smallcoilis introducedbetweenthe polesof an electromagnetso that itsaxis coincideswith the magneticfield direction. The cross-sectional area of the coilis equal to S = 3.0mm 2, the numberof turns is N its diameter,a

= 60.When the coilturns

process.

\177B

----\177F

\177--\177(\177

Fig,

\302\260

\177.

t80 about

through ballistic galvanometerconnectedto the coilindicates

= 4.5btC flowing throughit.Find the magneticinduction magnitudebetween the polesprovided the total resistanceof the electriccircuit equals R = 40 3.300.A square wire frame with sidea and a straight conductor carryinga constantcurrentI arelocatedin the sameplane(Fig.3.86). a chargeq

A conductingrod AB of mass m slides 3.302. without friction over two longconductingrails separatedby a distancel (Fig.3.88). At the left end the railsare interconnected by a resistance//. The to the system is locatedin a uniform magneticfield perpendicular = of A t the the moment t 0 AB starts the rod loop. moving to plane the right with an initialvelocity v 0. Neglectingthe resistances of the rails and the rod AB, as well as the self-inductance,find: (a) the distancecoveredby the rod until it comesto a standstill; (b) the amount of heat generatedin the resistance// during this

3.88.

Fig.

3.89.

3.303. A connectorAB can slidewithout friction along a Hconductor locatedin a horizontalplane (Fig.3.89). The conshaped nectorhas a length l, mass m, and resistance//. The whole system is a locatedin

uniform magneticfield of inductionB directedvertically. the moment t ----0 a constanthorizontalforce F starts acting on the connectorshifting it translationwiseto the right. Find how the varies with time t. The inductanceof the velocity of the connector and the resistance of the H-shapedconductorare assumed to loop

be negligible.

3.304.Fig.3.90illustratesplane figures made of thin

which

Fig.

arelocatedin

a uniform magneticfield

conductors

directedaway

from a

3.86.

The inductanceand the resistanceof the frame areequal to L and T/ The frame was turned through i80 about the axisOO' respectively. separatedfrom the current-carrying conductor by a distanceb. Find the electricchargehaving flown through the frame. 3.30i.A long straight wire carriesa current I0. At distancesa and b from it thereare two other wires, parallelto the former one, which are interconnectedby a resistanceR (Fig.3.87). A connector

(a)

(c)

(b)

(d)

\302\260

150

Fig.

3.90.

readerbeyond the plane of the drawing. The magnetic induction starts diminishing.Find how the currentsinducedin theseloopsare directed. t51)))

A plane loopshown in Fig.3.9iis shaped as two squares 3.305. i0 cm and is introducedinto a uni-

20 cm and b with sidesa form magneticfield at right anglesto the -----

\177

loop'splane.The magnetic with time as B Bo sin cot, where B0 = i0 mT induction varies-1 Find the amplitudeof and co = t00 s the loopif its resisin induced the current tance per unit length is equal to p = 50 m\177/m. The inductanceof the loopis to \177

\342\200\242

be neglected.

3.306.A

plane spiralwith a great numberN of turns wound tightly to one another is locatedin a uniform magneticfield perplane.The outside pendicularto the spiral's radius of the spiral'sturns is equal to a. The magneticinductionvaries with time as B = B0 sin cot, where are constants.Find the amplitude of emf induced in B0 and \302\242o

spiral. 3.307.A H-shapedconductor is locatedin

the

a uniform magnetic and varying with conductor the of the to plane perpendicular starts movconnector 0.t0T/s.A conducting time at the rate ---the cm/s w l0 along parallelbars of the ing with an acceleration conductor.The length of the connectoris equal to l 20 cm. Find s after the beginningof the the emf induced in the loopt----2.0 the t 0 moment looparea and the magnetic motion,if at the induction are equal to zero.The inductanceof the loop is to be field

\177

--\177

\177

\177--

\177

neglected.

radius a 3.308.In a long straight solenoidwith cross-sectional and number of turns per unit length n a current varies with a constant velocity A/s. Find the magnitudeof the eddy current field strengthas a function of the distancer from the solenoidaxis.Draw the approximate plot of this function. diameterd = A long straight solenoidof cross-sectional 3.309. its cm of one turns 20 n lengthhas a round 5 cm and with per mm tightly put areaS turn of copperwire of cross-sectional on its winding.Find the current flowing in the turn if the current with a constant velocity in the solenoidwinding is increased t00A/s. The inductanceof the turn is to be neglected. radius a has a thin insu3.3i0.A long solenoidof cross-sectional its on winding;one half of the ring has lated wire ring tightly put the resistance timesthat of the otherhalf. The magneticinduction B bt, where b is producedby the solenoidvaries with time as field electric the of strength in the a constant.Find the magnitude

--\177

-----

\177

--\177

\177.0

]

\177

\177l

\177-

ring.

3.3ii.A

non-conductingring of mass m carrying a chargeq can freely rotateabout its axis.At the initial moment the ring was uniform at rest and no magneticfield was present.Then a practically the to whichwas plane switched f ield was on, perpendicular magnetic i52

thin

to a certainlaw B (t). with time according of the ring and increased of the induction function a as ea the of Find the angularvelocity ring

B (t).

r is locatedinwire ring of radius a and resistance The coincide. their axes so that length of the side a long solenoid radius, to b. At a certain solenoidis equal to l, its cross-sectional was connectedto a sourceof a constantvoltage moment the solenoid of the circuitis equal to //. Assuming the resistance V. The total inductanceof the ring to be negligible,find the maximum value of

3.3i2.A thin

radialforce acting per unit length of the ring. loop with a resistance 3.3i3.A magneticflux through a stationary varies during the time interval as (I) at -- t). Find the amount of heat generatedin the loop duringthat time.The inductance of the loopis to be neglected. there is a coaxialring of 3.3i4.In the middleof a long solenoidmaterial with resistivity made of conducting square cross-section, outsideradii and its inside to is of the h, ring equal p.The thickness Find the current inducedin the are equal to a and b respectively. with ring if the magneticinductionproducedby the solenoidvaries of the inductance T he is constant. ring a where time as B is to be neglected. thin wire are requiredto manufac3.3|5.How many metresof at00 t.0mH cm and inductanceL of ture a solenoid length 10 diameteris considerably lessthan its cross-sectional if the solenoid's

the

\177

/\177

\177

('\177

--\177

\177t,

\177

-----

length?

3.3t6.Find

the inductanceof a solenoid of length l whose of copperwire of mass rn. The winding resistance is made winding is equal to /L The solenoiddiameteris considerably lessthan its

length.

3.3i7.A

mH coil of inductance L-----300

and resistance// How soon will source. t40 m\177 is connectedto a constantvoltage value? the of 50% steady-state c urrentreach the coil Calculatethe time constantx of a straightsolenoidof length 3.318. ire whose total l t.0m having a single-layerwinding of copperwdiameter of the cross-sectional The to rn is mass equal kg. solenoidis assumed to be considerablylessthan its length. Note.The timeconstantx is the ratioL/t\177, where L is inductance is active resistance. and of a cableconsisting 3.3i9.Find the inductanceof a unit length the radius of the outif of two thin-walledcoaxialmetalliccylinders one.The permeinside the of that times is 3.6 side cylinder to \177

\177--

\177

-\177

\177.0

/\177

ability of a medium between the cylinders is assumedto be equal unity. Calculatethe inductance of a doughnut solenoidwhose 3.320. has the form of a square insideradius is equal to b and cross-section o f N turns. The spaceinconsists with sidea.The solenoid winding uniform with having peris filled solenoid paramagnetic sidethe up meability \177l-\177

\177.

\17753)))

3.32t.Calculatethe inductanceof a unit length of a doubletape line (Fig.3.92)if the tapesare separatedby a distanceh which is lessthan theirwidth b, considerably

in the

sameplane,with

the

sideb beingclosestto the wire,separated

by a distanceI from it and orientedparallelto

it.

50. namely, b/h 3.322.Find the inductanceof a \177

length of a double line if the radius of each wire is timesless than the distancebetween the axes of the wires.The field inside the wires is to be neglected,the permeabilityis assumed to be equal to unity throughout, and >> 3.323. A superconductinground of radius a and inductanceL ring waslocatedin a uniform magneticfield of inductionB.The ring plane was parallelto the vectorB, and the currentin the ring was equal to zero.Then the ring was turned through 90 so that its plane became to the field.Find: perpendicular (a) the current inducedin the ring after the turn; (b) the work performed during the turn. 3.324.A current I0 = 1.9A flows in a long closedsolenoid. wire it is The wound of is in a superconductingstate.Find the current flowing in the solenoidwhen the length of the solenoidis increasedby 5%. 3.325. A ring of radius a 50 mm made of thin wire of radius b = i.0mm was locatedin a uniform magneticfield with induction to the vectorB. B = 0.50mT so that the ringplanewas perpendicular Then the ring was cooleddown to a superconductingstate, and the magneticfield was switchedoff. Find the ring currentafter that. Note that the inductanceof a thin ring along which the surface current flows is equal to L\177oa(ln\177--2 3.326. A closedcircuitconsistsof a sourceof constant emt $ anti The activeresistanco a chokecoilof inductanceL connected in series. of the wholo circuitis equal to R. At the moment t = 0 the choke coilinductancewas decroasedabruptly times.Find the current in unit

\177l

\302\260

\177-

\177l

-\177

a function of time t. Instruction. During a stepwisechange of

the

Fig.

3.94.

3.330.

Determinethe mutual inductanceof a doughnut coiland an infinite straight wire passingalongitsaxis.The coilhas a rectanits insideradius is equalto a and the outsideone, gularcross-section, to b. The length of the doughnut'scross-sectional sideparallelto the wire is equal to h. The coilhas N turns. The system is locatedin a uniform magneticwith permeability Two thin concentric wires shaped as circleswith radii a and b liein the same plane.Allowiug for a b, find: (a) their mutual inductance; (b) the magneticflux through the surface enclosed by the outside wire, when the insidewire carriesa current

3.33t.

\177.

\177<

3.332. A smallcylindricalmagnetM (Fig.3.95) is placedin

the

centreof a thin coilof radiusa consistingof N turns. The coilis connectedto a ballistic The activeresistance of the whole galvanometer. circuitis equal to R. Find the magneticmoment of the magnet if its removal from the coilresults in a chargeq flowing through the galvanometer. 3.333. Find the approximate formula expressing the mutual in-

ductanceof two thin coaxialloops of the sameradius a if their centresare separatedby a distancel, with l:>:>a.

\177l

circuit as

indu(\177tance

the

total

linkage) remains constant. 3.327.Find the time dopendenceof the current flowing through the inductanceL of the circuitshown in Fig.3.93after the switch Sw is shorted at the moment t 0. 3.328.In the circuitshown in Fig.3.94an emf $, a resistanceR, of and coilinductancesL1and L2 areknown. Theintornalresistance the sourceand the coilresistancesare negligible.Find the steadystate currents in the coilsafter the switch Sw was shortod. 3.329.Calculatethe mutual inductanceof a longstraight wire and

magneticflux

Fig.3.93.

(flux

Fig. :3.95.

Fig.

3.96.

-\177

rectangularframe

154

with

sidesa and

b. The frame and tho wire lie

3.334.There are two stationary loopswith mutual inductance starts to be varied as Ltd.. The current in one of the loops at, where a is a constant, t is time. Find the timedependence12(t) of the currentin the otherloop whoseinductanceis L2 and resistance R. \177

I\177

A coilof inductanceL -- 2.0 3.335. and resistance// i.0 is connectedto a sourceof constant emf $ ----3.0V (Fig.3.96).A \177tH

-----

\177

i55)))

Find the in parallelwith the coil. resistanceR0 ----2.0 is connected amount of heat generatedin the coilafter the switch Sw is disconnectis negligible. ed.The internal resistanceof the source An iron tore supports N -- 5{)0 turns. Find the magnetic 3.336. 2.0A producesa magneticflux across field energy if a current I cross-section the tore's equal to (1) ----i.0mWb. 3.337.An iron coreshaped as a doughnut with round cro\177s-secturns through cm carriesa winding of N tion of radiusa \177

-\177

\177

3.\177

i\177\302\2420

which a current I i.0A flows. \177he mean radius of the doughnut is b 32 cm.Usingthe plot in Fig.3.76,find the magneticenergy storedup in the core.A field strengthH is suppcsedto be the \177ame and equal to its magnitudein the centhroughout the cross-section \177

\177

tre of

the

cross-section.

A thin 3.338.

ring made of a magnetic has a mean dfameter 30 cm and supports a w\177nding of N = 8\1770 turns. The crosssectionalarea of the r\177ng is equal to S = 5.0cme. The ring has a cross-cutof width b = 2.0ram.When the winding carriesa certaia i\17700. Neglectof the magneticequals current, the permeability at the flux gap edges,find: ing the dissipationof magnetic the and in the magnetic; i n of ratio the gap magneticenergies (a) two do it in the of ways: using the flux the inductance system; (b) field. and using the energy of the A long cylinder of radiusa carrying a uniform surface charge rotates about its axis with an angularvelocity Find the magneticfield energy perunit length of the cylinder if the linearcharge

\177

3.339.

\177.

density equals and

\177

At 3.340.

\177

i.of the electricfield

what magnitude strengthin \177acuum as that of the magis the same field this of the \177olume energy density T (alsoin \177acuum). neticfield with inductionB cm rotates A thin uniformly chargedring of radius a = the raFind rad/s. iO0 about its axiswith an angularvelocity tio of volume energ\177 densitiesof magneticand electricfields on the axis of the ring at a point removed from its centreby a distance \177

3.34t.

i.0

\1770

\177

for volume density of magneticener3.342. Usingthe expression to of work

contributed

gy, demonstratethat the amount

magneti-

para-or diamagnetic,is equal to A = = -- JB/2. eachof inductanceL, areinterconnected Two identicalcoils, 3.343. (a) in series,(b) in parallel.Assuming the mutual inductanceof the coilsto be negligible,find the inductanceof the system in both cases. 3.344.Two solenoidsof equal length and almostequal crosssectionalareaare fully insertedinto one another.Find their mutual and inductanceif their inductancesare equal to interactionof 3.345.Demonstratethat the magneticenergy of as be can in vacuum l ocated represented two current-\177arrying loops = (i/\1770) and arethe magneticinductions dV, whero zation of a

unit volume of

L\177

W\177a

B\177B\177

B\177

L\177.

within a volume elementdV, of the first and the second

produced individually by the currents

looprespectively. 3.346. Find the interactionenergy of two loops carrying currents I1and 12if both loopsare shaped as circlesof radii a and b, with a << b. The loops'centresare locatedat the samepoint and their planes form an angle 0 between them. 3.347.The spacebetween two concentricmetallicspheresis filled up with a uniform

poorly conductingmedium of resistivity p and permittivity e. At the moment t = 0 the inside sphere obtains a

certaincharge.Find: currentden(a) the relationbetween the vectors of displacement sity and conductioncurrent density at an arbitrary point of the medium at the samemoment of time; current acrossan arbitrary closedsurface (b) the displacement the internal sphere,if wholly locatedin the medium and enclosiag at the given moment of timethe chargeof that sphereis equal to q. 3.348.A parallel-platecapacitoris formed by two discswith a uniform poorly conductingmedium between them. The capacitor was initially chargedand then disconnected from a voltagesource. Neglectingthe edge effects, show that there is no magneticfield between capacitorplates. 3.349.A parallel-plate air condenserwhose each plate has an area S = cm2 is connectedin seriesto an ac circuit.Find the electricfield strength amplitudein the capacitorif the sinusoidal current amplitude in lead wires is equal to Im= i.0mA and the current frequency equals = i.6.i07 s-1 3.350. The spacebetween the electrodes of a parallel-plate capacitoris filled with a uniform poorly conductingmedium of conducti\17700

\177o

and permittivity e.The capacitorplatesshaped as round discs separatedby a distanced. Neglectingthe edgeeffects, find the magneticfield strengthbetween the platesat a distancer from their axisif an ac voltageV = V m cos cot is appliedto the capacitor. 3.351. A long straight solenoid has n turns per unit length. An ---I current sin cot flows alternating Im through it.Find the displacement currentdensity as a function of the distancer from the solenoid axis.The cross-sectional radius of the solenoidequalsR. A point chargeq moves with a non-relativisticvelocity 3.352. v = const.Find the displacement currentdensity j d at a point located at a distancer from the chargeon a straight line

vity

\177

are

(a) coincidingwith the chargepath; to the path and passingthrough the charge. (b) perpendicular A thin wire ring of radius a carrying a chargeq approaches 3.353. the observationpoint P so that its centremoves rectflinearly with a constant velocity v. The plane of the ring remainsperpendicular to the motion direction.At what distancex\177nfrom the point P will the ring be locatedat the moment when the displacement current maximum?What is the magnitude of density at the point P becomes this maximum density?

\177

t57)))

i56

3.354.A point

charge q moves with a non-relativistic velocity = const.Applying the theoremfor the circulationof the vectorI4 around the dottedcircleshown in Fig.3.97,find H at the point A as a function of a v

radius vector r and

v of the

charge. (a) a time-dependent magneticfield cannotexistwithout an electric field; (b) a uniform electricfield cannot existin the presenceof a timevelocity 3.355. Using Maxwell'sequations,show that

dependent magnetic field; (c) inside an empty cavity a uniform electric(or magnetic)field can be time-dependent. 3.356. Demonstrate that the law of electricchargeconservation,

=-Op/Ot, follows from i.e.V.j 3.357.

Maxwell'sequations. Demonstratethat Maxwell'sequations V and V.B= 0 are compatible,i.e.the first one doesnot contradict the second one. In a certainregionof the inertialreferenceframe there is 3.358. magneticfield with inductionB rotating with angular velocity to. Find V x E in this region as a function of vectorsto and B. In the inertialreference 3.359. frame K there is a uniform magnetic field with inductionB.Find the electricfield strength in the frame K' which moves relativeto the frame K with a non-relativisticvethe forcesacting locity v, with v_LB. To solve this problem,consider on an imaginary chargein both reference frames at the momentwhen the velocity of the chargein the frame K' is equal to zero. \303\227E

\177B/\177t

3.363. Using Eqs.(3.6h),demonstratethat if in the inertialreferenceframe K there is only electricor only magneticfield, in any other inertialframe K' both electricand magneticfieldswill coexist simultaneously,with E'._LB'. 3.364.In an inertialreferenceframe K there is only magnetic field with induction B = b (yi --xj)/ (x where b is a constant, i and j are the unit vectorsof the x and y axes.Find the electric field strength E' in the frame K' moving relativeto the frame K with a constant non-relativistic velocity v -= vk; k is the unit vector of the z axis.The z' axis is assumed to coincidewith the z \177

\177-

y\177),

axis.What is the shape of the field E'? 3.365. In an inertialreference frame K there is only electricfield of strength E a (xi + yj)/(x2+ y2), where a is a constant, i and j are the unit vectorsof the x and y axes.Find the magneticinduction B' in the frame K' moving relativeto the frame K with a constant non-relativistic velocity v = vk; k is the unit vector of the z axis. The z' axisis assumed to coincide with the z axis.What is the shape of the magneticinductionB'? 3.366. Demonstratethat the\" transformation formulas (3.6h) -----

follow from the formulas (3.6i) at v 0 << c. 3.367.In an inertialreferenceframe K there is only a uniform electricfield E ----8 kV/m in strength. Find the modulus and direction (a) of the vector E', (b) of the vector B' in the inertialreference frame K' moving with a constant velocity v relativeto the frame K at an anglea ----45 to the vectorE. The velocity of the frame K' is equal to a ----0.60fraction of the velocity of light. 3.368. Solvea problem differing from the foregoingone by a magneticfield with inductionB = 0.8T replacingthe electricfield. 3.369.Electromagnetic field has two invariant quantities.Using the transformation formulas (3.6i), demonstratethat thesequantities are (a) EB; (b) E\177- ceB 3.370.In an inertialreferenceframe K there are two uniform mufields:an electricfield of strengthE = 40 kV/m tually perpendicular and a magnetic field induction B = 0.20mT. Find the electric strength E' (or the magneticinductionB') in the referenceframe \302\260

\177

\177.

Fig.

3.97.

Fig. 3.98.

3.360. A largeplate of non-ferromagnetic materialmoves with a constant velocity v 90cm/s in a uniform magneticfield with inductionB = 50 mT as shown in Fig.3.98. Find the surface density of electricchargesappearingon the plate as a result of its motion. 3.361. A long solidaluminum cylinder of radius a----5.0 cm rotates about its axis in a uniform magneticfield with induction B =- t0 mT. The augularvelocity of rotation equals ----45 rad/s, with B. Neglectingthe magneticfield of appearingcharges, find their spaceand surface densities. 3.362. A non-relativistic point charge q moves with a constant velocity v. Usingthe field transformationformulas, find the magnetic inductionB produced by this chargeat. the point whose position relativeto the chargeis determinedby the radius vector r. --=-

\302\242o

(\177

t58

\177

K' where only one field, electricor magnetic,is observed.

Instruction.Makeuse of the field invariants citedin the foregoing problem. 3.371. A point chargeq moves uniformly and rectilinearly with a relativisticvelocity equal to a fraction of the velocity of light (fi = v/c). Find the electricfield strength E producedby the charge at the point whose radius vector relativeto the charge is equal to r and forms an angle 0 with its velocity vector. \177

t59)))

3.7.MOTION AND

OF CHARGED PARTICLESIN ELECTRIC

MAGNETIC

, Lorentz force: , Motion

equation

F qE q- q [vB]. a relativistic particle: d mov

(3.7a)

----

]/t--(v/c)

(a) the angle0 between the

FIELDS

Period of revolution

----

(3.7b)

\302\260

a charged particle in a

uniform

T: qB '

magnetic field:

2nrn

relativistic mass is Betatron condition, , a betatron: a circular where m

the

that

of the particle, rn mo/l\177f\177 is the condition for an electronto move

orbit in

\177

Bo =

where B 0 is the magnetic induction of the induction inside the orbit.

3.372.At

(3.7c)

-- (v/c)

\177.

i (B),

(3.7d)

-\177-

along

orbit'spoint, (B) is

the mean value

0 an electronleavesone plateof a para negligiblevelocity. An accelerating t00V/s, is appliedbetween at, where a voltage,, varying as V The separationbetween the platesis I = 5.0cm.What the plates. is the velocity of the electronat the moment it reachesthe opposite the moment

--\177

allel-platecapacitorwith \177

-\177

plate?

3.373.A proton acceleratedby a potential difference V gets into the uniform electricfield of a parallel-plate capacitorwhose plates extend over a length l in the motion direction.The field strength varies with time as E = at, where a is a constant.Assumingthe profind the anglebetween the motion directon to be non-relativistic, tions of the proton before and after its flight through the capacitor; the proton gets in the field at the moment t = 0.The edgeeffects are to

be neglected.

3.374.A particlewith

specificchargeq/ra moves rectilinearlydue = E to an electricfield Eo ax, where a is a positive constant,x is the distancefrom the point where the particlewas initially at

rest. Find:

till

the moment it came (a) the distancecovered by the particle to a standstill; of the particleat that moment. (b) the acceleration 3.375.An electronstarts moving in a uniform electricfield of strength E = i0 kV/cm.How soon after the start will the kinetic energy of the electronbecomeequal to its rest energy? of a relativisticelectronmoving the acceleration 3.376.Determine E at the moment when its field of electric a uniform strength along kineticenergy becomesequal to T. 3.377.At the moment t = 0 a relativisticproton flies with a velocity vo into the regionwhere there is a uniform transverseelectric field of strength E, with vo _J_ E. Find the time dependenceof

proton's velocity vectorv

and the ini-

tial directionof its motion; (b) the projectionvx of the vector v on the initial directionof motion. 3.378.A proton acceleratedby a potentialdifference V = 500kV flies through a uniform transverse magneticfield with induction B ----0.5tT. The field occupiesa region of spaced = t0cm in thickness(Fig.3.99). Find the anglea through which the proton deviatesfrom the initialdirectionof itsmotion. 3.379.A chargedparticlemoves along a circle of radius r----t00mm in a uniform magnetic field with induction Fig. 3.99. B = t0.0mT. Find its velocity and period of revolution if that particleis proton; (a) a non-relativistic (b) a relativisticelectron. 3.380. A relativisticparticlewith charge q and rest mass mo moves alonga circleof radius r in a uniform magneticfield of induction B. Find: momentum vector; (a) the modulus of the particle's (b) the kineticenergy of the particle; (c) the accelerationof the particle. 3.381. Up to what values of kineticenergy does the period of revolutionof an electronand a proton in a uniform magneticfield velocitiesby rI = t.0%? exceedthat at non-relativistic An electronaccelerated 3.382. by a potentialdifference V = = i.0kV moves in a uniform magneticfield at an anglea = 30 to the vector B whose modulus is B = 29 mT. Find the pitch of the helicaltrajectory of the electron. A slightly divergentbeam of non-relativistic 3.383. chargedparfrom a point ticlesaccelerated by a potentialdifference V propagates The beam is brought into A along the axis of a straight solenoid. focus at a distance1 from the point A at two successivevalues of Find the specificchargeq/m of the magneticinductionB1 and particles. A non-relativistic electronoriginatesat a point A lying 3.384. on the axis of a straight solenoidand moves with velocity v at an angle a to the axis.The magneticinductionof the field is equal to B.Find the distancer from the axisto the point on the screeninto The screenis orientedat right anglesto which the electronstrikes. the axisand islocatedat a distancel from the point A. From the surface of a round wire of radius a carrying a 3.385. directcurrentI an electronescapeswith a velocity vo perpendicular to the surface.Find what will be the maximum distanceof the electron from the axisof the wire before it turns backdue to the action of the magneticfield generatedby the current. 11--9451 \302\260

B\177.

t6t)))

A non-relativistic 3.386. chargedparticleflies through the electric field of a cylindricalcapacitorand getsinto a uniform transverse In the capacitorthe magneticfield with inductionB (Fig.3.t00). in the of a arc the circle, magneticfield, along particlemoves along of radiusr. The potentialdifference appliedto the capaa semi-circle citorisequal to V, the radii of the electrodesare equal to a and b, b.Find the velocity of the particleand itsspecificcharge with a

3.39t.A beam of non-relativistic chargedparticlesmoves without deviationthrough the regionof spaceA (Fig.3.t03) where there are transversemutually perpendicular electricand magneticfields with

\177

q/m.

Fig.

3.i00.

Fig.

3.i01.

3.387.Uniform electricand magneticfields with strength E and inductionB respectivelyaredirectedalong the g axis (Fig.3.t0t). with specific A particle chargeq/m leavesthe originO in the direction

x axiswith an initialnon-relativistic velocity re.Find: it crossesthe g axis when of the coordinate the particle g n (a) for the nth time; velocity vector and the g (b) the angle between the particle's axis at that moment. A narrow beam of identicalions with specific 3.388. chargeq/m, possessingdifferent velocities,entersthe regionof space,where there are uniform parallelelectricand magneticfields with strength E The beam direction and inductionB, at the point O (seeFig. coincideswith the x axisat the point O.A plane screenorientedat right anglesto the x axisis locatedat a distancel from the point O. Find the equation of the trace that the ions leave on the screen. that at z l it is the equationof a parabola. Demonstrate of the

c\177

3.t0t).

<\177

A non-relativistic 3.389. proton beam passesWithout

deviation

muspacewhere there are uniform transverse = t20 with E fields kV/m and electric magnetic tually perpendicular and B = 50 roT. Then the beam strikesa groundedtarget. Find the force with which the beam actson the targetif the beam current is equal to I = 0.80mA. Nori-relativistic 3.390. protonsmove rectilinearlyin the regionof electricand perpendicular spacewhere there are =uniform mutually ---of 50 roT. T he and B 4.0 fields with E trajectory kV/m magnetic the protons liesin the plane xz (Fig. 3.t02)and forms an angle along (p = 30 with the x axis.Find the pitch of the helicaltrajectory which the protonswill move after the electricfield is switched off. through the regionof

\302\260

t62

3.i02.

Fig.

3.i03.

strengthE and inductionB.When the magneticfield is switched off, the trace of the beam on the screenS shifts by Ax. Knowing the distancesa and b, find the specificchargeq/m of the particles. 3.392. A particlewith specific chargeq/m moves in the regionof electricand spacewhere there are uniform mutually perpendicular magnetic fields with strength E and induc-

tion B (Fig. 3.t04).At the moment t the particlewas locatedat the point O and had zero velocity. For the non-relativistic casefind: (a) the law of motionx (t) and y (t) of the particle;the shape of the trajectory; x g (b) the lengthof the segmentof the trajectory between two nearestpoints at which the velocity of the particleturns into zero; Fig. 3.t04. veloc(c) the mean value of the particle's ity vector projectionon the x axis (the drift velocity). A system consistsof a long cylindricalanode of radius a and a coaxialcylindricalcathodeof radius b (b a).A filament locatedalong the axis of the system carriesa heatingcurrent proFind the leastpoducinga magneticfield in the surroundingspace. tentialdifference between the cathodeand anodeat which the thermal electronsleavingthe cathode without initial velocity start reaching the anode. 3.394.Magnetron is a deviceconsistingof a filament of radius a and a coaxialcylindricalanode of radius b which arelocatedin a uniform magneticfield parallelto the filament.An accelerating potential difference V is appliedbetween the filament and the anode. Find the value of magneticinductionat which the electronsleaving the filament with zero velocity reach the anode. A chargedparticlewith specific chargeq/m starts moving in the regionof spacewhere there are uniform mutually perpendicular electricand magneticfields.The magneticfield is constant and \177

3.393.

3.395.

ll*

i63)))

has an inductionB while the strengthof the electricfield varieswith time as E = E,\177 coso)t, where o) qB/m.For the non-relativistic casefind the law of motionx (t) and y (t) of the particleif at the moWhat is 0 it was locatedat the point O (seeFig.3.i04). ment t the approximate shape of the trajectoryof the particle? 3.396.The cyclotron'soscillatorfrequencyis equalto v ----i0MHz. Find the effective accelerating voltageappliedacrossthe deesof that cyclotron if the distancebetween the neighbouringtrajectoriesof t.0 cm, with the trajectoryradius protons is not less than Ar \177--

-----

\177

m. being equal to r----0.5 in a cyclotron so that the maximum 3.397.Protonsare accelerated of their curvature radius trajectoryis equal to r ----50 cm. Find: of the protons when the accelerationis the kinetic energy (a) in the cyclotron is B ----t.0T; induction i f the magnetic completed of the cyclotron'soscillatorat which (b) the minimum frequency the kineticenergy of the protons amounts to T 20 MeV by the \177--

end of acceleration. in a cyclotron so 3.398. Singly chargedions He+ are accelerated that their maximumorbitalradius is r = 60cm. The frequency of t0.0MHz, the effective acoscillatoris equal to a cyclotron's ---50 kV. Neglectingthe gap is V the dees celeratingvoltageacross between the dees, find: \177

\177

total time of accelerationof the ion; distancecoveredby the ion in the processof approximate its acceleration. mag3.399.Sincethe periodof revolutionof electronsin a uniform neticfield rapidly increaseswith the growth of energy, a cyclotron This is unsuitablefor their acceleration. drawback is rectified in a microtron AT in the (Fig.3.t05)in wi\177ich a change is electron an of revolution of period mademultiplewith the periodof accelerating field To.How many timeshas an electronto crossthe acceleratinggap of a microtronto acquirean energy W-\177 ----4.6MeV if AT To, the magnetic induction is equal to B t07 mT, and Fig. 3.105. the frequency of acceleratingfield to MHz? v ----3000 The illeffects associatedwith the variation of the period 3.400. of revolutionof the particlein a cyclotron due to the increaseof its by slow monitoring(modulating)the frequency energy areeliminated field.Accordingto what law o) (t) should this frequenof accelerating (a) the (b) the

\177

is equal to B and the cy be monitoredif the magneticinduction The chargeof the revolution? AW an per energy particleacquiresitsmass m. is and is particle q insidea round 3.40i.A particlewithr from specificchargeq/m is located switched into current the axis. With its solenoidat a distance

the winding, the magneticinduction of the field generatedby the solenoidamounts to B.Find the velocity of the particleand the curvature radius of itstrajectory,assumingthat during the increaseof current flowing in the solenoidthe particleshifts by a negligible

distance. 3.402.In a betatronthe magneticflux acrossan equilibriumorbit timeat practically of radiusr 25 cm growsduring the acceleration 5.0Wb/s. In the process,the electronsacquirean constantrate \177--

\177)

-----

25 MeV. Find the number of revolutionsmade by the energy timeand the correspondingdistance electronduringthe acceleration W

\177

coveredby

it.

3.403. Demonstratethat electronsmove

in a

betatron along a

round orbit of constantradius provided the magneticinductionon the orbit is equal to half the mean value of that inside the orbit

(the betatron condition). 3.404.Using the betatron condition,find the radius of a round orbitof an electronif the magneticinductionis known as a function of distancer from the axisof the field.Examinethis problemfor the specificcaseB ----B o ar where Bo and a are positive constants. that the strength 3.405. Usingthe betatroncondition,demonstrate of the eddy-current field has the extremum magnitudeon an equilib-

rium

orbit.

3.406.In a

\177,

betatron the magnetic induction on an equilibrium

orbit with radius r = 20cm varies during a time interval At = t.0ms at practicallyconstantrate from zero to B = 0.40T. Find the energy acquiredby the electronperrevolution. 3.407.The magneticinduction in a betatron on an equilibrium time at practically orbit of radius r varies during the acceleration constant rate from zero to B. Assuming the initial velocity of the electronto be equal to zero,find: (a) the energy acquired by the electronduring the acceleration time; distancecoveredby the electronif the acce(b) the corresponding lerationtimeis equal to \177

At.)))

x and the velocity Find the coordinate

PART FOUR

after that

moment.

v\177

of the

particlet

\177

2.40s

4.4.Find

OSCILLATIONS AND WAVES

the angular frequency and the amplitude of harmonic at distancesx1 and x2 from the equilibrium position its velocity equalsVl and v 2 respectively. A point performs harmonicoscillations along a straight line with a periodT 0.60s and an amplitudea ----t0.0cm.Find the mean velocity of the point averagedover the time interval during which it travels a distancea/2, starting from (a) the extremeposition; (b) the equilibrium position. 4.6. At the moment t 0 a point st.arts oscillatingalbng the x axis accordingto the law x a sin (or. Find: (a) the mean value of itsvelocity vector projection(v\177); (b) the modulusof the mean velocity vector l(v) I ; (c) the mean value of the velocity modulus (v) averagedover 3/8 of the periodafter the start. 4.7.A particlemoves along the x axis accordingto the law x ---a cos(ot. Find the distancethat the particlecovers during the time interval from t ----0 to t. 4.8.At the moment t 0 a particlestarts moving along the x 35 cos cm/s, axis so that its velocity projectionvaries as in seconds. Find the distancethat this particle where t is expressed covers during t 2.80s after the start. 4.9.A particleperforms harmonic oscillations along the x axis accordingto the law x a cos(or. Assuming the probability P of to \177a to be equal to the particleto fall within an interval from unity, find how the probability density dP/dx dependson x. Here dP denotesthe probability of the particlefalling within an interval from x to x dx.Plot dP/dx as a function of x. Usinggraphicalmeans,find an amplitudea of oscillations of the resultingfrom the superpositionof the following oscillations of a particleif oscillations

4.5.

\177

4.i.MECHANICAL

OSCILLATIONS

Harmonic motion equation

\342\200\242

\177'-t-

= 0,

a)o\177x

its solution:

and

x----acos

(\177)ot-t-

(4.ta)

\177),

oscillationfrequency. Damped oscillation equation and its solution:

a)o is the natural

whore \342\200\242

\"x'\177t-

where

\177

2\177 \177-

the damping

\177Oo\177X

----0,

x----aoe-I\177 cos(a)t

coefficient, a) is the frequency

\177-

(4.tb)

\302\242t),

of damped oscillations:

-----

\177

\177--

\177

V\177-\177--

$ Logarithmic damping decrement

\177.

\177'

\177.tc)

\342\200\242

and quality

factor

\177--

\177-

\177t

v\177

where

T 2n/\177. $ Forced oscillation equation \177

\177'+2\177+\177x=fo

\177

and

cos

\177t,

its steady-state solution:

x=a cos(\177t--\177),

\177

(4Ae)

where

a= \177(\177__\177)\177

\342\200\242

Maximum shift

tan +4\177 a\177

amplitude

\177. 2\177

\177=

(4.t[)

4.i0.

occurs at

Ores= \177-(4.tg) 4.i.A point--oscillatesalongthe x axisaccordingto the law x a cos a/4). Draw the approximateplots (a) displacement x, velocity projectionVx, and acceleration projectionWx as functions time t; (b) velocity projectionVx and ac\177leration projectionwx as Junctions the coordinatex. 4.2.A point moves along the x axis accordingto the law x = a sin (\177t --\177/4). Find: draw the plot x (t); (a) the amplitudeand periodel oscillations; the \177ordinate x; (b) the velocity projectionvx as a \177unction draw the plot vx (x). 4.3.A particleperforms harmonicoscillations along the x axis about the equilibriumpositionx 0. The oscillation\177requency is = 4.00s-\177. At a certainmoment of timethe particlehas a coordinate Xo = 25.0cm and its velocity is equal to = cm/s. 2\177'.

\177

(\177t

o\177

\177

o\177

\177

V\177o

\17766

\17700

--a

\177

same direction: (a) X 1 = 3.0cos (o)t -}- .\177/3), x2 =

8.0sin(o)t +

4.ti.

3.0coso)t, x2 5.0cos(o)t -t-

\177/6);

xa = 6.0sin A point participates simultaneouslyin two harmonicoscilx1 a coso)t and x2 = a cos2o)t. lations of the samedirection: Find the maximumvelocity of the point. 4.t2.The superpositionof two harmonicoscillations of the same of a point according to the law directionresultsin the oscillation x ----a cos where t is expressedin seconds.Find the cos50.0t, and the period angular frequenciesof the constituent oscillations with which they beat. A point A oscillates accordingto a certainharmoniclaw in the reference frame K' which in itsturn performs harmonic oscillations relativeto the reference frame K.Both oscillations occuralong the samedirection. at the frequency When the K' frame oscillates 20or 24 Hz, the beat frequency of the point A in the K frame turns

(b) xI

----

\177

\177/4),

\177

2.it

4.t3.

167)))

out to be equal to At what frequency of oscillation of the frame K' will the beat frequency of the point A becomeequal to A point moves in the plane xy accordingto the law x b coscot, where a, b, and co are positiveconstants. a sin cot, y

4.2t.A pendulum clockis mounted in an elevatorcarwhich starts

\177.

4.|4.

-----

2\177?

-\177

-----

Find:

of the point and the directionof (a) the trajectory w of the point as a function of itsradiusvector (b) the acceleration r relativeto the originof coordinates. 4.t5.Find the trajectoryequationy (x) of a point if it moves accordingto the following laws: asin2cot; (a) x----asincot,y (b) x ----a sin cot, y = a cos2cot. Plotthese trajectories. 4.t6.A particleof mass m is locatedin a unidimensionalpotential field where the potentialenergy of the particledependson the coordinate x as U (x) = Uo (i N cosax); Uo and a areconstants.Find the periodof smalloscillations that the particleperforms about the y (x) itsmotion along thisequation trajectory;

\177

w, with w \177 g. At a height h goingup with a constantacceleration the acceleration of the carreverses, its magnituderemainingconstant. How soon after the start of the motionwill the clockshow the right time again?

4.22.Calculatethe periodof smalloscillations of a \177

-\177

\303\227

removed from one of the endsby ----t/3 of the spring's length. Neglectingthe mass of the spring, find the periodof smalllongitudinal oscillations of the body. The force of gravity is assumedto be absent. \177

equilibriumposition.

4.|7.

Solvethe foregoingproblemif the potentialenergy has the form U (x) = a/x2 \177 b/x, where a and b are positive constants. 4.t8.Find the periodof smalloscillations in a verticalplane performed by a ball of massm ----40 g fixed at the middleof a horizontally stretchedstring l ---- m in length. The tensionof the string is assumed to be constantand equal to F i0 N. Determine of a mathematical the periodof smalloscillations ---that is a a ball thread l 20cm in length, pendulum, suspendedby if it is locatedin a liquid whose density is ----3.0timeslessthan that of the The resistanceof the is to be neglected. 4.20.A ball is suspendedby a threadliquid of length l at the point O on the wall, forming a smallangle a with the vertical(Fig.4.t).Then

t.0

-----

4.|9.

ball.

Fig. 4A.

\177l

Fig.

([\177

\177

4.3.

4.24.Determinethe periodof small longitudinaloscillations 'of a body with mass m in the system shown in Fig. The stiffness values of the springsare and The friction and the massesof the springs are negligible. 4.25.Find the periodof smallverticaloscillations of a body with mass m in the system illustratedin Fig.4.4.The stiffness values of the springsare and their massesare 4.26.A smallbody of mass m is fixed to negligible. the middleof a stretched string of length 2l.In the equilibriumpositionthe string tensionis of the equal to To.Find the angularfrequency of smalloscillations body in the transversedirection.The mass of the string is negligible, the gravitationalfield is absent.

4.3.

\303\2271

Fig. [\177

i68)))

Fig.

4.2.

the threadwith the ballwas deviatedthrough a smallangle a) and set free.Assuming the collision of the ball against the wall to be perfectly elastic, find the oscillation periodof such a pendulum.

hydrometer

(Fig.4.2) ,which was slightly pushed down in the verticaldirection. The massof the hydrometer is m 50 g, the radius of its tube is r = 3.2ram, the density of the liquid is p = i.00g/cm3. The resistance of the liquid is assumed to be negligible. 4.23.A non-deformedspring whose ends are fixed has a stiffness t3 N/re.A smallbody of mass m ----25 g is attachedat the point

\303\2272-

\303\2272,

4.4.

Fig.

4.5.

of mercury of mass 4.27.Determinethe periodof oscillations ----200g poured into a bent tube (Fig.4.5) whose right arm forms an angle 0 ----30 with the vertical.The cross-sectional area of the tube is S ----0.50cm2. The viscosity of mercury is to be neglected. m

\302\260

4.28.A

uniform

rod is placedon two spinningwheelsas shown in

Fig.4.6.The axesof the wheelsare separatedby a distancel =----20cm, 0.t8. the coefficient of frictionbetween the rod and the wheelsis k that in this casethe rod performs harmonicoscillaDemonstrate tions.Find the periodof these oscillations.

the plank when the amplitudeof oscillationof the plank becomes T ----t.0 4.33.Find the time dependenceof the angle of deviation of a mathematical pendulum 80cm in length if at the initialmoment the

less than

pendulum (a) was deviated through the angle 3.0 and then set free without push; (b) was in the equilibriumpositionand itslowerend was imparted the horizontal velocity 0.22m/s; (c) was deviated through the angle3.0and itslower end was imm/s directedtoward the equilibrium partedthe velocity 0.22 position. 4.34.A body A of massm 1 = t.00kg and a body B of mass m = 4.t0kg areinterconnected by a spring as shown in Fig.4.9.The body A performs free verticalharmonicoscillations with amplitudea t.6cm and frequency \177 co ----25 Neglectingthe mass of the spring, find the maximumand minimum values of force that this system exertson the bearingsurface. 4.35.A plank with a body of massm placed \177 on it starts moving straight up accordingto the law y ,'//////////\177 a (t coscot), where y is the \302\260

\302\260

Fig.

4.6.

4.29.Imaginea shaft going all the

poleto

way through the Earth from Earth to be a ho-

polealongitsrotationaxis.Assumingthe

mogeneousball and neglectingthe air drag, find: (a) the equationof motionof a body falling down ifito the shaft; (b) how long doesit take the body to reach the other end of the shaft; (c) the velocity of the body at the Earth'scentre. of a mathematical 4.30.Find the periodof smalloscillations pendulum of length l if itspoint of suspensionO moves relativeto the Earth'ssurface in an arbitrary directionwith a constantacceleration 2t cm, w = g/2, and the w (Fig.4.7).Calculatethat periodif l angle between the vectorsw and g equals = t20 -----

\302\260.

[\177

s-l. -----

\177q\177-

\177

displacementfrom the initial position,co Fig. 4.9. = tt Find: (a) the time dependenceof the force that the body exertson the plank if a = 4.0cm; plot this dependence; of the plank at which (b) the minimum amplitudeof oscillation the body starts falling behind the plank;

s-1.

\177

of the plank at which the body (c) the amplitudeof oscillation springsup to a height h = 50 cm relativeto the initial position(at M

Fig.

4.7.

@4.8.

Fig.

arrangementshown in Fig.4.8the sleeveM of mass is fixed between two identicalspringswhose combined

4.31.In the

ra -=

0.20kg

stiffness is equal to ----20N/re.The sleeve can slidewithout friction over a horizontalbar AB. The arrangementrotateswith a constant angular velocity co = 4.4rad/s about a verticalaxis passing through the middleof the bar. Find the periodof smalloscillations of the of the sleeve.At what valuesof co will there be no oscillations \303\227

sleeve?

on it performshorizontalharmonic 4.32.A plank with a bar placed oscillations with amplitudea = t0 cm. Find the coefficient of friction between the bar and the plank if the former startsslidingalong t70

the moment

4.36.A

t----0).

body of mass m was suspendedby a non-stretchedspring, and then sot free without push. The stiffness of the spring is Neglectingthe mass of the spring, find: of the body (a) the law of motiony (t), where y is the displacement

from the equilibriumposition; (b) the maximumand minimum tensionsof the spring in the process of motion. 4.37.A particleof mass m moves due to the force F -= amr, where is a positive constant,r is the radiusvector of the particle relativeto the originof coordinates. Find the trajectoryof itsmotion if at the initial moment r = roi and the velocity v = Voj, where i and j are the unit vectorsof the x and y axes. 4.38.A body of mass m is suspendedfrom a spring fixed to the ceilingof an elevatorcar.The stiffness of the springis At the mow. Neglecting ment t = 0 the carstartsgoingup with an acceleration the mass of the spring,find the law of motiony (t) of the body relative to the elevatorcar if y (0)= 0 and y (0)= 0. Considerthe fol-

\177z

\303\227.

lowing two cases:

t7])))

(a) w = const;

at, where a is a constant. 4.39.A body of massm = 0.50kg issuspendedfrom a rubbercord (b)

-\177

elasticitycoefficient k 50N/re.Find the maximumdistance over which the body can be pulled down for the body's oscillations to remain harmonic.What is the energy of oscillationin this case? 4.40.A body of mass m fell from a heighth onto the pan of a spring of the pan and the springarenegligible, balance(Fig. The masses the stiffnessof the latteris Havingstuckto the pan, the body starts in the verticaldirection.Find the performing harmonico.scillations amplitudeand the energy of theseoscillations.

with

\177

smallangleabout the verticalaxispassingthrough itsmiddlepoint C.The threads deviated in the processthrough an angle = 5.0 Then the rod was releasedto start performing smalloscillations. \302\242\177

\302\260.

Find:

(a) the oscillationperiod; (b) the rod'soscillationenergy.

4.i0).

\177\302\242.

4.t2. Fig. 4.13. Fig. 4.t4. 4.46.An arrangementillustratedin Fig.4.t4consistsof a horiFig.

zontal uniform discD of mass m and radius and a thin rod AO whose torsionalcoefficient is equal to k. Find the amplitudeand the if at the initial moment the energy of smalltorsionaloscillations discwas deviated through an angle from the equilibriumposition and then impartedan angular velocity oscil4.47.A uniform rod of mass m and length l performs.small lationsabout the horizontalaxispassingthrough itsupper end.Find the mean kineticenergy of the rod averagedover one oscillationperiodif at the initialmoment it was deflectedfrom the vertical by an angle 0o and then impartedan angular velocity 4.48.A physical pendulum is positionedso that its centreof gravFrom that positionthe pendulum ity is above the suspensionpoint. startedmoving toward the stableequilibriumand passedit with an angularvelocity 0). Neglectingthe friction find the periodof small /\177

\302\242Po

Fig.

4.i0.

Fig.

4.it.

4.4t.Solvethe foregoingproblemfor the caseof the pan having a mass M. Find the oscillationamplitudein this case. 4.42.A particleof massm moves in the plane xy due to the force convarying with velocity as F ----a \177j), where a is a positive At the initial the x and axes. i are the of and unit vectors stant, y j 0 the particlewas locatedat the point x = y ----0 and moment t a velocity Vo directedalongthe unit vector Find the law possessed of motionx (t), y (t) of the particle,and also the equationof itstra(\177i

-\177

jectory.

4.43.A

pendulum is constructedas a light thin-walledsphereof filled up with water and suspendedat the point O from a The distancebetween the point O and the light rigid rod (Fig. centreof the sphereis equal to How many times will the small of such a pendulum changeafter the water freezes?The oscillations viscosity of water and the changeof itsvolume on freezing are to radius/\177

4.it).

be neglected.

of a thin uniform 4.44.Find the frequency of smalloscillations verticalrod of mass m and length l hingedat the point O (Fig.4.t2). The combinedstiffness of the springsis equal to The massof the springs is negligible. 4.45.A uniform rod of massm i.5kg suspendedby two identical threads l-\177 90cm in length (Fig.4.t3)was turned through a \177\302\242.

\177

t72

(\177o-

of the pendulum. oscillations about the 4.49.A physical pendulum performs smalloscillations horizontalaxiswith frequency 0)1 t5.0S--1. When a smallbody of mass m = 50 g is fixed to the pendulum at a distancel -= 20 cm \177

below the axis, the oscillationfrequency becomesequal to ----t0.0s-1. Find the moment, of inertiaof the pendulum relative 0)\177

oscillationaxis. about physical pendulums perform small oscillations the samehorizontalaxiswith frequencieso)1 and 0)2. Their moments of inertiarelativeto the given axisare equal to I1and I2respectively. In a stateof stableequilibriumthe pendulums were fastened rigidly of the comtogether.What will be the frequencyof smalloscillations pound pendulum? 4.5t.A uniform rod of length l performs smalloscillationsabout to the rod and passingthrough))) the horizontalaxis00'perpendicular to the

4.50.Two

one of

itspoints.Find the distancebetween the centreof inertiaof

the rod and the

axis00'at which the oscillationperiodis the short-

est.What is it equal to? 4.52.A thin uniform plate shaped as an equilateraltriangle with a height h performs small oscillations about the horizontal axiscoincidingwith one of itssides. Find the oscillationperiodand the reducedlength of the given pendulum. 4.53.A smoothhorizontaldiscrotatesabout the verticalaxisO (Fig.4.i5)with a constantangularvelocity (o.A thin uniform rod AB of length l performs smalloscillations about the verticalaxisA fixed to the discat a distancea from the axisof the disc. Find the frequency (oo of these oscillations.

of the arrangepulley.Find the frequency of smalloscillations 4.56.A soliduniform cylinder of radius r rollswithout sliding alongthe insidesurface of a cylinder of radius performing small oscillations. Find their period. of the

ment.

/\177,

4.57.A soliduniform

cylinder of mass

performs smalloscilla-

tions due to the action of two springswhose combinedstiffness is in the equal to (Fig.4.t8).Find the periodof these oscillations absenceof sliding. 4.58.Two cubeswith massesrn I and m2 were interconnectedby a weightlessspring of stiffness and placedon a smooth horizontal surface.Then the cubeswere drawn closerto eachother and released simultaneously. Find the natural oscillationfrequency of the \303\227

\303\227

system.

4.59.Two ballswith massesm 1 ----t.0 kg and

slippedon a thin smooth horizontalrod

Fig.

\177

2.0kg

(Fig.4.t9).The

are

balls are

4.19.

24 N/m. The leftinterconnected by a light spring of stiffness hand ball is impartedthe initial velocity Vx = t2 cm/s.Find: (a) the oscillationfrequency of the system in the processof mo--\177

\303\227

Fig. 4.i6.

4.54.Find the frequency of smalloscillations of the arrangement illustratedin Fig.4.t6.Theradius of the pulley is R, itsmoment of inertiarelativeto the rotationaxisis I, the mass of the body is m, and the spring stiffness is The massof the thread and the spring is negligible,the thread doesnot slideover the pulley, there is no friction in the axis of the pulley. \303\227.

4.55.A

freely

uniform cylindricalpulley of mass M and radius R can

rotateabout the horizontalaxisO (Fig.4.t7).The free end of

tion;

(b) the

and the

oscillations.

energy amplitude of a system 4.60.Find the period of smalltorsionaloscillations

consistingof two discsslippedon a thin rod with torsionalcoefficient k. The momentsof inertia of the discsrelativeto the rod'saxisare equal to Ix and I2. 4.6t.A mock-upof a CO2moleculeconsistsof three ballsinterconnectedby identicallight springsand placedalonga straight line in the state of equilibrium.Such a system can freely perform oscillations of two types, as shown by the arrowsin Fig.4.20. Knowing the massesof the atoms,find the ratio of frequenciesof these oscilla-

tions.

a thread tightly wound on the pulley carriesa deadweightA. At a certainangle it counterbalances a point mass m fixed at the rim c\302\242

t74

Fig.

4.20.

Fig.

4.2i.

4.62.In a cylinder filled up with idealgas and closedfrom both areaS (Fig.4.2t). ends there is a piston of mass m cross-sectional a\177)d

175)))

In equilibriumthe piston divides the cylinder into two equal parts, eachwith volume Vo. The gaspressureis Pc.The piston was slighlty displacedfrom the equilibriumpositionand released.Find itsoscilin the gas to be adiabatic lation frequency, assumingthe processes and the friction negligible. 4.63.A smallball of massm = 2i g suspendedby an insulating thread at a height h = i2 em from a largehorizontal conducting After a chargeq had been (Fig.4.22). planeperformssmalloscillations to the oscillation the ball, periodchanged ----2.0times. imparted \177l

Find q.

(a) the angular velocity and the angular acceleration of the body at the moment t ----0; (b) the momentsof time at which the angular velocity becomes (\177

\177

maximum.

4.69.A

point performs damped

oscillations with

v\177

with frequency (o ---point performs damped oscillations ----25 1. Find the damping coefficient if at the initial moment the velocity of the point is equal to zero and its displacement from the equilibriumpositionis = t.020timeslessthan the amplitudeat

\302\256\177

\177

4.70.A

frequency

and damping coefficient accordingto the law Find the initial amplitudea o and the initialphase (z if at the(4.tb). moment t = 0 the of the point and its velocity projectionare equal to displacement = x (a) (0) 0 and vx (0) = Xo; (b) x (0) ----Xo and (0) ----O.

moment.

that

4.71.A

--\177

\177

\177l

point performs damped

oscillations with

frequency

and damping coefficient Find the velocity amplitudeof the point as a function of timet if at the momentt ----0 (a) its displacement amplitudeis equal to ao; of the point x (0) ----0 and its velocity pro(b) the displacement \177.

Fig.

4.22.

Fig.

4.23.

about an 4.64.A smallmagneticneedleperformssmalloscillations to the magneticinductionvector.On changing axis perpendicular decreased oscillation the magnetic induction the needle's = 5.0times.How much and in what way was the period magneticinduc-

\1771

tion changed?The

oscillationdamping is assumed to be negligible.

4.65.A loop(Fig.4.23)is formed by two parallelconductorscon-

nectedby a solenoidwith inductanceL and a conductingrod of mass The over the conductors. m which can freely (without friction)slide conductorsare locatedin a horizontalplane in a uniform vertical magneticfield with inductionB.The distancebetweenthe conductors is equal to l. At the moment t = 0 the rod is imparted an initialvelocity Vo directedto the right. Find the law of its motion x (t) if the electricresistanceof the loopis negligible. 4.66.A coilof inductanceL connectsthe upper endsof two vertical copperbars separatedby a distancel.A horizontalconductingconnectorof mass m starts falling with zero initial v'elocity along the bars without losingcontactwith them.The whole system is located to the in a uniform magneticfield with inductionB perpendicular the motion x of connector. the law of of bars. the Find (t) plane 4.67.A point performs damped oscillations accordingto the law x ----ace sin cot. Find: (a) the oscillationamplitudeand the velocity of the point at the moment t = 0; (b) the momentsof time at which the point reachesthe extreme -\177t

positions. 4.68.A body performs torsionaloscillations accordingto = coscot. Find: \302\242p

\177oe-\177t

the law

jection vx

(0).=

4.72.Thereare Xo. two damped oscillations with the followingperiods T and damping coefficients = t00 s-1 and T1 =-0.t0ms, T2 i0 ms, ----10s -1.Which of them decays faster? 4.73.A mathematicalpendulum oscillates in a medium for which \177:

\1771

\177

\1772

logarithmicdamping decrementis equal to = t.50.What will be the logarithmicdampingdecrementif the resistanceof the medium increasesn = 2.00times? How many times has the resistance of the medium to be increasedfor the oscillations to become the

\177,o

impossible? 4.74.A deadweight suspendedfrom a weightlessspringextendsit by Ax =- 9.8cm. What will be the oscillation periodof the dead\177veight when it is pushed slightly in the verticaldirection? Tbelogarithmicdamping decrementis equal to 3.t. 4.75.Find the quality factor of the oscillator whose displacement amplitude decreases ----2.0timesevery n = tt0 oscillations. 4.76.A particlewas displacedfrom the equilibriumposition by a distancel = t.0cm and then left alone.What is the distancethat the particlecovers in the processof oscillations tillthe complete stop,if the logarithmicdamping decrementis equal to = 0.020? 4.77. Find the quality factor cf a mathematical l =- 50 cm long if during the timeinterval v ----5.2minpendulum its total mechanical energy decreases = 4.0\"10 times. 4.78.A uniform discof radius R t3 cm can rotateabout a horizontal axisperpendicular to itsplane and passingthrough the edge of the disc. Find the periodof smalloscillations of that disc if the logarithmicdamping decrementis equal to t.00. \177,

\177

\177l

\177,

\177

\177l

\177--

\177,

\177)4\177

\177-

t77)))

thin uniform disc of mass m and radius R suspendedby an elasticthread in the horizontalplane performs torsional The moment of elasticforces emergingin the lationsin a

4.79.A

oscil-

liquid.

thread is equal to N

\177

(z(\177,

where a is a constantand

(\177

is the angle

force acting The resistance rotationfrom the equilibriumposition. is a constant where to is the disc of area F1 on a unit equal of

-----

\177lV,

\1771

and v is the velocity of the given elementof the discrelativeto the frequency of small oscillation. liquid.Find the A of 4.80.A disc radius R suspended by an elasticthread between two stationary planes (Fig. 4.24) performs torsionaloscillations about its axis OO'.The moment of inertia of the discrelativeto the clearancebetween the disc and each of that axis is equal to the planes is equal to h, with h \177 R. Find the viscosity of the gas period of the discequals T surroundingthe discA if the oscilla.tion decrement, and the logarithmic damping

\177.

4.84.A particleof mass m can perform undaml\177ed harmonicoscillationsdue to an electricforce with coefficient k. When the particle was in equilibrium, a permanentforce Fwas appliedto it for seconds.Find the oscillation amplitude that the particleacquired after the action of the force ceased.Draw the approximateplot x (t) of oscillations. cases. possible 4.85.A ball of massInvestigate m when suspendedby a spring stretchesthe latter by Al. Due to externalverticalforce varying accordingto a harmoniclaw with amplitude Fo the ball performs forced oscillations. logarithmicdamping decrementis equal to Neglecting the mass of the spring, find the angular frequency of the external force at which the displacement amplitudeof the ball is maximum. \177

\177[he

What

\177.

is the

of that

amplitude? 4.86.The magnitude forced harmonicoscillations have

equal displacement s -1 amplitudesat frequenciescol = 400 s-1 and co2----600 Find the resonancefrequency at which the displacement amplitude is maximum. 4.87.Thevelocity amplitudeof a particleis equalto half the maximum value at the frequenciescol and co2 of externalharmonicforce. Find: (a) the frequency correspondingto the velocity resonance; (b) the dampingcoefficient and the damped oscillation frequency co of the particle. 4.88.A certainresonancecurve describesa mechanicaloscillatt.60.For ing system with logarithmicdamping decrementk this curve find the ratio of the maximumdisplacement amplitude to the displacement amplitudeat a very low = F0 coscot a body 4.89.Due to the externalverticalforce Fxfrequency. a forced oscillations accordsuspendedby springperforms steady-state a cos(a)t -Find the work performed by ing to the law x the forceF during one oscillation 4.90.A ball of mass m.----50g is period. suspendedby a weightlessspring with stiffness \303\227-----20.0 N/re. Due to externalvertical harmonic force with frequency co ----25.0s the ball performs steady-state oscillations with amplitude a i.3cm. In this casethe displace3 ment of the ball lags in phase behind the externalforce by :\177u. Find: \177

g Fig.

4.24.

Fig.

4.25.

conductorin the shape of a squareframe with sidea suspended by an elasticthread is locatedin a uniform horizontalmagnetic field with inductionB. In equilibriumthe plane of the frame Having been displacedfrom is parallelto the vector B (Fig.4.25). about the frame performssmalloscillations the equilibriumposition, inertia of of The moment centre. its a vertical axispassingthrough R. i s resistance electric its to is that axis I, to relative the frame equal interval after Neglectingthe inductanceof the frame, find the time e-fold. decreases deviation angle which the amplitudeof the frame's with = horizontal a on 0.50 m plane of mass A bar 4.82. kg lying by means of a friction coefficient k ----0.t0is attached to the a horizontal non-deformed spring. The stiffness of the spring is so The barwas displaced equalto == 2.45N/cm, its massis negligible. Find: that the springwas stretchedby x0 ----3.0cm,and then released. (a) the periodof oscillationof the bar; that the bar performs until it (b) the total number of oscillations

4.8i.A

\177vall

\177

(\177).

-\177

\177-

(\177

(a) the quality factor of the given oscillator; (b) the work performedby the externalforce during one oscillation

\303\227

stopscompletely. oscilla4.83.A ball of mass m can perform undamped harmonic moAt the natural 0 with x coo. frequency the about tions point a force Fx Fo cos cot ment t = 0,when the ball was in equilibrium, to it. Find the law of forced -----

\177-

coincidingwith the x axiswas applied

oscillationx (t)

for that

ball.

period.

4.91.A ball of mass m suspendedby

a weightlessspring canperverticaloscillations with damping coefficient The natural oscillation is to a) Due to externalvertical the frequency equal 0. force varying as F F0 cosa)t the ball performs steady-state harmonic oscillations. Find: the mean (a) power (P),developedby the force F, averagedover one oscillationperiod; form

-----

t2.))) \17778

\177.

(b) the frequency co of the force F at which (P) is maximum;what is (P)maxequal to? 4.92.An externalharmonicforceF whosefrequency can be varied,

amplitudemaintainedconstant, acts in a verticaldirectionon ballsuspendedby a weightlessspring.The damping coefficient is timesless than the natural oscillationfrequency coo of the ball. How much, in per cent,doesthe mean power (P) developedby the resonancediffer from the force F at the frequency of displacement maximum mean power (P)ma:\177? Averaging is performed over one oscillationperiod. 4.93.A uniform horizontaldisc fixed at its centreto an elastic due to the moment verticalrod performsforced torsionaloscillations of forces Nz = Nrn coscot. The oscillations obey the law q) = = cos(cot -- Find: (a) the work performed by friction forces acting on the discduring one oscillationperiod; (b) the quality factor of the given oscillatorif the moment of inertia of the discrelativeto the axis is equal to I. with

\177q

\302\242z).

\302\242Prn

4.2.ELECTRICOSCILLATIONS \342\200\242

Damped oscillation in a circuit

q=qme-fit cos

where

(o= 1/\177-\177 ,

\1770=

\177

$ Logarithmic damping decrement and Eqs.(4.1d).When damping is low: \177

quality

\177

(4.2a)

factor Q of a circuit am

defined by

4.94.Due to a certaincausethe free electronsin a plane copper plate shifted over a smalldistancex at right anglesto itssurface. As a result, a surface charge and a corresponding restoring force Find the emerged,giving rise to so-calledplasma oscillations. if the free electron concentangularfrequency of these oscillations ration in copperis n = 0.85.t0m -1. 4.95.An oscillatingcircuitconsistingof a capacitorwith capacitanceC and a coilof inductanceL maintains free undamped oscillations with voltage amplitudeacrossthe capacitorequal to For an arbitrary moment of time find the relationbetween the current I in the circuitand the voltage V acrossthe capacitor.Solve this problemusingOhm'slaw and then the energy conservationlaw. 4.96.An oscillatingcircuitconsistsof a capacitorwith capacitance C, a coilof inductanceL with negligibleresistance,and a the capacitorwas chargedto switch. With the switch disconnected, a voltage Vrn and then at the momentt = 0 the switch was closed. Find: (a) the currentI (t) in the circuitas a function of time; \1779

(b) the emf of self-inductancein the

\177.5

\177F

Steady-stateforcedoscillationin a circuit with a connectedin series: \177

I = Imcos((ot-

where Irn

(4.2c)

\177

C\177

t) i

(b) the peak value of the current flowing through the

coil.

(4.2d)

\177_2\177 of curren[ \177,q?Z

tan

Fig.

\177

Fig.

4.26.

The correaponding vector diagram for voltagesis shown in Fig. 4.26. Power generated in an ac circuit: P = VI cos T, (4.2e) where V and I are the effective values of voltage and current: I = Irn/lf'\177. V = Vm/I/'Z (4.2f) 180

\177F.

\177

V/-R\177-(o\177L

\342\200\242

coilat the momentswhen the

electricenergy of the capacitoris equal to that of the current in the coil. 4.97.In an oscillatingcircuitconsistingof a parallel-platecapacitor and an inductancecoilwith negligibleactive resistancethe with energy W aresustained.The capacitorplateswere oscillations slowly drawn apart to increasethe oscillationfrequency \177q-fold. What work was done in the process? 4.98.In an oscillatingcircuitshownin Fig.4.27the coilinductance is equal to L = mH and the capacitorhave capacitances C1 -------2.0 and 3.0 The capacitorswere chargedto a voltage Find: V = i80V, and then the switch Sw was closed. (a) the natural oscillationfrequency;

4.27.

Fig.

4.28.

4.99.An electriccircuitshown in Fig.4.28has a negligiblysmall was chargedto a voltageVo The left-hand capacitor activeresistance. and then at the moment

-----

0 the

Find the switch Sw was closed.

time dependenceof the voltagesin left and right capacitors. 4.t00.An oscillatingcircuitconsistsof an inductancecoilL and C.The resistanceof the coiland the lead a capacitor with capacitance t8t)))

The coilis placedin a permanent magneticfield wires is negligible. sothat the total flux passingthroughall the turns of the coilis equa} to (I). At the moment t = 0 the magnetic field was switched off. Assuming the switching off time to be negligiblecomparedto the natural oscillation periodof the circuit,find the circuitcurrent as a function of

time t.

4o10|.The free

are maintained in a circuit, damped oscillations such that the voltageacrossthe capacitorvariesas V = Vrne cos Find the momentsof time when the modulusof the voltage across -\177t

the

capacitorreaches

(a) peak values; (b) maximum(extremum)values.

4o102. A certainoscillatingcircuitconsistsof a capacitorwith capacitanceC, a coilwith inductanceL and active resistance the capacitorwas and a switch. When the switch was disconnected, set in.Find the charged;then the switch was closedand oscillations ratio of the voltage acrossthe capacitorto its peakvalue at the after closingthe switch. moment immediately A circuitwith capacitance C and inductanceL generates free damped oscillations with current varying with time as I = = Irne-\177 t sin o)t. Find the voltageacrossthe capacitoras a function of time, and in particular,at the moment t = 0. An oscillating circuitconsistsof a capacitorwith capacitance C----4.0 and a coilwith inductanceL = 2.0mH and active resistanceR = l0 Find the ratio of the energy of the coil's electricfield at the moment magneticfield to that of the capacitor's 4.|03\302\260

4.i04\302\260

\177F

\177.

when the current has the maximum value. An oscillatingcircuitconsistsof 4.105o

is/\177

= 1.0

\177.

Find the energy of

certainmoment

the switch Sw was disconnected.

oscillations in the circuit

immediatelyafter the switch was disconnected; = 0.30s after switch was disconnected. 4.110. are inducedin a circuitwhose quality Damped oscillations factor is Q -- 50 and natural oscillation frequency is = 5.5kHz. How soon will the energy storedin the circuit decrease = 2.0 (a)

(b) t

th\177

\1770

\177i

times?

4.ill.

oscillatingcircuit incorporatesa leaking capacitor. C and active resistanceto R. The coil inductanceis L. The resistanceof the coiland the wires is negligible. Find: frequency of such a circuit; (a) the damped oscillation its factor. (b) quality C 4.i12. Find the quality factor of a circuitwith capacitance = 2.0 and inductance L 5.0mH if the maintenanceof undamped oscillations in the circuit with the voltageamplitudeacrossthe capacitorbeing equal to V,n 1.0Vrequires a power (P)---= 0.10roW. The dampingof oscillations is sufficiently low. 4.113. What mean power should be fed to an oscillatingcircuit with active resistance 0.45 to maintain undamped harmonic An

Its capacitanceis equal to

\177-

\177

\177F

\177

/\177

\177

= 30mA? oscillations with current amplitude 4.ii4.An oscillatingcircuitconsists of a capacitorwith capacitanceC 1.2nF and a coil with inductanceL = 6.0 and active resistance = 0.50 What mean power should be fed to with volthe circuitto maintain undamped harmonicoscillations I\177n

\177

\177H

/\177

\1772.

= l0V? tage amplitude acrossthe capacitorbeing equal to circuit shown of the 4.115. Find the damped oscillation frequency in Fig.4.30. The capacitance C, inductanceL, and active resistance/\177 V\177n

coilsconnectedin serieswhose inductancesare L1 and L2, active resistancesare and and mutual inductanceis negligible.Thesecoilsare to be two

/\1772,

replacedby one, keeping the frequency and the quality factor of of the circuitconstant.Find the inductanceand the active resistance such a coil. 4.106.How soon does the current ampli\177tude in an oscillating

circuitwith quality factor Q = 5000decrease = 2.0timesif the oscillationfrequency is = 2.2MHz? 4.107.An oscillatingcircuitconsistsof capacit\177ceC l0 1.0 How many inductanceL = 25 mH,and activeresistanceR oscillation periodsdoesit takefor the current \177 amplitudeto decreasee-fold? How much (in per cent)doesthe free 4.108. oscillation frequency o) of a circuitwith quality factor Q = 5.0differ from the natural T5 \177 t of that circuit? oscillation Fig. 4.29. 4.i09.Infrequency a circuit shown in Fig.4.29the battery emf is equal to $ = 2.0V, its internal resistanceis r 9.0 the capacitanceof the capacitoris C = l0 the coilinductanceis L = 100mH, and the resistance

are supposed to be known.Find how must C,L, and be interrelated to make

l=l

oscillations possible.

\177l

\177

\177--

\177.

o)\177

\177

\177F,

t82

\177,

\177F,

Fig.

(a)

4.30.

4.i16.There are two

Fig.

4.3i.

oscillatingcircuits(Fig. 4.3i)withcapaci-

tors of equalcapacitances. How must inductancesand active resistancesof the coilsbe interrelatedfor the frequenciesand damping of free oscillations in both circuitsto be equal? The mutual inductance of coilsin the left circuitis negligible. A circuitconsistsof a capacitorwith capacitance 4.117. C and a coilof inductanceL connectedin series,as well as a switch and a

resistanceequalto the criticalvalue

for this circuit.With the switch)))

the capacitorwas chargedto a voltage V o, and at the disconnected, moment t ----0 the switch was closed. Find the currentI in the circuit as a function of timet. What is I,\177ax equal to? A coilwith active resistance and inductanceL was con4.118. nectedat the moment t ----0 to a sourceof voltage V ----V,n coscot. Find the current in the coilas a function of timet. A circuitconsistingof a capacitor 4.119. with capacitance C and a resistance connectedin serieswas connectedat the moment t 0 to a sourceof ac voltage V V,n coscot. Find the current in the circuit as a function of time t. 4.120. A long one-layersolenoid tightly wound of wire with resistivity p has n turns per unit length. The thicknessof the wire insulation is negligible.The cross-sectional radius of the solenoid is equal to a.Find the phasedifferencebetween current and alternating voltagefed to the solenoidwith frequency 5.121. A circuitconsistingof a capacitor and an active resistance ll0 connected in seriesis fed an alternating voltage with amplitude ----ll0 V. In this casethe amplitudeof steady-state ----0.50A. Find the phase differencebetween current is equal to the current and the voltage fed. 4.i22.Fig.4.32illustratesthe simplestripple filter. A voltage V V o (l coscot) is fed to the left input. Find:

resistanceis connectedto an oscillatorwhose frequency can be varied without changing the voltage amplitude.Find the frequency at which the voltageamplitudeis maximum (a)

/\177

\177

/\177

\177--

\177.

\177

V\177n

I\177n

\177-

\303\267

(a) the output voltage V' (t); (b) the magnitudeof the product/\177C at which the output amplitude of alternating voltage componentis = 7.0 timesless than the directvoltagecomponent,if co 314

s-l.

\177l

\177

Fig.

4.32.

L,,\177

\177

Fig.

L,R

4.33.

4.123.Draw

the approximatevoltage vector diagrams in the electriccircuitsshown in Fig.4.33a, b. The externalvoltage V is assumed to be alternating harmonicallywith frequency co. 4.124. A seriescircuitconsistingof a capacitorwith capacitance C 22 and a coilwith active resistance 20 and inductance L 0.35H is connectedto a sourceof alternating voltage \177

/\177

\177F

with amplitude

V,\177

\177

180V

\177-

s-l. Find:

and frequency co ----314

(a) the current amplitudein the circuit; (b) the phase differencebetween the current and the

tage;

\177

externalvol-

(c) the amplitudesof voltageacrossthe capacitorand the coil. A seriescircuitconsistingof a capacitor 4.125. with capacitance C, a resistanceR, and a coilwith inductanceL and negligibleactive 184

acrossthe capacitor;

(b) acrossthe

coil.

alternating voltage with frequency co----314 180V is fed to a seriescircuitconsistingof and amplitude 40 and induca capacitorand a coilwith active resistance will tance L 0.36H.At what value of the capacitor'scapacitance the voltageamplitude acrossthe coilbe maximum? What is this amplitude equal to? What is the correspondingvoltageamplitude acrossthe condenser?

4.126.An

s-\177

V\177n

\177-

/\177

\177

--\177

C whose interelectrode with capacitance A capacitor 4.127. space is filled up with poorly conductingmedium with activeresistanceR is connectedto a sourceof alternating voltage V Vm coscot. Find the timedependenceof the steady-state currentflowing in lead wires.The resistanceof the wires is to be neglected. C circuitconsistsof a capacitorof capacitance An oscillating 4.128. and a solenoidwith inductanceL1.The solenoidis inductively con\177

coilhaving an inductance and a neglinectedwith a short-circuited Their mutual inductancecoefficient is equal resistance. gibleactive Find the natural frequency of the given oscillatingcirto L\177

Li\177.

cuit. Find the 4.129.

quality factor of an oscillatingcircuitconnected seriesto a sourceof alternatingemf if at resonancethe voltageacross the capacitoris n timest.hat of the source. circuitconsistingof a coiland a capacitor An oscillating 4.130. connectedin seriesis fed an\" alternating emf, with coilinductance beingchosento provide the maximumcurrentin the circuit.Find the quality factor of the system, provided an n-fold increaseof inductance resultsin an \177l-fold decreaseof the currentin the circuit. A seriescircuitconsistingof a capacitorand a coilwith 4.131. to a sourceof harmonicvoltagewhose active resistanceis connected the voltage amplitudeconstant. be can varied, keeping frequency the current amplitudesare n timesless At frequenciescol and than the resonanceamplitude.Find: (a) the resonancefrequency; (b) the quality factor of the circuit. 4.132. Demonstratethat at low damping the quality factor Q is approximately equal of a circuitmaintainingforced oscillations to coo/A(o, where coo is the natural oscillationfrequency, A(o is the curve I (co)at the \"height\" which is V times width of the resonance lessthan the resonancecurrent amplitude. and a coilconnectedin A circuitconsistingof a capacitor 4.133. in

(o\177

\177

serids is fed two alternatingvoltagesof equal amplitudesbut diffeThe frequency of one voltageis equal to the natural rent frequencies. oscillationfrequency (coo)of the circuit,the frequency of the other voltage is timeshigher. Find the ratio of the current amplitudes \177l

t85)))

(Io/I)generatedby the two voltagesif the quality factor of the system is equal to Q.Calculatethis ratio for Q l0 and 100,if = 1.10. 4.134. It takest o hours for a directcurrentIo to chargea storage battery. How long will it take to chargesuch a battery from the mains using a half-wave rectifier,if the egectivecurrent value is alsoequal to 4.i35.Find the effective value of current if its mean value is I -\177

\1771

and its time dependence is (a) shown in Fig.4.34;

(b)

\177, I

4.142.A coiland

an inductance-freeresistanceR

\177

are

connectedin parallelto the acmains.Find the heat power generated A is drawn from the mains. in the provided a current ---A and and the resistanceR carry currents = The \177

coil coil 0.60A respectively.

4.t43.An

0.90

----.

0.50 frequency (o ----314 I\177

alternating current of

I\177

is fed

s-\177

73 C to a circuitconsistingof a capacitorof capacitance = the F ind in connected R 100 resistance an active parallel. dance of the circuit. \177

t\177F

\177

and

impe-

4.144.Draw the approximatevector diagramsof currents in theA ']?he voltage appliedacrossthe points circuitsshown in Fig.4.35. of eachcircuitare the parameters and B is assumed to be sinusoidal; the external behind in I so chosenthat the total current o lags phase

sin

voltageby an angle %

Fig.4.34.

4.136.

A solenoid with inductanceL ----7 mH and activeresistance R ----44 is first connectedto a sourceof directvoltageV o and then to a sourceof sinusoidal voltage with effective value V ----V o. At what frequency of the oscillatorwill the power consumedby the solenoidbe ----5.0timeslessthan in the former case? \177

4.137. A coilwith inductiveresistance XL = 30 and impedance Z----50 is connectedto the mains with effective voltage value V = 100 V. Find the phase difference between the current and the voltage,as well as the heat power generatedin the coil. \177l

\1772

\177

A coilwith 4.138.

inductanceL

----0.70H and

active resistance

20 is connectedin serieswith an inductance-freeresistanceR. An alternatingvoltagewith effective value V = 220 V and frequency ----314s-1 is appliedacrossthe terminalsof this circuit.At what r ----

\177

a\177

value of the resistance R will the maximumheat power be generated in the circuit?What is it equal to? A circuitconsistingof a capacitorand a coilin seriesis connectedto the mains.Varying the capacitanceof the capacitor, the heat power generatedin the coilwas increased n = 1.7times. How much (in percent)was the value of cos changedin the process? 4.t40.A sourceof sinusoidalemf with constant voltage is connectedin serieswith an oscillatingcircuitwith quality factorQ 100.At a certainfrequency of the externalvoltagethe heat power generatedin the circuitreachesthe maximumvalue. How much the power (in per cent) should this frequency be shifted to decrease generated n ----2.0times? A seriescircuitconsistingof an inductance-free resistance R 0.16k\177 and a coilwith active resistanceis corinectedto the mains with effective voltage V = 220V. Find the heat power generated in the coilif the effective voltagevalues acrossthe resistance R = 180V respectively. and the coilare equal to V1 = 80 V and

4.139.

\177

4.141. \177

V\177

186

(a)

(c\177

Fig.

4.35.

and a coilwith C ----t.0 A capacitorwith capacitance 4.145. = 1.0 mH are conL inductance and active resistanceR 0.10 = Find: t\177F

\177

nectedin parallelto a sourceof sinusoidalvoltage V 3t V. sets in; (a) the frequency at which the resonance in current resonance,as well as the fed of (b) the effective value currentsflowing through the coiland through the the corresponding capacitor. C and a coilwith active resiswith capacitance A capacitor 4.146. tance R and inductanceL are connectedin parallelto a sourceof sinusoidalvoltageof frequency Find the phasedifference between the current fed to the circuitand the sourcevoltage. capacitanceC and 4.147.A circuitconsistsof a capacitorwith connected in paralL R and inductance a coilwith active resistance of (o at circuit, alternating the o f frequency lel.Find the impedance voltage. A ring thin wire with activeresistance/\177and inductanceL 4.148. rotateswith constant angular velocity in the externaluniform to the rotationaxis. I'n the process,the magneticfield perpendicular the ring varies with flux of magneticinductionof externalfield across a\177

a\177.

o\177

a\177

time as

(1) ----

cos

Demonstratethat

as I---(a) the inductive current in the ring varies with time

Imsin(o\177t

eLC;

(D\177

o\177t.

\177 \177),

where

Im= O)\177o/WR

\177

\177\"

o\177L

with

tan(\177----

externalforcesto (b) the mean mechanicalpower developed by -P +))) formula the is defined rotation maintain by \177

\177/ao)\177\302\242P\177oR/(R

4.i49.A

wooden core(Fig.4.36)supports two

coils:coil1 with

inductanceL1 and short-circuited coil 2 with active resistanceR and inductanceL2. The mutual inductanceof the coilsdependson

4.i50.How long will it takesound waves to travel the distancel between the points A and B if the air temperaturebetween them varies linearly from T1 to T27 The velocity of sound propagationin is a constant. air is equal to v V T, where 4.15i.A plane harmonic wave with frequency co propagatesat a velocity v in a directionforming angles 6, ? with the x, y, z axes. Find the phase difference between the oscillations at the points of medium with coordinates xl, Yl, zl and y2, 4.i52.A plane wave of frequency co propagatesso that a certain phase of oscillationmoves along the x, y, z axeswith velocities Find the wave vectork, assumingthe unit vectors v2, va respectively. of the coordinate axesto be assigned. ex, \177

\177z

\177,

x\177,

Fig.

v\177,

4.86.

the dist.ancex between them according to the law Lt2 (x). Find the mean (averaged over time) value of the interactionforce between the coilswhen coil carriesan alternating current ---- o coscot.

4.3.ELASTICWAVES.

z\177.

11 I

4.I53.A e\177

e\177,

(cot-

a cos plane elasticwave kx) propagatesin a medium K.Find the equationof this wave in a reference frame K' moving in the positive directionof the x axis with a constant veoblocity V relativeto the medium K. Investigate the expression \177

\177

tained.

ACOUSTICS

4.i54.Demonstratethat any differentiable function /(t ax), where is a constant,provides a solutionof wave equation.What is the physical meaningof the constant The equationof a travellingplane sound wave has the form 4.155. = 60cos (t800t t 5.3x),where is expressedin micrometres, in seconds, and x in metres.Find: amplitude,with which the par(a) the ratio of the displacement ticles of medium oscillate, to the wavelength; (b) the velocity oscillationamplitudeof particlesof the medium and itsratioto the wave propagationvelocity; amplitudeof relativedeformation of the medium (c)the oscillation and its relationto the velocity oscillationamplitudeof particlesof \303\267

\342\200\242

of plane

Equations

----\177

spherical waves:

and

a cos(o\177t-- kx),

\177

\177z

= aor cos

(o\177t--

kr).

(4.3a)

case of a homogeneous absorbing medium the factorse and e resis the wave damping coefficient. pectively appear in the formulas, where Q Wave equation: In

the

-\177x

-\177r

\177

0\177

4\177

Ox\177

0\177 \"

0\177

-- t

Oz\177

Oy\177

a Phasevelocity of longitudinal verse waves in a string (vz):

0\177

waves in

(4.3b)

Ot\177

v\177

elasticmedium

(vii)

and

trans-

\177

the

where E is Young's modulus, p is the density of the medium, the string, is its linear density. a Volume density of energy of an elastic wave:

T is the

tension

\1771

= palm

Ener\177 flow

Standing

sin (rot kx), (w) 1/2pa2\177. or the Umov vector for a density, \177

wave

\177

a Acoustical Dopplereffect:

188

= Vo

(Ap)m:

(4.3f)

\177.

O\177/Ot,

O\177/Ox

(ap)\177/2pv.

\177

\1771

v \177 rob

a Loudnesslevel (in bels): L = log (I/Io). a Relation between the intensity I of a sound wave amplitude

\177

= a cos kx.cos mr.

\177

= a cos(cot --kx) propagatesin a homogeneouselasticmedium.For the moment t = 0 draw and vs x; (a) the plotsof (b) the velocity directionof the particlesof the medium at the points where ----0,for the casesof longitudinaland transversewaves; plot of density distributionp (x)of the medium (c)the approximate for the caseof longitudinalwaves. = ae-\177 4.157. A plane elasticwave in a homogeneous medium.Find co, and k are constants,propagates the phase difference between the oscillations at the points where the particles'displacement amplitudes differ by = t.0%, if = ----0.52m -1and the wavelength is = 50 cm. 4.158. Find the radiusvectordefining the positionof a point source of spherical waves if that sourceis known to be locatedon the straight line between the points with radius vectorsrl and at which the oscillationamplitudes of particlesof the medium are equal to and a2.The damping of the wave is negligible, the medium is homogeneous. plane wave

\177

wave:

equation: \177

lation

(4.3d) travelling

medium.

4.I56.A

\177,

\177

(4.3g)

and the

(4.3h) pressureoscil(4.3i)

\177

r\177

i89)))

4.i59.A point with isotropicsourcegeneratessound oscillations -- i.45 ktiz.At a distancero 5.0m from the source

frequency

\177

the displacement amplit\177de of particlesof the medium is equal to a o -- 50 \177tm, and at the point A locatedat a distancer t0.0m from the sourcethe displacement 3.0timesless amplitude is \177

--\177

than

ao. Find:

\177l

\177h

(a) the damping coefficient ? of the wave; (b) the velocity oscillationamplitudeof particlesof the medium at the point A. Two plane waves propagatein a homogeneous elasticmedium, one along the x axis and the other along the y axis: a cos(o)t a cos(o)t kx), ky). Find the wave motion pattern of particlesin the plane xy if both waves (a) are transverseand their oscillationdirections coincide;

4.|60.

\177

-\177-

\177

are

longitudinal. (b) A plane undamped harmonicwave propagates 4.i6i. in a medium.

Find the mean spacedensity of the total oscillationenergy if at any point of the medium the spacedensity of energy becomes

w o one-sixthof an oscillation periodafter passingthe displacementmaximum. A point isotropic 4.162. sound sourceis locatedon the perpendicular to the plane of a ring drawn through the centreO of the ring. The distancebetween the point O and the sourceis l ----|.00m, the radius of the ring is/\177 0.50m. Find the mean energy flow across the areaenclosed by the ring if at the point O the intensity of sound is equal to Io = 30\177W/m The damping of the waves is negligible. 4.i63.A point isotropicsourcewith sonicpower P 0.i0W is locatedat the centreof a round hollow cylinder with radius --- i.0m and heighth ----2.0m. Assumingthe sound to be completely absorbed by the walls of the cylinder, find the mean energy flow reachingthe lateral surface of the cylinder.

equal to

\177

\177.

-----

/\177

4.164. The

equation of a plane standing wave in a homogeneous form a coskx.cos (ot. Plot: as functions of x at the momentst 0 and t T/2,

elasticmedium has the

--\177

\177

(a) and O\177/Ox where T is the oscillationperiod; (b) the distributionof density p (x) of the medium at the moments \177--

\177

0 and t T/2 in the caseof longitudinaloscillations; (c) the velocity distributionof particlesof the medium at the moment t T/4; indicatethe directionsof velocitiesat the antinodes, both for longitudinal and transverse oscillations. A longitudinalstandingwave 4.i65. a coskx.cos (or is maintained in a homogeneous medium of density p. Find the expressions t

\177

-\177

\177

for the

spacedensity of

(a) potentialenergy wp (x, t); (b) kinetic energy (x, t). Plot the spacedensity distributionof the total energy w in the space between the displacement nodesat the momentst 0 and t T/4, w\177

\177--

where T is the oscillationperiod. 190)))

4.i66.A string i20cm in length sustains a standingwave, with the pointsof the string at which the displacement amplitudeis equal cm.Find the maximumdisplaceto 3.5mm beingseparatedby 15.0 ment amplitude.To which overtone do theseoscillations correspond? 4.i67.Find the ratio of the fundamental tone frequenciesof two 2.0%and identicalstrings after one of them was stretchedby the other,by ----4.0%.Thetensionis assumed to be proportional

-\177

\177

\177l\177

elongation. Determinein what 4.i68.

to the

way and how many

timeswill the

fun-

damentaltone frequency of a stretchedwire changeif its length is shortenedby 35% and the tensionincreased by 70%. 4.169.To determinethe sound propagationvelocity in air by acousticresonancetechniqueone can use a pipe with a piston and a sonicmembraneclosingone of its ends.Find the velocity of sound if the distancebetween the adjacentpositionsof the piston at which 2000Hz is equal to 1 resonanceis observed at a frequency =8.5cm. of air coloscillations 4.i70.Find the number of possiblenatural---i250Hz.The length liebelow umn in a pipe whosefrequencies 85 cm. The velocity of sound is v = 340 m/s. of the pipe is l Considerthe two cases: (a) the pipe is closedfrom one end; (b) the pipe is openedfrom both ends. The open ends of the pipe are assumed to be the antinodesof dis\177

\177

-\177

\177o

\177

placement. 4.i7i.A copperrod of lengthl 50 cm is clampedat its midpoint. of the rod in Find the number of natural longitudinaloscillations the frequency rangefrom 20 to 50kHz.What are those frequencies --\177

equal to? 4.172.A string of mass m is fixed at both ends.The fundamental are excitedwith circularfrequency (o and maximum tone oscillations amplitudea,\177. Find: displacement (a) the maximum kineticenergy of the string; (b) the mean kineticenergy of the string averaged over one oscillation period. o)t is maintained in a a sin kx.cos 4.i73.A standing wave areaS and density p.Find the rod with cross-sectional homogeneous totalmechanical energy confined between the sectionscorresponding nodes. to the adjacent displacement i000Hz with frequency v o 4.174. A sourceof sonicoscillations 0.17m/s. moves at right angles to the wall with a velocity u arelocatedon a straight line, Two stationary receivers and coincidingwith the trajectoryof the source,in the following succession:/\177-source-R\177-wall. Which receiverregistersthe beatings and what is the beat frequency? The velocity of sound is equal to v \177

\177

R\177

= 340 m/s.

4.t75.A

tuning

/\177e

\177

from two stationary observerreceivessonicoscillations and the other recedeswith forks one of which approaches,

the same velocity.As this takesplace,the observerhearsthe beatings frequency v ----2.0Hz. Find the velocity of each tuning fork if Cheir oscillation frequencyis v o ----680Hz and the velocity of sound in air is v 340 m/s. 4.i76.A receiverand a sourceof sonicoscillations of frequency Hz arelocatedon the x axis. The sourceswings vo ----2000 harmonically alongthat axiswith a circularfrequency (o and an amplitude a----50 \177m. At what value of o) will the frequency bandwidth registeredby the stationary receiverbe equalto Av 200Hz?The velocity of sound is equal to v ----340 m/s. 4.177.A sourceof sonicoscillations with frequency Vo ----i700Hz and a receiverare locatedat the same point.At the moment t ----0 the sourcestarts receding from the receiverwith constant acceleration w = 10.0 m/s Assuming the velocity of sound to be equal to v = 340m/s, find the oscillation frequency registered by the stations after the start of motion. ary receivert ----10.0 4.178.A sourceof sound with natural frequency t.8 kHz moves uniformly along a straight line separatedfrom a stationary observer by a distancel ----250m. The velocity of the sourceis equal to ----0.80fraction of the velocity of sound.Find: (a) the frequency of sound receivedby the observerat the moment when the sourcegets closestto him; (b) the distancebetweenthe sourceand the observerat the moment when the observerreceives a frequency ---4.i79.A stationary source sends forth monochromatic sound. A wall approaches it with velocity u = 33cm/s. The propagation velocity of sound in the medium is v ----330m/s. In what way and how much, in per cent,doesthe wavelength of sound changeon reflection from the wall? 4.180. A sourceof sonicoscillations with frequency Vo ----1700Hz and a receiverare locatedon the samenormal to a wall. Both the sourceand the receiverare stationary, and the wall recedesfrom the sourcewith velocity u ----6.0cm/s.Find the beat frequencyregistered by the receiver.The velocity of sound is equ,al to'.v ----340 m/so Find the damping coefficient 7 of a sound wave if at distancesrl t0m and ----20 m from a point isotropicsourceof sound the sound wave intensity values differ by a factor 4.182. A plane sound wave propagatesalongthe xaxis.The dampm At the point x ----0 ing coefficient of the wave is ? 0.0230 the loudnesslevel is L ----60 dB.Find: (a) the loudnesslevel at a point with coordinatex = 50 m; x of the point at which the sound is not heard (b) the coordinate any more. At a distance ro 20.0m from a point isotropic source of sound the loudnesslevel L o 30.0dB.Neglectingthe damping of the sound wave, find: m from the source; (a) the loudnesslevel at a distancer = 10.0 (b) the distancefrom the sourceat which the sound is not heard. with

\177

\177.

\177o

\177

\1771

\177

\177o.

4.i81. \177

r\177

\177

\1771

\177

4.i83.

\177

-1.

4.i84.An observerA locatedat a distancerA m from a ringingtuning fork notesthe sound to fade away = 19 s later than an observerB who is located at a dis[ance = 50 m from the tuning fork. Find the damping coefficient of oscillations of the tuning fork. The sound velocity v = 340 rots. \177---5,0

\177

rt\177

[\177

4.i85.A

plane longitudinalharmonicwave propagatesin a me-

dium with density p. The velocity of the wave propagationis v. Assumingthat the density variationsof the medium,inducedby the propagating wave, Ap \177 p, demonstratethat

(a) the pressure increment in the medium Ap------pv (O\177/Ox), \177

where 8\177/Ox is the relativedeformation; (b) the wave intensity is defined by Eq. (4.3i). A ball of radius R ----50 cm is locatedin the way of pro4.i86. pagationof a planesound wave. The sonicwavelength is ----20cm, the frequency is v ----1700Hz, the pressureoscillation amplitude in air is(Ap)m 3.5Pa.Find the mean energy flow, averaged over an oscillation period,reachingthe surface of the ball. 4.i87.A point A is locatedat a distancer ----1.5m from a point 600Hz. The sonicpower isotropicsourceof sound of frequency of the sourceis P 0.80W. Neglectingthe dampingof the waves and assumingthe velocity of sound in air to be equal to v 340 m/s, find at the point A: (a) the pressureoscillationamplitude (Ap)m and itsratio to the \177

-----

\177

-----

air

pressure;

(b) the oscillationamplitudeof particlesof the medium;compare it with the wavelength of sound. 4.i88.At a distancer ----100m from a point isotropic sourceof sound of frequency 200Hz the loudnesslevel is equal to L ----50 dB. The audibility thresholdat this frequency corresponds to the sound The damping coefficient of the sound intensity Io ----0.t0nW/m wave is 7 ----5.0\"10-\177 m-\177. Find the sonicpower of the source. \177.

4.4.ELECTROMAGNETIC \342\200\242

4.5.

Phase velocity of \177

\342\200\242

In a

. \177

Flow density

electromagneticwave:

= dV\177

wh\177re

\177

(\177.\177)

\177/V\177-\177o.

electromagneticwave:

travelling

Space density

RADIATION

WAVES. an

o\177

the energy

o[ an electromagneticfield: (4.4c)

w=\177+\177. of electromagneticener\177,

Ener\177 flow density

S = [EH].

the Poynting

of electricdipol\177 radiation

\177

sin 0, \177

\177

far

vector: fi\177ld

(4.4d) zone: (4.4e)

\17792

13\1779451 t93)))

where r is the distancefrom the dipole,0 is the angle between the radius r and the axis of the dipole. Radiation power of an electricdipolewith moment p (t) and of a q, moving with accelerationw: \342\200\242

\177

vector charge

2q\177w

-----

4.i91.A

wave of frequency t0 MHz plane electromagnetic poorly conductingmedium with conductivity (r = i0 mS/m and permittivity 9. Find the ratio of amplitudes of conductionand displacement current densities. 4.i92.A plane electromagneticwave E = E,\177cos ((ot--kr) propagatesin vacuum.Assumingthe vectorsEm and k to be known, find the vector H as a function of time t at the point with radius

propagatesin a

\177

vector r

4.i93.A

\177

wave E = plane electromagnetic

where E m ----Eme

E,\177

cos

(\177ot

--kr),

k are the unit vectors of the x, ke\177, y axes,propagatesin vacuum.Find the vector H at the point with at the moment (a) t ---- (b) t = to. Consider radius vectorr when m x 7.7 m, and to ---the case V/m, k E,\177 \177

-----

ns.

4.194.A

\177

\177,

e\177,

e\177

xe\177 \177

\17760

-----

0.5i -1,

wave plane electromagnetic

\177

E,\177

cos(o)t --kx)

propagating vacuum inducesthe emf \177ind in a square frame with side The orientation of the frame is shown in Fig.4.37.Find the amplitude value ein,if Em -- 0.50 mV/m, the frequency u =5.0MHz

4.196. Find the mean

2 i P--4\177e 3ca o

\177

this plate?

(4.4f) 3ca 4.i89.An electromagnetic wave of frequency 3.0MHz passes from vacuum into a non-magneticmedium with permittivity e = 4.0.Find the incrementof its wavelength. 4.190. A plane electromagnetic ,\177'ave falls at right anglesto the surface of a plane-parallel plate of thickness The plate is made of non-magneticsubstancewhose permittivity decreases exponentially from a value el at the front surface down to a value e 2 at the rearone.How long doesit take a given wave phase to travel across 4\177teo

relationshold:

vacuum the following

andl = 50cm.

netic wave E = cuum.

E,\177

OB c\177

Poynting vector (S)of a plane electromag-

cos(o)t-kr)

if the wave

propagatesin va-

4.197.A plane harmonicelectromagnetic wave with plane polarization propagatesin vacuum.The electriccomponentof the wave has a strength amplitude E,\177 50 mV/m, the frequency is \177

100MHz. Find:

\177

--\177

(a) the efficient value of the displacement current density; (b) the mean energy flow density averaged over an oscillation

period.

4.t98.A ball of radius 50 cm is locatedin a non-magnetic medium with permittivity e = 5.0.In that medium a plane electromagneticwave propagates,the strength amplitudeof whose electric componentis equal to E,\177 = 200 V/re. What amount of energy reachesthe ball during a time interval t i.0rain? /\177

\177

-----

4.i99.A standingelectromagnetic wave with electriccomponent E= cos

kx.coso)t is sustainedalongthe x axisin vacuum.Find E,\177 the magneticcomponentof the wave B (x, t). Draw the approximate distributionpattern of the wave'selectricand magneticcomponents (E and B) at the momentst = 0 and t T/4, w\177ere T is the oscillation period. 4.200.A standing electromagnetic wave E coskx.cos (or is sustainedalong the x axis in vacuum.Find the projection of the Poynting vector on the x axis (x, t) and the mean value of that projectionaveraged over an oscillationperiod. -----

-----

E,\177

S\177

4.20i.A parallel-plateair capacitorwhose electrodesare shaped

as discsof radius/\177

\177

6.0cm is connectedto a sourceof an alternat-

= i000s with frequency Find ratio of peak values of magnetic and electricenergieswithin capacitor. ing

sinusoidalvoltage

\177o

-\177.

the the

4.202.An

alternating sinusoidal current of frequency (o = in the winding of a solenoidwhose crosssectionalradius is equal to R 6.0straight cm. Find the ratio of peak values of electricand magneticenergieswithin the solenoid. \177

1000

s-\177

flows

--\177

Fig,

4.37.

Fig.

4.38.

4.195.Proceedingfrom Maxwell'sequatious show that

case of

wave a plane electromagnetic

the

(Fig. 4.38) propagatingin

4.203. A parallel-plate capacity whose electrodesare shaped as round discs is chargedslowly. Demonstratethat the flux of the Poynting vector acrossthe capacitor'slateralsurface is equal to the incrementof the capacitor'senergy per unit time. The dissipation of iield at the edge is to be neglectedin calculations. 4.204.A current I flows along a straight conductorwith round cross-section. Find the flux of the Poynting vector acrossthe lateral surface of the conductor'ssegmentwith resistanceR. 4.205. Non-relativistic protons accelerated by a potentialdifferenceU form a round beam with currentI. Find the magnitudeand ta* \17795)))

directionof the Poynting vectoroutside the beam at axis.

distancer A currentflowing in the winding of a longstraight solenoid 4.206. is increasedat a sufficiently slow rate. Demonstrate that the rate at which the energy of the magneticfield in the solenoidincreases is equal to the flux of the Poynting vectoracrossthe lateralsurface of the solenoid. 4.207.Fig.4.39illustratesa segmentof a double line carrying directcurrent whose directionis indicatedby the arrows. Taking into account that the potential and making use of the Poynting vector,establishon which side (left or right) the source from its

\302\242p\177

of the current is

\177

\302\242P1,

located.

1.0T. Find the ratio of the energy lostby the proton due to radiationduring its motion in the field to itsinitialkinetic

equalsB = energy.

A non-relativistic 4.216. chargedparticlemoves in

a transverse

magneticfield with inductionB. Find the timedependence of the particle's kineticenergy diminishingdue to radiation.How soon will its kineticenergy decreasee-fold?Calculatethis time interval for the case (a) of an electron,(b) of a proton. 4.2i7.A chargedparticlemoves alongthe y axisaccordingto the law y = a cos(ot, and the point of observationP is locatedon the x axisat a distancel from the particle(1>> a).Find the ratioof electroat the point P at the moments magneticradiationflow densities = a. Calculate when the coordinateof the particleyl 0 and uniform

S\177/S\177

ratio if

4.2i8.

\177 \1770

Fig.

3.3.106 s -1

-----

and l 190m. A chargedparticlemoves uniformly with velocity v along a circleof radius R in the planexy (Fig.4.40). An observeris located

that

\177

y\177

\177

4.39. P

The energy is transferred from a sourceof constantvoltage 4.208.

to a consumerby means of a long straight coaxialcable with negligibleactive resistance.The consumedcurrent is I. Find the of the cable.The conductive energy flux acrossthe cross-section sheath is supposedto be thin. 4.209.A sourceof ac voltage V = Vo coscot deliversenergy to a consumerby meansof a longstraight coaxialcablewith negligible active resistance.The current in the circuit varies as I = ----Io cos cot -- q)). Find the time-averaged energy flux through the of the cable.The sheath is thin. cross-section 4.2i0.Demonstratethat at the boundary between two mediathe normal componentsof the Poynting vector are continuous, i.e. Sin = that a closedsystem of charged non-relati5.941. Demonstrate vistic particleswith identicalspecificeSargesemitsno dipolo raV

S\177,\177.

diation. \302\243949\177.

Find the mean radiationpower of an electronperforming 0.i0nm and frequency

with amplitude a = harmonicoscillations \177o

s 6.5.1014 -\177

4.213. Find the

radiationpower developedby a non-relativistic particlewith chargee and massm, moving along a circularorbit in the field of a stationary point charge q. of radius 4.2i4.A particlewith chargee and mass m flies with non-relativistic velocity v at a distanceb past a stationary particlewith charge q. Neglectingthe bending of the trajectoryof the moving particle,find the energy lostby this particledue to radiationduring the total flight time. 4.2i5.A non-relativisticproton enters a half-spacealong the /\177

normal to the transverseuniform magnetic field whose induction t96

Fig.

4.40.

x axis at a point P which is removed from the centreof the circleby a distance much exceedingR. Find: (a) the relationshipbetween the observed values of the y projection of the particle's acceleration and the y coordinate of the particle; radiation flow densities (b) the ratio of electromagnetic at the point P at the momentsof timewhen the particlemoves, from

on the

S\177/S\177

the standpointof the observerP, toward him and away from him, as shown in the figure. 4.2\1779. An electromagnetic wave emittedby an elementary dipole propagatesin vacuum so that in the far field zone the mean value of the energy flow density is equal to So at the point removed from the dipoleby a distance r along the perpendicular drawn to the axis. Find the mean radiation power of the dipole. dipole's 4.220. The mean power radiatedby an elementary dipoleis equal to Po.Find the mean spacedensity of energy of the electromagnetic field in vacuum in the far field zone at the point removed from the dipoleby a distancer alongthe perpendiculardrawn to the dipole's

axis.

\302\24322|.

electricdipolewhose modulus is constant and whose

moment is equal to p rotates with constant angular velocity about the axis drawn at right anglesto the axis of the dipoleand passing through its midpoint.Find the power radiated by such a dipole. \302\242\177

t97)))

4.222. A free electron is locatedin the field of a planeelectromagneticwave. Neglectingthe magnetic componentof the wave disturbing its motion,find the ratio of the mean energy radiated

the

oscil-

lating electronper unit time to the mean value of theby energy flow density of the incident wave. 4.223. A plane electromagnetic wave with (o falls upon an elasticallybonded electronwhose naturalfrequency frequency equals (Oo. Neglectingthe damping of oscillations, find the ratio of the mean

energy dissipatedby the electronper unit time to the mean value of the energy flow density of the incident wave. 4.224.Assuming a particleto have the form of a ball and to sorb all incident light, find the radius of a particlefor which abits gravitationalattractionto the Sun is counterbalancedby the force that exertson it.The power of light radiatedby the Sun equals 26 P = light 5.10 W, and the density of the particleis p = i.0 g/cma.

FIVE

PART

OPTICS

5.i.PHOTOMETRYAND

Spectralresponseof an eye V

\342\200\242

OPTICS

GEOMETRICAL

(\177)

shown in

Fig.

5.t.

(11\177

=l.SrnW/Im

\177

\17770

\1778\177

Fig. Luminous

\342\200\242

intensity

Illuminance

produced by a

a is the

source. \342\200\242

\177

angle between the normal to

Luminosity

and luminance

M----d(l)emit

\342\200\242

isotropic source: 1cos

point

E= where

For a Lambert

(5.1a)

E-\177

d---\177'

\342\200\242

5.t.

I and illuminance E: d(l)iac I d(D dS -\177

sourceL

----

\177

the

surfaceand

the

(5.ib) direction to

the

: L

-\177

d(l)

dOAS cos0

(5.ic)

const and luminosity M

gL.

(5.td) t99)))

Relation between refractive angle 0 of a prism angle ct: 0 0 sin ct-4- = n sin \342\200\242

\177

where n \342\200\242

the

Equation

and

(5.1e)

\177-,

\342\200\242

is the

are removed from the crest points O and Principal planes H and of surfacesof a thick lens (Fig. 5.3) by the following distances: d d \342\200\242

01 ' X'= --7-5\"

(I)\177

x=\177

\342\200\242

(5.1f)

curvature radius of the mirror. for aligned optical system (Fig.

Equations

5.2):

s'--s =i,

(5.ig)

\177=0,

Fig.

nyu \342\200\242

5.3.

Lagrange-Helmholtzinvariant:

\342\200\242

Magnifying

power

of an

= const.

an-----\177

Fig.5.2. and

\342\200\242

Optical power of a

thin

\342\200\242

where d two thin

(5.th)

the

refractive index of

Optical power of a

thick

(5.ii)

lens in a medium

O=(n--no) R1 where n

Fig. 5.1),find:

Optical power of a sphericalrefractive surface: \342\200\242

with

refractive index no:

(5.tj)

lens. lens:

the

the thickness of the lens.This equation is also valid for a system lensesseparatedby a medium with refractive index n.

(5.in)

\177p

5.t.Making use of the

optical power:

n n' ]' n' ],=-\177-' /= O' =---h'\342\200\242

where and are the anglessubtended at the eye by an image formed by the optical deviceand by the correspondingobjectat a distancefor convenient viewing (in the case of a microscopeor magnifying glass that distanceis equal to l o = 25 cm). \177p'

Relations between focal lengths

(5.1m)

optical device:

F= tan t

\342\200\242

refractive index of the prism. of spherical mirror:

I I s'{s R' where R

least deviation

spectralresponsecurve

for an eye (see

1.0

lm (a) the energy flux correspondingto the luminousflux of at the wavelengths 0.51and 0.64\177m; (b) the luminousflux correspondingto the wavelength interval from 0.58to 0.63\177m if the respective energy flux, equal to (I)e = = 4.5mW, is uniformly distributed over all wavelengths of the interval. The function V (g) is assumed to be linear in the given

spectralinterval. 5.2.A point isotropicsourceemits a luminous flux (I) = 10lm with wavelength g = 0.59 Find the peak strength values of electricand magneticfields in the luminous flux at a distancer = = 1.0m from the source.Make use of the curve illustrated in Fig. 5.1. 5.3.Find the mean illuminanceof the irradiatedpart of an opaque \177m.

spherereceiving (a) a parallelluminousflux resulting in illuminanceEo at the point of normal incidence; (b) light from a point isotropicsourcelocatedat a distancel = 100cm from the centreof the sphere;the radius of the is R----60cm and the luminous intensity is I----36cd. sphere -----

200 20i)))

5.4.Determinethe luminosity of a surface whose luminance depends on directionas L = Lo cos0, where 0 is the angle between radiationdirectionand the normal to the surface. certainluminous surface obeys Lambert'slaw. Its luminance is equal to L. Find: (a) the luminousflux emittedby an elementAS of this surface into a cone whose axis is normal to the given elementand whose

the

5.5.A

apertureangle is equal to 0; (b) the luminosity of such a source. cm An illuminantshaped as a planehorizontaldiscS = 100 5.6. in areais suspended over the centreof a round tableof radius R = = 1.0m. Its luminancedoesnot depend on directionand is equal to L = 1.6.104 cd/m At what height over the table should the illuminant be suspendedto provide maximum illuminanceat the circumferenceof the table? How great will that illuminancebe? The illuminant is assumed to be a point source. 5.7.A point sourceis suspendedat a height h = 1.0m over the centreof a round tableof radiusR = t.0m.The luminousintensity I of the sourcedependson directionso that illuminanceat all points of the tableis the same.Find the function I (0),where 0 is the angle between the radiation directionand the vertical,as well as the luminous flux reaching the table if I (0) = I o = t00cd. 5.8.A verticalshaft of light from a projectorforms a light spot S ----t00cm in areaon the ceilingof a round roomof radius R = = 2.0m. The illuminanceof the spot is equal to E ----t000lx. Find The reflectioncoefficient of the ceilingis equal to p = 0.80. the maximumilluminanceof the wall producedby the light reflected from the ceiling.The reflection is assumed to obey Lambert'.s law. 5.9.A luminousdome shaped as a hemisphererests on a horizontal plane.Itsluminosity is uniform. Determine the illuminanceat the centreof that plane if its luminanceequalsL and is independent \177

\177.

\177

direction.

5.10.A

Lambert sourcehas the Iorm of an infinite plane. Its luminanceis equal to L. Find the illuminanceof an areaelement

orientedparallelto

5.11.An

the given

source.

illuminant shaped as a plane horizontaldisc of radius B = 25 cm is suspended over a table at a height h = 75 cm. The illuminanceof the tablebelow the centreof the illuminant is equal to Eo 70 lx. Assuming the sourceto obey Lambert'slaw, find its luminosity. 5.12. A smalllamp having the form of a uniformly luminoussphere of radius R ----6.0cm is suspendedat a height h = 3.0m above the floor. The luminanceof the lamp is equal to L = 2.0.i04 cd/m and is independentof direction.Find the illuminanceof the floor \177

\177

directlybelow the lamp. in

5.13.Write

the law of reflectionof a light beam from a mirror vector form, using the directingunit vectorse and e' of the inci-

202

dent and reflected beams and the unit vectorn of the outsidenormal to the mirrorsurface. Demonstrate that a light beam reflected from three mutually

5.14.

perpendicularplane mirrors in successionreverses its direc-

tion.

5.15.At

what value of the angle of incident is a shaft of light the surface of water perpendicular to the refracted

reflectedfrom shaft?

0\177

5.16.Two opticalmediahave a plane boundary between them. is the criticalangle of incidenceof a beam and is Suppose the angle of incidence at which the refractedbeam is perpendicular to the reflected one (the beam is assumed to comefrom an optically denser medium).Find the relativerefractive index of these media 0\177c\177

sin

0\177

= = t.28.

O\177/sin A light beam falls upon a plane-parallel 5.17, glassplated=6.0cm O\177

\1771

in thickness.The angle of incidence is 0 = 60 Find the value of deflection of the beam which passedthrough that plate. 5.18.A man standing on the edge of a swimming pool looksat a stone lying on the bottom.The depth of the swimming poolis equal to h. At what distancefrom the surface of water is the image of the stone formed if the line of vision makesan angle 0 with the normal to the surface? 5.19.Demonstratethat in a prism with smallrefractingangle 0 \302\260.

the shaft of light deviatesthrough the angle a (n t) 0 regardlessof the angleof incidence,provided that the latter is also small. 5.20.A shaft of light passesthrough a prism with refractingangle0 and refractive indexn. Let a be the diffraction angle of the shaft. Demonstratethat if the shaft of light passesthrough the prism \177

symmetrically,

(a) the angle a is the least; (b) the relationship between the angles a and 0 is defined by

Eq. (5.1e).

5.21.

The least deflection angle of a certainglass prism is equal to its refracting angle.Find the latter. 5.22.Find the minimum and maximum deflection angles for a light ray passing through a glass prism with refracting angle 0 = 60 A trihedralprism with refractingangle 60 provides the least deflection angle 37 in air.Find the least deflection angle of that prism in water. 5.24.A light ray composedof two monochromatic components passesthrough a trihedralprism with refracting angle 0 60 Find the angle Ace between the components of the ray after itspassage through the prism if their respectiveindicesof refractionare and t.520. The prism is orientedto provide the least equal to

5.23. \302\260.

\302\260

\177

\302\260.

1.515 5.25.Using Fermat'sprinciplederive the

deflection angle.

laws of deflection and

refraction of light on the plane interface between two media. 2O

3)))

5.26.By

means of plotting find: the path of a light ray after reflectionfrom a concaveand (a) convexsphericalmirrors(see Fig.5.4,where F is the focal point, OO' is the opticalaxis);

5.31.

A point sourceis locatedat a distance of 20 cm from the The lens is 5.0cm front surface of a symmetrical glassbiconvexlens. thick and the curvature radius of its surfaces is 5.0cm. How far beyond the rearsurface of this lens is the imageof the sourceformed? An objectis placedin front of convexsurface of a glass piano-convexlens of thicknessd 9.0cm.The imageof that object is formed on the plane surface of the lens servingas a screen.Find: (a) the transverse magnification if the curvature radius of the

5.32.

\177

O\"

\177

lens'sconvexsurface is

O\177

2.5cm;

(b) the imageilluminanceif the luminanceof the objectis L ---= 7700 cd/m and the entrance aperture diameterof the lens is of light are negligible. D = 5.0ram; losses Find the opticalpower and the focal lengths with refractive index n o = (a) of a thin glass lens in liquid = (I) if its opticalpower in air is 0 D; (b) of a thin symmetrical biconvexglasslens,with air on one side and water on the other side,if the opticalpower of that lens in air \177

Fig.

5.33.

5.4.

of the mirrorand its focal point in the cases positions illustratedin Fig.5.5,where P and P' are the conjugatepoints. op O' Or 0 0 op

(b) the

\177

o\177

\177

Fig.

5.27.Determinethe

5.5.

[\177

--0.50

= --0.25.

5.28.A

= +t0 D.

5.34.By means of \1770

plotting find: a of of the ray light beyond thin convergingand diverging path (a) is the opticalaxis,F and F' are the front lenses(Fig.5.7,where and rear focal points);

00'

--2.0;

5.0

point sourcewith luminous intensity

[\177

I 0 = 100cd

positionedat a distances = 20.0cm from the crestof mirrorwith focal length ] = 25.0cm.Find the luminous intensity of the reflected ray if the reflectioncoefficient of the mirroris p

focal length of a concavemirrorif:

(a) with the distancebetween an objectand its imagebeingequal to l = cm, the transversemagnification = (b) in a certainpositionof the objectthe transverse magnification is == and in anotherpositiondisplaced with respectto the former by a distancel ---cm the transverse magnification [\1771

i.7

--5.0

a concave

-- 0.80. 5.29.Proceedingfrom Fermat'sprinciple

derive the refraction formula for paraxial rays on a sphericalboundary surface of radius B between mediawith refractive inand n'. 5.30.A parallelbeam of light falls from

Fig,

5.7.

(b) the positionof a thin lens and its focal points if the position

of the cojugate points opticalaxis 00' and the positions P, P' (seeFig.5.5)are known; the mediaon both sidesof the lenses are identical; (c) the path of ray 2 beyond the convergingand diverginglenses of the lens and of its (Fig.5.8)if the path of ray 1and the .positions of the

O' 0

dicesn

5.6.

Fig. vacuum on a surface enclosinga medium with refractiveindex n (Fig.5.6).Find the shape of that surface, x (r), if the beam is brought into focus at the point F at a distance from the crestO. What is the maximumradiusof a beam that can

]

stillbe focussed? 204

Fig.

5.8.

opticalaxis00'areall known; the mediaon both sidesof the lenses are identical. 5.35. A thin converginglens with focal length f 25 cm projects the imageof an objecton a screenremoved from the lens by a dis-----

205)))

tance /=5.0m. Then the screenwasdrawn closerto the lens by a distance Al ----i8 cm. By what distanceshould the objectbe shifted for its image to becomesharp again? 5.36.A sourceof light is locatedat a distancel =- 90cm from a screen,A thin converginglens provides the sharp image of the sourcewhen placedbetween the sourceof light and the screenat two Determinethe focal length of the lens if positions. (a) the distance between the two positionsof the lens is A1-------30

cm;

(b) the transverse dimensionsof the imageat one positionof the lens are 4.0greaterthan those at the other position. 5.37.A thin converginglens is placedbetween an object and a screenwhose positionsare fixed.There are two positions of the lens at which the sharp imageof the objectis formed on the screen. Find the transversedimensionof the objectif at one positionof the lens the image dimensionequals h' = 2.0mm and at the other, \177--

\177l

h\"

4.5mm.

----

5.38.A thin

converginglens with aperture ratio D : ] t :3.5 (D is the lens diameter,] is its focal length) provides the imageof a sufficiently distant objecton a photographicplate.The object luminanceis L = 260cd/m2. The losses of light in the lens amount to a =: 0.10. Find tim illuminanceof 5.39.How does. the luminanceof a therealimage. image depend on diameterD of a thin converginglens if that image is observed (a)

directly;

(b) on a white screenbackscattering accordingto Lambert'slaw? two thin symmetrical lenses: one is converging, with refractive index n 1 and the other is divergingwith t.70, refractive iudex n 2 t.51.Both lenseshave the samecurvature radius of their surfaces equal to // ----t0 cm. The lenseswere put closetogetherand submergedinto water.What is the focal length of this system in water?

5.40.There are

\177

5.41.

Determinethe focal length of a concavesphericalmirror which is manufactured in the form of a thin symmetric biconvex glass lens one of whose surfaces is silvered.The curvature radius of the lens surface is 40 cm. 5.42.Figure 5.9illustratesan alignedsystem consistingof three thin lenses.The system is locatedin air. Determine: /\177

\177

5..0cm---)

\"(---5.0 crn

(a) the position of the point of convergenceof a parallelray incomingfrom the left after passing through the system; (b) the distancebetween tile first lens and a point lying on the axisto the left of the system, at which that point and its imageare locatedsymmetrically with respectto the lens system.

5.43.A Galileantelescopeof 10-fold magnification has the length of 55 cm when adjusted to infinity. Determine: (a) tim focal lengths of the (elescope's objectiveand ocular; to adjust tile (b) by what distancethe ocularshould be displaced telescopeto the distance of 50 m. 5.44.Find the magnification of a Kepleriantelescopeadjusted to if the mounting of the objective has a diameterD and the infinity ocularhas a diaimage of that mounting formed by the telescope's meterd.

5.45.On passingthrough a telescope a flux of Iight increases its d.0.l0a times.Find the angulardimensionof a distant objectif its image formed by that telescopehas an angulardimension 2.0. 5.46.A Kepleriantelescopewith magnificatioa F 15was intensity

\177

\177l

\302\260

\177--

\177'

\177

\177:

-100L\177

Fig.

+\177

5.9.

027

1.50.

\177

-\177

\177

5.50.

\177-10.0D

s\177ib-

To make mergedinto water which filled up the insideof the telescope. the system work as a telescope again withia the former dimensions, the objective was replaced .What has the magnification of the telescope becomeequal to? The refractive index of the glass of which the ocularis made is equal to n 5.47.At what magnification F of a telescopewith a diameterof the objectiveD 6.0cm is the illmninanccof tim image of an The pupil objecton the retina not lessthan without the telescope? diameteris assumed to be equal to do 3.0mm.The lossesof light ia the telescopeare negligible. 5.48. The opticalpowersof the objective and the ocularof a microThe microscope scopeare equal to 100and 20 D respectively. magnification is equal to 50.What will the magnification of tile microscope be when the distancebetweenthe objective and the ocularis increased by 2.0cm? 5.49.A microscope has a numericalaperturesin a 0.\1772, where is the apertureanglesubtended by the entrancepnpil of tile microthe diameterof an e)es pupil to be equal to do scope. = 4.0Assuming ram, determinetbc microscope magnification at which (a) the diameterof the beam of hight comingfrom the microscope is equal to the diameterof tile eye'spupil; (b) the illuminanceof the imageon the retina is independentof magnification (considerthe case when the beam of light passing is bounded by the mounting through the system \"microscope-eye\" of the objective). Find the positionsof the principal planes,the focal and nodal points of a thin biconvexsymmetric glasslens with curvat\177re radius of its surfaces equal to R = 7.50cm. There is air on one sideof the lens and water on the other.)))

5.51. of focalpoints and By meansof plotting find the positions principalplanesof alignedopticalsystems illustratedin Fig.5A0: (a) a telephotolens, that is a combinationof a convergingand a divergingthin lenses(fl = 1.5a, ]2 = --t.5a); 0!

Fig.

(b) a system of two thin converginglenses = t.5a, = 0.5a); lens(d = 4 cm,n = 1.5,(D1 = -{-50D, (c)a thickconvex-concave =--50D). (/\177

]\177

5.52.An opticalsystem is locatedin air. Let OO' be itsoptical H and H' are the axis,F and F' are the front and rearfocal points, front and rear principalplanes,P and P' are the conjugatepoints. \1772

By means of plotting find:

(a) the positions F' and H' (Fig. (b) the position of the point conjugateto the

S' 5.11a);

(Fig. 5.1tb);

P Y

o'o

o'o.

point S

\177H

B\177

\177

B\177

f\177,

f\177

(c)

(b)

of the principalplanes and focal 5.55.Calculatethe positions glasslens if the curvature radius points of a thick convex-concave 10.0cm and of the concave of the,.convexsurfaceis equal to 5.0cm and the lens thickness is d 3.0cm. surface to 5.56.An aligned opticalsystem consistsof two thin lenseswith the distancebetween the lensesbeingequal focal lengths and to d. The given system has to be replacedby one thin lens which, would provide the sametransverse at any position of an object, magnification as the system. What must the focal length of this lens be equal to and in what position must it be placedwith respect to the two-lenssystem? 5.57.A system consistsof a thin symmetricalconvergingglass lens with the curvature radius of itssurfacesB ----38 cm and a plane mirrororientedat right anglesto the opticalaxis of the lens.The distancebetween the lens and the mirroris l 12cm. What is the opticalpower of this system when the spacebetween the lens \177

and the in

mirror is

5.58.At air

filled up with water? what thicknesswill a thick convex-concave glass lens

provided the curvature radius of itsconvex (a) serveas a telescope 1.5cm greaterthan that of its concavesurface? surface is A// D if the curvature (b) have the optical power equal to and 7.5cm radii of its convexand concavesurfaces are equal to 10.0 \177

--i.0

respectively?

of the principalplanes,the focal length 5.59.Find the positions of a thick convex-concave of the and the

to?

glass sign opticalpower (a) whosethicknessis equal to d and curvature radiiof the surfaces are the same and equal to B; (b) whose refractivesurfaces are concentricand have the curvature radii //1 and \177//1)consistsof two glass balls with radii 5.60.A telescopesystem ----i.0em.What are the distancebetween the R1 = 5.0cm and eentresof the ballsand the magnification of the system if the bigger ball servesas an objective? 5.61.Two identicalthick symmetricalbiconvexlensesare put close together.The thickness of each lens equals the curvature radius of itssurfaces,i.e.d = R = 3.0em. Find the opticalpower of this system in air. medium with 5.62.A ray of light propagatingin an isotropie refractive index n varying gradually from point to point has a curvature radius p determinedby the formula t__ a (Inn) ' ON 9 where the derivative is taken with respectto the principalnormal to the ray. Derivethis formula, assumingthat in such a medium Here0 is the angle bethe law of refractionn sin 0 = const holds. tween the ray and the directionof the vector Vn at a given point.

208

]4-9451

(a)

(e)

\177(b)

Fig.5,ii.

5.ttc,

where the path of the and H' (Fig. (c) the positionsF, ray of light is shown before and after passingthrough the system). SupposeF and F' are the front and rear focal points of an s optical ystem, and H and H' are itsfront and rearprincipalpoints. By meansof plottingfind the positionof the image, of the point S of the points F, H, for the following relative positions and (a) FSHH'F';(b) HSF'FH';(c) H'SF'FH;(d) F'H'SHF. 5.54.A telephotolensconsistsof two thin lenses,the front converg-

F',

5.53.

H':

S, F',

optical powers (I)1 = and (I)\177---of principalaxesof that (a) the focal length and the positions d ----4.0em; a distance are lenses if the by separated system ratio of a focal lenses at which the (b) the distanced between the l between the a distance to of the converginglens and system lengthf the rear principalfocal point is the highest.What is this ratioequal ing

lens and the rear diverging lens

= q-t0D

--i0D. Find:

with

lens

//\177

(//\177

R\177

209)))

5.63.Find

the curvature radius of a ray of light propagating horizontal directioncloseto the Earth's surface where the to approximately of the refractive index in air is gradient would the ray of light 3.i0-s m -1.At what value of that gradientequal propagate all the way round the Earth? in a

5.2.INTERFERENCE Width

\342\200\242

of a

fringe:

(5.2a)

\177,,

-\177

distancefrom

sourcesto

the

screen,d is the distancebetween

the

o Temporaland spatial coherences.Coherencelength lcoh where thin

\177-\177

--\177

coherenceradius:

\177

Pcoh

and

(5.2b)

is the angular dimension of the source. $ Condition for interference maxima in the case of light reflectedfrom a plate of thickness b: \177

2b]/'n2 -- sin 01 = \302\260\"

1/2)

\177-

(5.2c)

\177,

is an integer. Newton's rings producedon reflection of light from the surfacesof an air interlayer formed between a lens of radius and a glassplate with which the convexsurface of the leas is in contact. The radii of the rings: where k

/\177

= k = t, r

..... of

and dark if k the central dark

spot.

(a) have the

samedirectionand

the phasedifferencebetween the

are incoherent,and

all the values

are equally probable; oscillations

have the same frequency and (b) are mutually perpendicular, an arbitrary phase difference. 5.65.By means of plotting find the amplitude of the oscillation

of the resultingfrom the addition of the following three oscillations

samedirection: \1771

= acoscot, = 2asincot, \177

\177

5.66.A certainoscillationresultsfrom

oscillations of

the

t.5acos(cot

\303\267

\177/3).

...,

the addition of coherent

= a cos[cot -;-(k -- 1) oscillation 2, (k = N), is

same direction

\177

i, (k-

find: (a) the angles 0 at which the q\177

(q\177

\177

<:\177),

radiation intensity is maximum; (b) the conditionsunderwhich the radiationintensity in the direction0 = n is maximumand in the

14 \177

oppositedirection,minimum. 5.68.A stationary radiating system consistsof a Fig. 5.t2. linearchain of paralleloscillators separatedby a distance d, with the oscillationphase varying linearly along the chain.Find the timedependence of the phasedifference between the neighbouringoscillatorsat which the principalradiation maxiAq\177

of the system will be \"scanning\" the surroundingswith the constant angular velocity co. 5.69.In Lloyd'smirrorexperiment(Fig.5.t3)a light wave emitted directlyby the sourceS (narrow slit) interfereswith the wave reflected from a mirrorM. As a result, an interferencefringe pattern is

mum

....

are added, when two harmonicoscillations the time-averagedenergy of the resultant oscillationis equal to if both of the sum of the energiesof the constituent oscillations, of

source1 by

= 2, 4, 6

that 5.64.Demonstrate

them

\177,.

(5.2d)

\177/\177,Rk/2,

with the rings being bright if 3, 5 k 0 correspondsto the middle

The value

momentsareorientedat right anglesto that plane.The sourcesare separatedby a distanced, the radiationwavelength is equal to Taking into account that the oscillations of source2 lag in phase behind the oscillations of

OF LIGHT Ax

where l is tile the \"sources.

5.67.A system illustratedin Fig. 5.12consistsof two coherent point sources1 and 2 locatedin a certainplane so that their dipole

where k is the number of the the phase difference between the kth and l)th Find the amplitude of the resultant oscillation.

q\177l,

oscillations. q\177

Fig.

5.t3.

formed on the screenSc.The sourceand the mirrorare separatedby a distance1 = 100cm.At a certainpositionof the sourcethe fringe width on the screenwas equalto Ax= 0.25mm, and after the source was moved away from the mirror plane by Ah = 0.60mm, the times.Find the wavelength of light. fringe width decreased = 5.70.Two coherentplane light waves propagating with a divergenceangle << I fall almost normally on a screen.The amplitudes of the waves are equal..Demonstrate that the distancebetween the where neighbouringmaximaon the screenis equal to Ax = is the wavelength. \1771

1.5

\177

\177,/\177,

\177,

5.71.Figure 5.t4illustratesthe interferenceexperimentwith Fresnelmirrors.The affgle between the mirrorsis a = 12',the distancesfrom the mirrors'intersectionline to the narrow slit S and the screenScare equal to r = t0.0cm and b 130cm respectively. The wavelength of light is = 0.55pm. Find: (a) the width of a fringe on the screenand the number of possible maxima; -----

\177,

2t0 t\177*

2tl)))

the shift of the interferencepattern on the screenwhen the along the arc of radius r with centreat the .point O; of the slitthe interferencefringes (c)at what maximumwidth on the screenare stillobserved sufficiently sharp.

(b) slit is.displacedby 5l ----i.0 mm 5\177nax

fringe pattern is formed on a screenplacedat a distancel i00cm behind the diaphragm.By what distanceand in which directionwill these fringes be displacedwhen one of the slitsis coveredby a glassplate of thicknessh---used in measure5.77.Figure 5.t6illustratesan interferometer ments of refractive indicesof transparent substances.Here S is

\177

Fig. Fig.

5.t4.

5.72.A plane light wave ----

falls on Fresnelmirrorswith an angle a 2.0'between them. Determinethe wavelength of light if the width of the fringe on the screenAz = 0.55mm. 5.73.A lens of diameter5.0cm and focal length f----25.0cm was cut alongthe diameterinto two identicalhalves.In the process, i.00mm in thicknesswas lost.Then the the layer of the lens a halves were put together to form a compositelens.In this focal plane a narrow slitwas placed,emitting monochromaticlight with wavelength = 0.60\177tm. Behind the lens a screenwas locatedat \177-

it.

\177,

Find: \177--50 cm from screenand the number of a on the the of width fringe (a) distanceb

possible maxima; at which the fringes on the, (b) the maximumwidth of the slit screenwill be stillobserved sufficiently sharp. 5.74.The distancesfrom a Fresnelbiprism to a narrow slitand 25 cm and b = t00 cm respectively. a screenare equal to a 5\177nax

\177

The refractingangleof the glass biprism is equal to 0 = 20'.Find the wavelength of light if the width of the fringe on the screenis Ax 0.55mm. 5.75.A plane light wave with wa0.70 falls normally velength on the base of a biprismmade of glass (n t.520)withtherefracting angle 0---Fig. 5.i5. . Behind biprism(Fig.5.15) ----5.0 there is a plane-parallel plate, with the Find the spacebetween them filled up with benzene (n' = t.500). width of a fringe on the screenSc placedbehind this system. 5.76.A plane monochromaticlight wave falls normally on a diaphragmwith two narrow slitsseparatedby a distanced ----2.5mm. \177

\177,-\177

\177

\302\260

2i2

\177tm

5.t6.

a narrow slitilluminatedby monochromatic light with wavelength ----589 nm, and 2 are identicaltubes with air of length 1 10.0cm each,D is a diaphragmwith two slits.After the air

\177,

\177-

\177

tube 1 was replacedwith ammoniagas, the interferencepattern

on the screenSc was displacedupward by N ----17 fringes.The rethe refract.000277. Determine fractive index of air is equal to n tive index of ammonia gas. wave falls normally on the boundary 5.78.An electromagnetic between two isotropicdielectricswith refractiveindicesnl and Making use of the continuity condition for the tangential components,E and I-I acrossthe boundary, demonstrate that at the interface the electric field vector E (a) of the transmitted wave experiences phase jump; (b) of the reflectedwave is subjectedto the phase jump equal to if it is reflected from a medium of higher opticaldensity. 5.79.A parallelbeamof white light falls on a thin film whose The ,angleof indicesis 01 refractiveindexis equal to n 52 What must the film thicknessbe equal to for the reflected light to be colouredyellow ----0.60 \177tm) most intensively? 5.80.Find the minimum thicknessof a film with refractiveindex maximum at which light with wavelength 0.64 experiences reflectionwhile light with wavelength 0.40 is not reflected at The incidenceangle of light is equal to 30 due to reflectionfrom the glass To decreaselight losses surface the latter is coatedwith a thin layer of substancewhose refractive indexn' \177where n is the refractive indexof the oscillations of electromagnetic glass.In this case the amplitudesare reflectedfrom both coatedsurfaces equal.At what thicknessof that coating is the glassreflectivity in the directionof the normal equal to zero for light with wavelength Diffused monochromatic light with wavelength ----0.60 Determinethe falls on a thin film with refractive indexn \177-

n\177.

\177o

\177

\177--

\177-

t.33.

-\177

\302\260.

(\177,

t.33 all. 5.81.

\177tm

\302\260

-\177

5.82.

\177,?

\177-

t.5.

\177,

\177tm

213)))

film

thickness if the angular separationof

maxima

neighbouring observed in reflectedlight at the angles close to 0 = 45 to the normal is equal to 60-=3.0. \302\260

\302\260

5.83. Monochromatic light passesthrough

(Fig. 5.i7)and being reflected from a producesfringes of equal inclination on the screen.The thickness of the plate is equal to d, the distance be-

screenSc transparent plate P

an orifice in a

thin

r\177.

r\177,

wave with wavelength falls on the surface of a glass wedge whose faces

Fig.

\177,

1. The

5.t7.

0\177.

\177,

\177

\177tm

\177

\177tm.

\177

incidenceis perpendicular to plane the edge,the angle of incidenceis Find the distancebetween the neighbouringfringe maximaon the screen placed at right angles to reflected light. 5.85.Light with wavelength = 0.55 from a distant point sourcefalls normally on the surfaceof a glasswedge.Afringe pattern whose neighbouringmaximaon the surface of the wedgeare separated by a distanceAx = 0.21 mm is observed in reflected light.Find: (a) the angle between the wedge faces; if the fringes (b) the degreeof light monochromatism disappearat a distance1 1.5cm from the wedge'sedge. 5.86.The convexsurface of a piano-convexglass lens comesinto contactwith a glassplate.The curvature radius of the lens's convex c\302\242

\177,

light taking into accountthat h << l. 5.84.A plane monochromatic light angle

glass lensesare brought into contact their sphericalsurfaces.Find the opticalpower of such a system reflected light with wavelength 0.60 the diameterof the fifth bright ring is d = 1.50mm. 5.91.Two thin symmetric glass lenses,one biconvexand the other biconcave,are brought into contactto make a system with opticalpower = 0.50D. Newton'srings are observed in reflected 0.61 Determine: light with wavelength (a) the radius of the tenth dark ring; (b) how the radius of that ring will changewhen the spacebetween the lensesis filled up with water. 5.92.The sphericalsurface of a piano-convexlens comesinto contactwith a glassplate.The spacebetween the lens and the plate is filled up with carbondioxide. The refractiveindicesof the lens, carbon dioxide,and the plate are equal to 1.50, -= 1.63, The curvature radius of the spherical and na =-1.70respectively. surface of the lens is equal to R =- 100cm. Determinethe radius of the fifth dark Newton'sring in reflected light with wavelength = 0.50 5.93.In a two-beam interferometerthe orange mercury line \177

tween the plateand the screenis l, the radii of the ith and kth dark rings are r i and Find the wavelength of

form an

5.90.Two piano-convexthin

with if in

\177tm

(A\177,/\177,)

\177

surface is R, the wavelength of light is equal to Find the width Ar of a Newton ring as a function of itsradius r in the regionwhere \177,.

Ar<<r. 5.87.The convexsurface of a piano-convexglass lens with curvature radius R = 40 cm comesinto contactwith a glass plate. A certainring observed in retlected light has a radius r = 2.5ram.

n\177

\177

= 579.03 of two wavelengths = 576.97nm and nm composed is employed.What is the least orderof interferenceat which the sharpnessof the fringe pattern is the worst? the yellow sodium line com5.94.In Michelson'sinterferometer = 589.6nm was posed of two wavelengths = 589.0nm and used. In the processof translationaldisplacementof one of the mirrorsthe interferencepattern vanished periodically(why?). Find the displacement of the mirrorbetween two successiveappearances of the sharpest pattern. 5.95. \177talon is illuminatedby monochromatic When a Fabry-Perot light with wavelength an interferencepattern,the system of con\177x

1\177

\177

\177=

1\177

Watching the given ring, the lens was gradually removed from the plate by a distanceAh = 5.0\177m. What has the radius of that ring becomeequal to? 5.88.At the crest of a sphericalsurface of a lens there is a ground-off plane spot of radius r0 =piano-convex 3.0mm through which the lens comesinto contactwith a glass plate.The curvature radius of the lens'sconvexsurface is equal to R = 150cm. Find the radius of the sixth bright ring when observed in reflected light with

655 nm.

wavelength 5.89. A piano-convex glasslens with \177,

surface R = of the tenth

equal to of light.

d\177

\177

curvature radius of

spherical i2.5cm is pressedagainst a glass plate.The diameters and fifteenth dark Newton'srings in reflected light are 1.00mm and 1.50mm. Find the wavelength

\177

d\177

Fig.

centricrings, appearsin thickness of the

\177talon

5.t8.

the focal plane of a lens

is equal to

The (Fig.5.18).

d. Determinehow

2t4 2t5)))

(a) the position of rings; (b) the angular width of fringes depends on the order of interference. 5.96.For the Fabry-Perot\177talon of thicknessd = 2.5cm find: (a) the highest order of interferenceof light with wavelength \177,

= 0.50 (b) the dispersionregion \177m;

A\177,,

i.e.the

spectralinterval

of wave-

lengths, within which there is stillno overlap with other ordersof interferenceif the observation is carried out approximately at wavelength

\177,

the condition

where

b a--'\177--

....

Fresnel zone:

' k = t, 2, 3

additional

= l, 2, .

Angular

..,

(5.3a)

dispersion

diffraction

diffraction 1\177

x\177/2/b\177.,

the number

(5.3e)

grating:

(5.3f)

the grating.

D -- t.22 k '

(5.3g)

is the least angular separationresolvedby the objective,D is the diameter of the objective. $ Bragg'sequation. The condition of diffraction maxima: 2d sin a = +__.kk, (5.3h) where d is the interplanar distance,a is the glancing angle, k t, 2, 3, . where

... .

objective \177,

-----

0.5

(5.3b)

grating:

= \177 = kN,

of lines of

Resolving power of an /\177

slit.

(5.3d)

\177,,

\302\261

exceptfor 0, N, 2N,

$ Resolving power of a

where N

...,

0= k[,

-- k D--50 5L d cos0

\177.5

minima:

Cornu'sspiral (Fig. 5.t9).The numbers along that spiral correspondto where x values of parameter v. In the case of a plane wave \177'=

from

the width of the slit, 0 is the diffraction angle. $ Diffraction grating, with light falling normally. The main Fraunhofer maxima appear under the condition (5.3c) dsin 0 +k\177., k 0, 2,

d sin

kth

normally

i, 2, 3

___kL,

falling

where b

\177in.

the

bsin0=

= 0.50

5.3.DIFFRAETION OF LIGHT I Radius of the periphery of the

.....

$ Fraunhofer diffraction produced by light minima: Condition of intensity

.. .

5.97.A plane light wave falls normally on a diaphragm with round apertureopeningthe first N Fresnelzones for a point P on a screenlocatedat a distanceb from the diaphragm.The wavelength of light is equal to Find the intensity of light I o in front of the diaphragmif the distributionof intensity of light I (r) on the screenis known. Here r is the distancefrom the point P. 5.98.A point sourceof light with wavelength = 0.50[tm is locatedat a distancea = 100em in front of a diaphragm with mm.Find the distanceb between round aperture of radius r = the observation and the diaphragm point for which the number of Fresnelzones in the aperture equals k = 5.99.A diaphragm with round aperture,whose radius r can be varied during the experiment,is placedbetween a point sourceof from the diaphragm to the source light and a screen.The distances cm.Determine em and b -and the screenare equal to a = at the centreof maximum if the of the wavelength light intensity the diffraction pattern of the screenis observed at -- \177.00 mm = \177.29 mm. and the next maximumat \177,.

2..8

\177,

1.0

Fig.

5.i9.

\17700

are the distancesdefining the position of the element dS of a wavefront relative to the observationpoint P as shown in the upper left corner of the and b figure.

\17725

r\177

216

217)))

5.i00.A plane monochromatic light wave with intensity I 0 falls normally on an opaquescreenwith a round aperture.What is the intensity of light I behind the screenat the point for which the aperture (a) is equal to the first Fresnelzone; to the internal half of the first

zone;

(b) was made equal to the first Fresnelzone and then half of it was closed(along the diameter)? A plane monochromatic light wave with intensity I falls normally on an opaque disc closingthe first Fresnel zone for the observationpoint P. What did the intensity of light I at the

5.i0i.

point P becomeequal to after (a) half of the disc (along the diameter) was removed; (b) half of the externalhalf of the first Fresnelzone was removed (along the diameter)? 5.i02.A plane monochromaticlight wave with intensity I0

falls normally on the surfaces of the opaque screensshown in Fig.5.20.Find the intensity of light I at a point

5.104. A plane light wave with wavelength and intensity I o falls normally on a large glassplate whose oppositeside serves as an opaque screenwith a round aperture equal to the first Fresnel zone for the observationpoint P. In the middleof the aperture there is a round recessequal to half the Fresnelzone. What must the depth h of that recessbe for the intensity of light at the point P to be the highest?What is this intensity equal to? 5.i05.A plane light wave with wavelength = 0.57 falls normally on a surface of a glass (n t.60)disc which shuts one and a half Fresne]zones for the observatio\177 point P. What must the minimum thicknessof that disc be for the intensity of light at the point P to be the highest?Takeinto accountthe interference of light on its passingthrough the disc. A plane light wave with wavelength \177,= 0.54 5.106. goes through a thin converginglens with focal length f = 50 cm and an aperture stop fixed immediately after the lens, and reaches \177,

\177,

\177tm

\177

\177tm

screenplacedat a distanceb 75 cm from what aperture radii has the centreof the on the screenthe maximumilluminance?

\177

the aperture stop. diffraction pattern

5.i07.A

plane monochromaticlight wave falls normally on aperture.At a distanceb 9.0m from it there is a screen showing a certaindiffraction pattern. The aperture diameterwas decreased = 3.0times. Find the new distanceb' at which the screenshould be positionedto obtain the diffraction pattern similar to the previous one but diminished tithes. a round

\177

\177]

5.i08.An

\177l

opaque ball of diameterD 40 mm is placedbetween a sourceof light with wavelength 0.55 and a photographic plate.The distancebetween the sot\177rce and the ball is equal to a '12m and that between the ball and the photographicplate is equal to b = i8 m. Find: \177

\177

Fig.

5.20.

(a) locatedbehind the cornerpoints of screens1-3and behind the edge of half-plane 4; (b) for which the rounded-off edge of screens5-8coincideswith the boundary of the first Fresnelzone. Derive the general formula describing the results obtained for screens 1-4;the same, for screens5-8. A plane light wave with wavelength \177,-----0.60\177m falls normally on a sufficientl$ largeglassplatehaving a round recesson the opposite side (Fig. For the observationpoint P that recesscorresponds to the first one and a half Fresnel /////i//////////\177 zones. Find the depth h of the recessat which the \177ntensity Fig. 5.21. light at the point P is

5.i03.

5.2i).

o\177

(\177)

maximum;

(b) minimum; (c) equal to the

2t8

\177

\177tm

\177

(a) the imagedimensiong' on the plateif the transversedimension of the source is g = 6.0mm; (b) the minimum height of irregularities,coveringthe surface of the ball at random,at which the ball obstructslight. Note.As calculationsand experienceshow, that happens when the height of irregularitiesis comparable with the width of tile Fresnelzone along \177 which the edge of an opaque screenpasses. A point sourceof monochromatic # light is positionedin front of a zone plate at a distancea m from The image of the source is formed at a distance b = m from the plate. Find the focal of length the zone plate. A plane light wave with waveFig. 5.22. length = 0.60\177m and intensity I o falls normally on a large glass plate whose side view is shown in \177

5.i09.

\177

/\177

\177

i.5

it.

t.0

5.||0. \177,

i\177tess\177ty

of incident light.

2\1779)))

Fig.5.22.At

what height h of the

ledgewill the intensity

of light

at points locateddirectly below be (a) minimum; due to reflectionareto be neglect(b) twiceas low as I o (the losses

ed).

A plane monochromatic 5.111. light wave falls normally on an opaque half-plane.A screenis locatedat a distanceb ---- 100cm behind the half-plane. find: Makinguseof the Cornu spiral(Fig.5.19),

(a) the ratio of

ing

intensitiesof the

minimum;

first

maximumand the neighbour-

(b) the wavelength of light if the first two maximaare separated by a distanceAx =-0.63mm. A plane light wave with wavelength 0.60\177m falls normally 5.112. on a long opaque strip 0.70mm wide. Behind it a screenis placed

5.19,

at a distance100cm. Using Fig. find the ratio of intensities of light in the middleof the diffraction pattern and at the edge of shadow. the geometrical A plane monochromatic light wave falls normally on a long rectangularslitbehind which a screenis positionedat a distance b 60 cm.First the width of the slitwas adjusted so that in the middleof the diffraction pattern the lowest minimum was observed. After wideningthe slitby Ah ----0.70mm, the next minimum was obtainedin the centreof the pattern.Find the wavelength of light. A plane light wave with wavelength \177,----0.65 falls normally on a largeglassplatewhose oppositesidehas a longrectan-

5.113.

\177

5.114.

\177tm

gular recess0.60mm wide. UsingFig.5.19, find the depth h of the recessat which the diffraction pattern on the screen77 cm away from the plate has the maximum

illuminanceat

\177m

side has large glassplate whose opposite a ledge and an opaque strip of width A screen a is 0.30mm (Fig.5.23). i10 cm fromplaced the at a distance b is h of the such The ledge plate. height that the intensity of light at point 2 of the Fig. 5.23. screenis the highest possible. Making nse of Fig. 5.19, the ratio of intensitiesat points 1 and 2. find A plane monochromatic 5.|16. light wave of intensity I o falls normally on an opaque screenwith a long slithaving a semicircular \177

\177

220

5.24.

\177/////////////////////////////////\177

Fig.

coincideswith

the

1.07timesless than

the

the cut

zone, with the hole diameterbeing

\177

width of the strip.Using Fig.5.t9,find the intensity of light at the point P provided that the intensity of the incident light is equal to

o. 5.119. Light with

\1771

wavelength k falls normally on a longrectangu-

lar slitof width b. Find the angular distribution of the intensity of light in the caseof Fraunhofer diffraction, as well as the angular positionof minima.of 5.120. Making use the result obtainedin the foregoingproblem, find the conditionsdefining the angular position of maximaof the first, the second,and the third order. 5.121. Light with wavelength k----0.50 falls on a slit of to its normal.Find the at an angle 0o----30 width b 10 located on both sidesof the the first of minima angular position centralFraunhofer maximum. \177tm

\302\260

\177

\177m

with

wavelength

\177

0.60

falls

normally on the face of a glasswedgewith refractingangleO ----15 The oppositeface of the wedge is opaque and has a slitof width b----t0\177m parallelto the edge.Find: (a) the angleA0 between the directionto the Fraunhofer maximum of zeroth orderand that of incident light; (b) the angular width of the Fraunhofer maximum of the zeroth \177tm

\302\260.

5.1i5.A plane light wave with wavelength \177,----0.65 falls normally on a

Fig.

side (Fig.5.24).The edge of

boundary line of the first Fresnelzone for the observation point P. The width of the slitmeasures0.90of the radius of the cut. Using find the intensity of light at the point P. Fig.5.19, A plane monochromatic 5.117. light wave falls normally on an a slit whose with screen shape is shown in Fig.5.25. long opaque the ratio of intensitiesof light at find of use 5.19, Fig. Making points 1,2, and 3 locatedbehind the screenat equal distancesfrom it. For point 3 the rounded-off edgeof the slitcoincideswith the boundary line of the first Fresnelzone. A plane monochromatic 5.118. light wave falls normally on an a screen as longstrip with a round hole in the middle. shaped opaque to half the Fresnel For the observationpoint P the hole corresponds

A plane light wave 5.122.

its centre.

\"/////////\177p\177-'\177/////////// \177'/////////////I///////////I///,

cut on one

5.25.

order.

5.t23.A monochromaticbeam falls on a reflectiongrating with

periodd 1.0mm at a glancingangle o ----1.0%When it is diffracted at a glancingangle \177z----3.0a Fraunhofer maximum of second orderoccurs.Find the wavelength of light. diffraction pattern originating in 5.124. Draw the approximate the case of the Fraunhofer diffraction from a grating consisting of three identicalslitsif the ratio of the grating periodto the slit --\177

\177z

\302\260

width is equal to (a) two; (b) three. With 5.125.

light falling normally on a diffraction grating, the secondorderis equal to 45 for a wavelength

angle of diffraction of

\302\260

221)))

\177,1

= 0.65 Find the angleof diffraction = 0.50

length

\177m.

5.|26.

third orderfor a wave

Light with wavelength 535 nm falls normally on a diffraction grating. Find itsperiodif the diffraction angle 35\302\260corresponds to one of the Fraunhofer maximaand the highest orderof spectrum is equal to five. Find the wavelength of monochromatic light falling normally on a diffraction grating with periodd = 2.2 if the angle between the directionsto the Fraunhofer maximaof the first and the secondorder is equal to A0 75 Light with wavelength 530 nm falls on a transparent diffraction grating with period7.50 \177m. Find the angle,relative to the grating normal,at which the Fraunhofermaximumof highest order is observed provided the light falls on the grating (a) at right angles; (b) at the angle 60 to the normal. falls normally on Light with wavelength \177,-= 0.60 a diffraction grating inscribedon a plane surface of a piano-convex cylindricalglasslens with curvatureradius R ----20cm. The period of the grating is equal to d = 6.0i\177m. Find the distancebetween the principalmaximaof first orderlocatedsymmetrically in the focal plane of that lens. A plane light wave with wavelength \177,----0.50 falls normally on the face of a glasswedgewith an angleO ---- \177mOn the face of the wedge a transparent diffraction grating with opposite periodd----2.00 is inscribed,whose linesare parallelto the wedge'sedge.Find the angles that the directionof incident light forms with the directionsto the principalFraunhofer maximaof the zero and the first order.What is the highest orderof the spectrum? At what angle to the directionof incidentlight is it observed? A plane light wave with wavelength falls normally on a phase diffraction grating whose side view is shown in Fig.5.26. The grating is cut on a glassplate with refractiveindex n. Find the depth h of the linesat which the intensity of the centralFraunhofer maximumis equal to zero.What is in this casethe diffraction angle correspondingto the first maximum?

5.i27.

i\177m

\302\260

-----

5.i28.

\302\260

5.i29.

i\177m

5.i30.

30.

i\177m

5.i3i.

\177,

Fig.

5.26.

222

-----

water.

To measurethe angulardistance 5.i33.

between the components of a double star by Michelson's method, in front of a telescope's lens a diaphragm was placed,which had two narrow parallelslits separatedby an adjustabledistanced. While diminishingd, the first smearingof the pattern was observed in the focal plane of the objectiveat d ----95 cm. Find assumingthe wavelength of light to be equal to = 0.55\177tm. A transparentdiffraction gratinghas a period d ----7.50\177tm. Find the angulardispersionD (in angularminutes per nanometres) correspondingto the maximum of highestorderfor a spectralline = 530 nm of light falling on the grating of wavelength at right angles; (a) (b) at the angle 0o = 45 to the normal. Light with wavelength falls on a diffraction grating at Find the angulardispersionof the grating as a function angles. right of diffraction angle 0. Light with wavelength \177----589.0nm falls normally on a diffraction grating with periodd =-2.5\177tm, comprisingN = ----t0000lines. Find the angularwidth of the diffraction maximum of second order. 5.t37.Demonstratethat when light falls on a diffraction grating at right angles,the maximum resolvingpower of the grating cannot exceedthe value I/\177, where l is the width of the grating and is the wavelength of light. Using a diffraction grating as an example,demonstrate that the frequency difference of two maximaresolvedaccordingto Rayleigh's criterionis equal to the reciprocalof the difference of propagationtimesof the extremeinterferingoscillations, 5v =\177

\177,

5.i34.

\177

\302\260

5.i35.

\177,

5.t36.

\177,

5.138.

i.e.

5.139. Light

composedof two spectrallines with wavelengths falls normally on a diffraction grating t0.0mm wide. At a certaindiffraction angle0 theselines areclose to being resolved (accordingto l\177ayleigh's criterion).Find 0. 5.140. Light falls normally on a transparent diffraction grating of width l = 6.5cm with 200lines per millimetre.The spectrum under investigationincludes a spectralline with \177----670.8nm = 0.0t5rim. Find: consistingof two componentsdiffering by in what order of the these will be resolv(a) spectrum components

600.000 and 600.050 nm

5\177

Fig.

\177.27.

5.i32.Figure 5.27illustratesan arrangementemployed in observations of diffraction of light by ultrasound. A plane light wave with wavelength = 0.55 passesthrough the water-filledtank T \177,

standing ultrasonic wave is sustained at a frequency 4.7 MHz. As a result of diffraction of light by the optically inhomogeneousperiodicstructure a diffraction spectrum can be observed in the focal plane of the objectiveO with focal length f = 35 cm. The separationbetween neighbouringmaximais Ax ----0.60mm.Find the propagationvelocity of ultrasonicoscillations in which a

i\177m.

\177'2

i\177m

ed;

(b) the least difference of wavelengths that can be resolved this grating in a wavelength region 670 nm. \177

b\177

\177

223)))

With 5.141.

light falling normally on a transparent diffraction wide, it was found that the componentsof the yellow and 589.6rim) are resolvedbeginningwith line of sodium (589.0

grating l0 mm

the fifth orderof the spectrum.Evaluate: (a) the periodof this grating; (b) what must be the width of the grating with the sameperiod nm nm whose componentsdiffer by for a doublet ----460.0 to be resolvedin the third orderof the spectrum. A transparent diffraction grating of a quartz spectrograph is 25 mm wide and has 250linesper millimetre.The focal length of an objectivein whose focal plane a photographicplate is located is equal to 80cm. Light falls on the grating at right angles.The spectrumunder investigationincludesa doublet with components nm. Determine: and 310.i84 of wavelengths (a) the distanceson the photographicplatebetweenthe components of this doublet in the spectraof the first and the secondorder;

0.13

2\177

5.142.

310.i54

(b) whether these componentswill be resolved in these orders

spectrum. The ultimate resolvingpower 5.143.

of the

of the spectrograph's trihedralprism is determinedby diffraction of light at the prism edges(as in the caseof a slit).When the prism is orientedto the least deviation angle in accordancewith Bayleigh'scriterion, Z/5)\177

and dn/d\177 is the where b is the width of the prism'sbase(Fig.5.28), this formula. Derive its material. of dispersion

5.146. There is a

the telescope,which it can resolve (assume = 0.55\177m). 5.147.Calculatethe minimum separationbetween two points on the Moon which can be resolvedby a reflectingtelescopewith mirrordiameter5 m. The wavelength of light is assumed to be equal = to 2\177

2\177

0.55

5.148. Determinethe \177tm.

minimum multiplicationof a telescope diameterof objectiveD = 5.0cm with which the resolving power of the objectiveis totally employed if the diameterof the eye's pupil is do --= 4.0mm. 5.i49.Thereis a microscope whose objective's numericalaperture is sin a 0.24, where a is the half-anglesubtendedby the objective'srim.Find the minimum separationresolvedby this microscope when an objectis illuminatedby light with wavelength 0.55 5.i50. Find the minimum magnification of a microscope, whose numericalapertureis sin a ----0.24,at which the resolvobjective's ing power of the objectiveis totally employed if the diameterof the eye's pupil is d o = 4.0ram. 5.151. A beam of X-rays with wavelength falls at a glancing angle 60.0on a linear chain of scatteringcenlreswith perioda. Find the angles of incidence corresY. ponding to all diffraction maxima = 2a/5. if with

-----

)\177

\177tm.

)\177

5.152.A 2\177

beam of X-rays with wavelength = 40 pm falls norreally on a planerectangulararray of scatteringcentresand produces a system of diffraction maxima

...... . . . .

(Fig. 5.29)on a plane screenremoved from the array by a distance l = iO cm. Find the array periodsa and b along the x and y axesif the

5.28.

---=

\302\260

\177,

Fig.

has a diameter

whose

D objective 5.0cm. Find the telescope resolvingpower of the objectiveand the minimum separation between two points at a distancel = 3.0km from ----

Fig.

\177

a\177

5.29.

distances between symmetrically locatedmaximaof secondorderare equal to Ax 60mm (along the x axis) and Ay = 40 mm (along the y axis). 5.153. A beam of X-rays impingeson a three-dimensional rectangular array whose periodsare a, b, and c.The directionof the incident beam coincides with the directionalongwhich the array period is equal to a.Find the directionsto the diffraction maximaand the wavelengths at which these maximawill be observed. A narrow beamof X-rays impingeson the natural facet 5.154. of a NaC1singlecrystal, whose density is p = 2.i6g/cma at a glancing ang|ea 60.0 The mirrorreflectionfrom this facet produces a maximum of second order. Find the wavelength of radiation. 5.i55. A beam of X-rays with wavelength 174pm falls on the surfaceof a singlecrystal rotating about its axiswhich is paral-----

5.144.A spectrograph'strihedral prism refractiveindexvaries with

is mhnufactured from wavelengthas n = A B/\177 2, glasswhose where A and B are constants, with B being equalto 0.0t0\177m 2. Making use of the formula from the foregoingproblem,find: (a) how the resolvingpower of the prism dependson =calculate ----434 nm and 656 nm in the vicinity of the value of if the width of the prism'sbase is b = 5.0cm; (b) the width of the prism'sbasecapableof resolvingthe yellow and 589.6 doublet of sodium (589.0 nm). How wide is the baseof a trihedralprism which has the 5.145. same resolving power as a diffraction grating with 10000lines \177-

2\177;

2\177/\1772\177

in the

224

secondorder of

2\177

\177.1

the spectrumifldn/d\177

0.10\177m-l?

\177

\302\260.

\177,

15---9451

-----

225)))

to the directionof the incident lelto itssurface and perpendicular to the maximaof secondand third

beam.In this casethe directions orderfrom the system of planesparallelto the surface of the single crystal form an angle ----60 between them.Find the corresponding interplanar distance. \302\260

\177

5.t56.On transmitting a

beam of X-rays with wavelength a system of diffracthrough a polycrystalline specimen a distancel = cm at screenlocated a tion rings is producedon of the radius t he Determine bright ring corresfrom the specimen. of planeswith the from of reflection order system to second ponding

pm. interplanardistanced-----i55

Malus'slaw:

-Imfn

(5.4a)

2 I Iocos q).

(5.45)

\177---

Brewster'slaw:

\342\200\242

tan \342\200\242

Fresnel equations two

tween

dielectrics:

Ix and I

0 of

for intensity 2

I\177

= I.L sin sin2

(01-03) '

(01q- 03) intensities of

I'll

reflectedat

light

the boundary

tan 2 tan 2 (01q- 02)

(01--03) '

be-

are the

(5.4d)

P'.

O0'

\342\200\242

Polarizers P and P'

6----2\177k

\302\260,

I = max \177

crossed

= min

... I.\177

[

I = min \177

I.\177

(5.4e)

= max

(5.4f)

\177)rtat

where

a is the

rotation

\177 tY,

constant,

q2mag

\177'--

VIH,

is Verdet'sconstant.

wave of natural light with intenA plane monochromatic 5.|57. of two touchingPolaroid on a screen

Io falls normally composed half-planes.The principaldirectionof one Polaroidis parallel, sity

=4.0 roW.

of energy of the incident ray is equal to (I)o

flux

beam of natural light falls on a system oiN 6 Nicol prisms whose transmissionplanes are turned eachthrough an angle

5.t60.A

---=

30 with respectto

that of the foregoingprism.What fraction passesthrough this system? 5.t61.Natural light falls on a system of three identicalin-line Polaroids,the principaldirectionof the middlePolaroidforming The maximum an angle = 60 with those of two other Polaroids. 0.81when transmissioncoefficient of eachPolaroidis equal to will the times How falls on them. many light plane-polarized intensity of the light decreaseafter its passing through the \302\260

-----

flux

\302\260

q\177

---=

\342\200\242

system?

5.t62.The degreeof polarizationof partially polarizedlight is 0.25.Find the ratio of intensitiesof the polarizedcomponent

component.

the way of partially polarized beam of light.When the prism is turned from the positionof maximum transmissionthrough an angle = 60 the intensity of transmitted light decreasedby a factor of = 3.0.Find the degreeof \302\260,

\177l

lution if the

q\177

Here 5 = 2n (n o --.ne)d/\177. is the phase difference between the ordinary 2, rays, k = 0, and magnetic rotation of the plane of polarization: \342\200\242

first half

5.t63.A Nicolprism is placedin

and

extraordinary Natural

hole;

\302\242o

of this light and the natural

\177=(2k-\177i)\177 I

parallel

the

Fresnelzone. 5.t59.A beam of plane-polarized light falls on a polarizerwhich rotatesabout the axisof the ray with angularvelocity = 21rad/s. Find the energy of light passingthrough the polarizerper one revo--

of luminous

incident light whose electricvector oscillations are respdctively perpendicularand parallel to the plane of incidence. If the angle between A crystalline plate between two polarizersP and of the plate is equal to 45 the plane of polarizer P and the optical axis of light which passesthrough the polarizer turns out to be the intensity either maximum or minimum under the following conditions: where

diameterof

of the

Imax ImaxW lmln

p.\177.

\342\200\242

(a) along the

OF LIGHT

inten-

sity Io falls normally on an opaquescreenwith round hole corresponding to the first Fresnelzone for the observation point P. Find the intensity of light at the point P after the hole was coveredwith two identicalPolaroidswhose principaldirectionsare mutually and the boundary between them passes perpendicular (b) along the circumferenceof the circlelimiting the

Degreeof polarization of light:

\342\200\242

5.t58.A plane monochromaticwave of natural light with

)\177-----

i7.8pm

----

5.4.POLARIZATION

to the boundary between them. and of the other perpendicular, What kind of diffraction pattern is formed behind the screen?What is the intensity of light behind the screenat the points of the plane perpendicularto the screenand passing through the boundary between the Polaroids?

polarizationof incident light. 5.t64.Two identicalimperfect polarizersare placedin the way of a natural beam of light. When the polarizers' planesare parallel, the system transmits \177----t0.0 times more light than in the case of crossedplanes.Find the degreeof polarizationof light produced (a) by each polarizerseparately: (b) by the whole system when the planes of the polarizersare parallel. beams of light of equal intenTwo parallelplane-polarized 5.165. form a smallangle between))) planes and sity whoseoscillation N\177

N\177

q\177

fall on a Nicolprism.To equalizethe intensities them (Fig.5.30) of the beamsemergingbehind the prism,itsprincipaldirectionN must be alignedalong the bisectingline A or B. Find the value of the angleq) at which the rotation of the Nicolprism through a small angle << q) from the positionA results in the fractional change of intensities of the beamsAI/I by the value = 100 timesexceeding that resulting due to rotation through the same angle from the \177q)

\177

positionB.

5.166.l\177esorting to the Fresnelequations, demonstratethat light reflectedfrom the surface of dielectricwill be totally polarized if the angleof incidence 01satisfiesthe conditiontan {)1----n, where n is the refractive index of the dielectric. What is in this casethe angle between the reflectedand refractedrays?

(b) the degreeof polarizationof the transmitted light if the light falling on the plate is natural. A narrow beam of natural light falls on a set of N thick 5.172. plane-parallelglassplates at the Brewster angle. Find: (a) the degreeP of polarizationof the transmitted beam; 2, 5, and 10. (b) what P is equal to when N 5.173. Using the Fresnelequations, find: (a) the reflection coefficient of natural light falling normally on the surface of glass; (b) the relativelossof luminous flux due to reflectionsof a paraxial ray of naturallightpassingthrough an alignedopticalsystemcomprising five glasslenses(secondaryreflectionsof light areto be neglected). 5.i74.A light wave falls normally on the surface of glasscoated with a layer of transparentsubstance. Neglectingsecondary reflections, demonstratethat the amplitudes of light waves reflected from tho two surfaces of such a laver will be equal under the condition n' \177where n' and n arethe refractive indicesof the layer and the glass respectively. 5.i75.A beam of natural light falls on the surface of glassat an angle of 45 Usingthe Fresnelequations,find the degreeof polari-\177

-----

\302\260.

zation of

(a) reflectedlight;

refracted

light. (b) 5.i76. UsingHuygens'sprinciple,construct the

Fig. 5.30. 5.i67.Natural light falls at the Brewsterangle on the surface of glass.Using the Fresnelequations,find (a) the reflectioncoefficient; (b) the degreeof polarizationof refractedlight. 5.168. A plane beam of natural light with intensity Io falls on the surface of water at the Brewsterangle.A fraction p ----0.039 of luminous

beam.

flux

is reflected.Find the intensity of the refracted

5.t69.A beam of plane-polarized light falls on the surface of water

at the Brewsterangle.The polarization plane of the electricvector of the electromagnetic wave makesan angle 45 with the incidence plane. Find the reflectioncoefficient. 5.170.A narrow beam of natural light falls on the surface of a thick transparent plane-parallelplate at the Brewster angle. As a result,a fraction p 0.080of luminous flux is reflectedfrom itstop surface.Find the degreeof polarizationof beams 1-4(Fig.5.3t) A narrow beam of light of intensity o falls on a planeparallelglass plate (Fig. 5.34) at the Brewsterangle. Using the Fresnelequations, find: if the oscillation (a) the intensity of the transmitted beam plane of the incident plane-polarized light is perpendicularto the \302\260

\177

--\177

5.171.

,/\177

incidenceplane; 228

the propagationdirectionsof the ordinary and in a positiveuniaxialcrystal whose

wavefronts and

extraordinaryrays

opticalaxis to the inci(a) is perpendicular dence plane and parallel to the surface of the crystal; (b) liesin the incidenceplane and is parallel to the surface of the crystal; (c) liesin the incidenceplane at an angle of 45 to the surface of the crystal,and light falls at right angles to the opticalaxis. 5.177.A narrow beam of na\302\260

tural light

with

wavelength

\177.

Fig.

5.32.

589 nm falls normally on the surface of a Wollaston polarizing The opticalaxes prism made of Icelandspar as shown in Fig. Find the of the two parts of the prism are mutually perpendicular. the o f the beams behind 6 the directions between prism if the angle angle 0 is equal to 30 What kind of polarizationhas a plane electromagnetiC of the vectorE on the x and y axesare perwave if the projections to the pendicular propagationdirectionand are defined by the

5.32.

\302\260

5.178.

following equations:

229)))

Ex = E cos --kz), Ey = E sin (cot -kz); (b) Ex = E cos(cot --kz), Ey = E cos(cot --kz 3/5); = E cos(cot --kz n)? (c) = E cos(cot --kz), 5.179. One has to manufacture a quartz plate cut parallelto its 0.50mm in thickness.Find the maxiopticalaxisand not exceeding mum thickness of the plate allowing plane-polarized light with = (a)

(o\177t

-\177-

E\177

wavelength

\177-

E\177

\177,

589 nm

(a) to experienceonly rotation of polarizationplane; (b) to acquire circularpolarization after passing through that plate.

\177tm

parallelto the opticalaxis.The axisof the plateforms an angle of 45 with the principaldirectionsof the Nicolprisms.The light transmittedthrough that system was splitinto the spectrum.How many dark fringes will be observed in the wavelength interval from 0.55 to 0.66\177tm? The difference of refractive indicesfor ordinary and extraordinary rays in that wavelength interval is assumed to be equal to 0.0090. A crystallineplatecut parallelto its opticalaxisis 0.25 mm thick and serves as a quarter-wave plate for a wavelength = = 530 rim. At what other wavelengths of visible spectrum will it also serve as a quarter-wave plate?The difference of refractive indicesfor extraordinaryand ordinary rays is assumed to be constant and equal to at all wavelengths of the visible no = 0.0090 \302\260

5.182.

\177,

ne-

A quartz plate cut parallelto its opticalaxis is placed 5.183. between two crossedNicolprisms so that their principledirections form an angleof 45 with the opticalaxisof the plate.What is the minimum thicknessof that plate transmittinglight of wavelength = 643 nm with maximum intensity while greatly reducingthe intensity of transmitting light of wavelength = 564 rim? The difference of refractiveindicesfor and ordinary rays = 0.0090 is assumedto be equal to ne no extraordinary for both wavelengths. A quartz wedge with refractingangle O= 3.5is inserted 5.184, between two crossedPolaroids. The optical axis of the wedge is parallelto its edge and forms an angle of 45 with the principal directionsof the Polaroids. On transmissionof light with wavelength = 550nm through this system, an interference fringe pattern is formed.The width of eachfringe is Ax = 1.0mm. Find the dif\177,1

\177,2

\302\260

\177,

2q\177.

(q\177

\302\260.

\302\260

\177

2q\177'

-\177-

5.180. A quartz plate cut parallelto the opticalaxis is placed between two crossedNicolprisms.The angle between the principal directionsof the Nicolprisms and the plateis equalto 45 Thethicknessof the plateis d = 0.50ram.At what wavelengths in the interval from 0.50to 0.60 is the intensity of light which passed through that system independentof rotation of the rear prism? The difference of refractiveindicesfor ordinary and extraordinary rays in that wavelength interval is assumed to be An-= 0.0090. 5.181. White natural light falls on a system of two crossedNicol prisms having between them a quartz plate t.50mm thick, cut

spectrum.

ferenceof refractive indicesof quartz for ordinary and extraordinary rays at the wavelength indicatedabove. 5.t85.Natural monochromatic light of intensity-Io falls on a system of two Polaroidsbetween which a crystallineplateis inserted, cut parallelto its opticalaxis. The plate introducesa phase difference between the ordinary and extraordinaryrays. Demonstrate that the intensity of light transmittedthrough that system is equalto I = t I0[cos2 --q)') --sin sin sin2 (6/2)],

are the anglesbetween the\302\260optical axisof the crystal In particular,consider of the Polaroids. and the principaldirections the casesof crossedand parallelPolaroids. falls normMonochromaticlight with circularpolarization ally on a crystallineplatecut parallelto the opticalaxis.Behind the plate there is a Nicolprism whose principaldirectionforms an that the angle with the opticalaxis of the plate.Demonstrate intensity of light transmitted through that system is equal to

where and q\177

q\177'

5.186. q\177

I = Io (t

-4- sin

2q\177.sin

8),

where 6 is the phase difference between the ordinary and extraordinary rays which is introducedby the plate. 5.187.Explainhow, using a Polaroidand a quarter-wave plate made of positive uniaxial crystal (ne no), to distinguish (a) light with left-hand circular polarizationfrom that with

right-hand polarization; (b) natural light from light with circularpolarizati-on and from of natural light and that with circularpolarization. the composition

5.188. Light with wavelength falls on a system of crossedpolarizerP and analyzer A between which a Babinetcompensator C The compensais inserted (Fig. 5.33). tor consistsof two quartz wedges with the optical axis of one of them being parallel to the edge, and of the other, perpendicularto it. The principaldirections of the polarizerand the analyser form an angleof 45 with the opticalaxes The refractingangle of the compensator. of the wedges is equal to 0 (O<<t) and the difference of refractive indices of \177

\302\260

--no. The insertion of quartz is investigatedbirefringentsampleS, with the optical axis oriented as shown in the figure, results in n\177

dis-

the fringes upward by 5x ram. Find: width of the fringe Ax; the (a) (b) the magnitude and the sign of the optical path difference of ordinary and extraordinary rays, which appears due to the

placementof

sample

S.)))

5.189.

Using the tablesof the Appendix, calculatethe difference of refractiveindicesof quartz for light of wavelength = 589.5 nm \177

and left-hand circularpolarizations. of wavelength 0.59Dm falls on a trihedralquartz prism P (Fig. 5.34) with refractingangle O -------30Inside the prism light propagatesalong the optical axis whose directionis shown by hatching.Behind

right-hand 5.190. Plane-polarizedlight

with

\302\260.

the

PolaroidPol an interference pattern

bright and dark fringes of width Ax----=- 15.0 mm is observed. Find the specificrotation constan-t of quartz and the distribution of intensity of light behind the Polaroid. 5.19l.Natural monochromaticlight falls on a system of two crossedNicol prisms between which a quartz plate cut at right angles to its optical axis is inserted.Find the minimum thicknessof the plate at which this system will transmit ----0.30of luminous Fig. 5.34. flux if the specific rotation constant of quartz is equal to = 17 ang.deg/mm. 5.|92.Light passesthrough a system of two crossedNicolprisms between which a quartz plate cut at right anglesto itsopticalaxis is placed.Determinethe minimum thickness of the plate which allows light of wavelength 436 nm to be completely cut off by the system and transmitshalf the light of wavelength 497 nm. The specific rotation constant of quartz for these wavelengths is equal to 41.5and 31.1angular degreesper mm respectively. 5.193. Plane-polarized light of wavelength 589 nm propagates alongthe axisof a cylindricalglass vesselfilled with slightly turbid

5.195. Monochromatic plane-polarized light with

angularfrequen-

certainsubstancealong a uniform magnetic field H. Find the difference of refractiveindicesfor right-handand left-hand componentsof light beam with circularpolarizationif the Verdetconstant is equal to V. 5.|96.A certainsubstanceis placedin a longitudinalmagnetic field of a solenoid locatedbetween two Polaroids. The length of the tube with substance is equal to l-----30 cm. Find the Verdet constant if at a field strength H 56.5kA/m the angle of rotation of polarizationplane is equal to = -} for one directionof the field and to \177------3\302\26020 for the opposite direction. 5.t97.A narrow beam of plane-polarized light passesthrough dextrorotatory positivecompound placedinto a longitudinalmagnetic field as shown in Fig.5.35. Find the angle through which the cy

\177o

passesthrough

-----

\1771

-5\302\260I0'

\177

s\302\242

sugar solutionof concentration500g/1.Viewing from the side,one can seea system of helicalfringes,with 50 cm between neighbouring dark fringes alongthe axis.Explainthe emergence of the fringes and determinethe specificrotationconstant of the solution.

5.194. A Kerrcell is positioned between two crossedNicolprisms so that the directionof electricfield E in the capacitorforms an

angleof 45 with the principaldirectionsof the prisms.The capacitor has the length l = 10cm and is filled up with nitrobenzene. Light of wavelength =-0.50Dm passes through the system. Taking into account that in this case the Kerr constant is equal to B \302\260

\177,

2.2.10 cm/V 2, find: -\177\302\260

strength of electricfield E in the capacitorat which the intensity of light that passesthrough this system is independent of rotationof the rear prism; (b) how many timesper secondlight will be interrupted when a sinusoidalvoltage of frequency v 10MHz and strength amplitude \177-50 kV/cm is appliedto the capacitor. Note.The Kerrconstantis the coefficient B in the equationne -(a) the

minimum

E,\177

B\177E

\177.

polarizationplane of the transmittedbeam will

turn

the length

of the tube with the compound is equal to the specificrotation constant of the compound is equal to so, the Verdetconstant is V, and the magneticfield strength is H. A tube of length l = 26 cm is filled with benzeneand placed in a longitudinalmagneticfield of a solenoid positionedbetween two Polaroids. The angle between the principledirectionsof the Polaroidsis equal to 45 Find the minimum strength of the magnetic field at which light of the wavelength 589 nm propagates through that system only in one direction(opticalvalve). What happensif the directionof the given magneticfield is changedto the opposite

5.198.

\302\260

one?

5.199. Experienceshows that a body irradiatedwith light with circularpolarizationacquiresa torque.This happensbecause such

a light possesses an angular momentum whose flow density in vacuum is equal to M I/o), where is the intensity of light, (o is the angular oscillation frequency. Suppose light with circular polarizationand wavelength = 0.70Dm falls normally on a uniform blackdiscof mass m = l0 mg which can freely rotate about itsaxis. How soon will its angular velocity becomeequal to (oo ----

-----

\177

= 1.0rad/s provided I = l0

W/cm\177?)))

5.5.DISPERSIONAND Permittivity

\342\200\242

ABSORPTION OF LIGHT of substanceaccordingto elementary n\177e2

e=t\177-

/me\302\260

\177 \177_\177

theory

of dispersion:

(5.5a)

is the concentration of electronsof natural frequency cook. Relation between refractive index and permittivity of substance:

where n h \342\200\242

Phase velocity

\342\200\242

v and group v

(5.5b)

\".

\"l]f\177

velocity u:

= co/k,

.==

(5.5c)

dco/dk.

Rayleigh's formula:

\342\200\242

(5.5d)

\177,

d--\177\"

Attenuation

of a

narrow

beam

I where and scattering. -\177

\303\227

\177t

\177-

\303\227',

\177,

\303\227' \303\227,

--\177

electromagneticradiation:

Ioe

are the coefficientsof linear attenuation,

A free electronis locatedin 5.200. co

3.4.i0 s -t.

(5.5e)

\177td,

wave. The intensity of light is light -la is

Find:

the field of a

absorption,

monochromatic

I = 150W/m 2, itsfrequency

electron's oscillationamplitudeand its velocity amplitude; (b) the ratio FroC,,where Fm and Fe are the amplitudesof forces (a) the

magneticand electriccomponentsof the light wave field act on the electron;demonstrate that that ratio is equal to l v/c, where v is the electron's velocity amplitude and c is the velocity of light. Instruction.The action of the magneticfield componentcan be in the equationof motionof the electronsincethe calcudisregarded lations show it to be negligible. with which the

-\177

5.201. An

wave electromagnetic

of frequency

\177)

propagatesin

dilute plasma.The free electronconcentrationin plasmais equal to n o. Neglectingthe interactionof the wave and plasmaions, find: (a) the frequency dependenceof plasma permittivity; wave depends (b) how the phase velocity of the electromagnetic on its wavelength in plasma. 5.202. Find the free electronconcentrationin ionosphereif its refractive index is equal to n = 0.90for radiowaves of frequency \177,

= 100MHz.

5.203. Assuming electronsof substanceto

be free when subjected

to hard X-rays, determineby what magnitudethe refractiveindex of graphitediffers from unity in the caseof X-rays whose wavelength in vacuum is equal to X = 50 pm. 234

and a \"friction force\" in the field of electromagnetic radiation.The E-component of the field varies as E ----Eo coscot. Neglectingthe action of the magnetic componentof the field, find: (a) the motion equation of the electron; (b) the mean power absorbed by the electron;the frequency at which that power is maximum and the expression for the maximum mean power. 5.205. In somecases permittivityofsubstanceturns out to be a complexor a negativequantity, and refractive index,respectively, a complex(n' = n -+- iu) or an imaginary (n' = quantity. Write the equation of a plane wave for both of thesecasesand find out the physical meaning of such refractive indices. 5.206. A soundingof dilute plasma by radiowaves of various frequencies reveals that radiowaves with wave\177x

i\303\227)

u----v-\342\200\242

An electron 5.204. a quasi-elastic force kx experiences

lengths exceeding = 0.75m experiencetotal internal reflection.Find the free electron concentration in \177o

that

plasma.

5.207.Using the definition of the group velocity u, derive Rayleigh's formula (5.5d).Demonstrate that in the

the vicinity of velocity u is equal to the segment v' cut by the tangent of the curve v at the point (Fig. 5.36). 5.208. Find the relationbetween the group velocity u and phase velocity v for the following dispersionlaws: \177+

\177

\177+'

(\177+)

v \177 \177/V\177 v \177 k; (c) v \177 l/co (\177)

(b)

\177.

Here

are the wavelength, wave number, and angular 5.209.In a certainmedium the relationshipbetween the group and phase velocitiesof an electromagnetic wave has the form uv = =c \177,

and

\302\242o

frequency.

+, where c is the velocity of light in vacuum.Find the dependence of permittivity of that medium on wave frequency, a The refractiveindex of carbon dioxideat the wavelengths 509,534, and 589 nm is equalto 1.647, and 1.630 1.640, respectively. Calculatethe phaseand group velocities of light in the vicinity of = 534 rim. A train of plane light waves propagatesin the medium where the phase velocity is a linearfunction of wavelength:v ----= a -fwhere a and b aresomepositiveconstants.Demonstrate +chat in such a medium the shapeof an arbitrary train of light waves is restoredafter the time interval + = l/b. A beam of natural light of intensity Io falls on a system of two crossedNicolprisms betweenwhich a tube filled with certain)))

5.2t0.

(\302\242o).

5.2tl. \177+

\177

b\177+,

5.212.

solution is placedin a longitudinalmagneticfield of strength H. The length of the tube is l, the coefficient of linearabsorption of solution is and the Verdet constant is V. Find the intensity of light transmitted through that system. 5.213. A plane monochromatic light wave of intensity Io falls normally on a plane-parallelplate both of whose surfaces have a reflection coefficient p. Taking into account multiplereflections, find the intensity of the transmitted light if (a) the plate is perfectly transparent, i.e.the absorption is \303\227,

absent;

(b) the coefficient of linearabsorptionis equal to

thickness is

5.214.

\303\227,

and the plate

Two plates,one of thicknessdl ----3.8mm and the other of thicknessd2 ----9.0 ram, are manufactured from a certainsubstance. When placed alternately in the way of monochromatic light, the first transmits %-\177 0.84fraction of luminousflux and the second, 0.70.Find the coefficient of linear absorptionof that substance.Light falls at right anglesto the plates. The secondare to be neglected. ary reflections A beam of monochromatic light passesthrough a pileof N----5identicalplane-parallelglass plates each of thickness 1 ----0.50 cm.The coefficient of reflectionat each surface of the plates is p 0.050. Theratio of the intensity of light transmittedthrough the pileof plates to the intensity of incident light is Neglectingthe secondary reflectionsof light, find the absorption coefficient of the given glass. A beam of monochromatic light falls normally on the surfaceof a plane-parallel plateof thickness The absorpticncoefficient of the substancethe plate is madeof varies linearly along the normal to its surface from to The coefficient of reflection at each surface of the plate is equal to p. Neglectingthe secondary reflections,find the transmissioncoefficient of such a plate. A beam of light of intensity o falls normally on a transparent plane-parallel plateof thickness The beam contains all the wavelengthsin the interval from to of equal spectralintensity. Find the intensity of the transmitted beam if in this wavelength interval the absorptioncoefficient is a linear function of with extremevalues and The coefficient of reflectionat each surface is equal to p. The secondary reflectionsare to be neglected. A light filter is a plate of thicknessd whose absorption coefficient dependson wavelength as \1772

5.215.

\177

5.2i9.A point sourceof monochromatic light emittinga luminous

(1) is positionedat the centreof a sphericallayer of substance. The insideradius of the layer is a, the outsideone is b. The coefficientof linearabsorptionof the substanceis equalto the reflection coefficient of the surfaces is equal to p. Neglectingthe secondary find the intensity of light that passesthrough that layer. reflections, 5.220. How many times will the intensity of a narrow X-ray beam of wavelength 20 pm decreaseafter passing through a lead mm if the mass absorption coefficient plate of thicknessd = for the given radiationwavelength is equal to \177/p---3.6 cm\177/g? A narrow beam of X-ray radiationof wavelength 62 pm penetratesan aluminium screen2.6cm thick. How thick must a lead screenbe to attenuate the beamjust as much? The mass absorptioncoefficientsof aluminiumand lead for this radiationare equal to 3.48and 72.0cm\177/g respectively. 5.222. Find the thicknessof aluminiumlayer which reducesby half the intensity of a narrow monochromatic X-ray beam if the 0.32cm\177/g. correspondingmass absorption coefficient is 5.223. How many 50%-absorption in the plate are there layers reducingthe intensity of a narrow X-ray beam vI 50 times? flux

\303\227,

1.0

5.22|.

\177

\177/p

-\177

5.6.OPTICSOF

\342\200\242----0.55.

5.2i6.

\303\2272.

\303\227\177

5.217.

I l.

\342\200\242

\177,

-- cm-l, = where a and Xo are constants.Find the passband of this light filter, that is the band at whose edgesthe attenuation of light is timesthat at the wavelength Xo. The coefficient of reflectionfrom the surfacesof the light filter is assumed to be the sameat all wave\303\227

(\177,)

o)---o)0

\"q

lengths.

236

the

source'smotion

i-- cos 0'

(5.6b)

\177

where

\177---\342\200\242

\342\200\242

If 0

ulc. -=--

the

Dopplereffect is called radial, and effect:

Vavilov-Cherenkov

el2, transverse.

cos O=\177

(5.6c)

O is the angle between city vector of a particle. where

the radiation

propagation

direction

and

the

velo-

\177

\177/\177o)\177

h\177,

(5.6a)

\342\200\242

\177

(\177

a is the velocity of a source, 0 is the angle between direction and the observation line. Dopplereffect in the general case:

\303\2272.

5.218.

\177

where

\177,,

\303\2271

Dopplereffect for

SOURCES

\177-----cos 0

\1772

\1771

MOVING

5.224.In the

Fizeauexperimenton measurementof the velocity light the distancebetween the gearwheel and the mirroris 1 = -= 7.0 kin, the number of teethis z = 720.Two successive disappearances of light are observed at the following rotation velocities of the wheel: = 283rps and n 2 \177--313 rps. Find the velocity of of

light.

n\177

237)))

A source 5.225. of light moveswith velocity v relativeto a receiver. Demonstratethat for v << c the fractional variation of frequency

of light is defined by Eq. (5.6a). 5.226. One of the spectrallinesemittedby excitedHe+ ions has a wavelength = 410nm. Find the Dopplershift of that line when observed at an angle 0 = 30 to the beam of moving ions A\177,

\177

\302\260

possessingkinetic energy

5.227.When

T----l0MeV.

0.59pm is obspectralline of wavelength served in the directions to the opposite edges of the solardisc along its equator,there is a difference in wavelengthsequalto = 8.0pm. Find the periodof the Sun's revolution about its own axis. 5.228. The Dopplereffect has made it possible to discoverthe doublestars which are so distant that their resolutionby meansof a telescope is impossible. The spectrallinesof such starsperiodically becomedoubletsindicatingthat the radiationdoescome from two a

-----

\177

A sourceemitting electromagnetic 5.236. signals with proper

s-1 moves at a constant velocity v-= frequency =- 0.80c alonga straight lineseparatedfrom a stationary observerP Find the frequency of the signalsperceived by a distance1 (Fig.5.37). by the observerat the moment when (a) the sourceis at the point O; (b) the observerseesit at the point O. \302\242o0----3.0-101\302\260

5\177

stars revolving about their centreof mass.Assuming the masses of the two stars to be equal,find the distancebetween them and their massesif the maximumsplitting of the spectrallinesis equal -4 and occurs ----t.2.10 to every = 30 days. 5.229. A planeelectromagnetic wave of frequency o falls normally on the surface of a mirrorapproachingwith a relativisticvelocity V. Making use of the Doppler formula, find the frequency of the reflected wave. Simplify the obtained expressionfor the caseV << c. 5.230. A radar operatesat a wavelength = 50.0cm. Find the (A\177/\177)m

Fig.

5.37.

Fig.

\302\242o

\177

velocity of an approachingaircraft if the beat frequency between the transmitted signal and the signal reflected from the aircraft is equal to 5v ------1.00kHz at the radar location. 5.231. Taking into account that the wave phase o)t-kx is an invariant, i.e.it retainsits value on transitien from one inertial frame to another, determinehow the frequency and the wave numberk enteringthe expression for the wave phase aretransformed. Examinethe unidimensionalcase. 5.232. How fast doesa certainnebula recedeif the hydrogen line = 434 nm in itsspectrumis displaced by t30nm toward longer \302\242o

5.237.A narrow beam of electronspassesimmediately over the surface of a metallicmirrorwith a diffraction grating with period inscribedon it. The electronsmove with velocity v, d ----2.0 \177tm

comparableto c, at right angles to the lines of the grating. The trajectoryof the electronscan be seen in the form of a strip, whose Interpret colouringdependson the observationangle 0 (Fig.5.38). this phenomenon.Find the wavelength of the radiationobserved at an angle 0

= 45%

of atoms of mass m beingin A gas consists 5.238.

wavelengths? 5.233. How fast should a car move for the driver to perceivea red traffic light \177 0.70\177tm) as a green one \177 0.55 pm)? 5.234. An observermoves with velocity vl = c alonga straight (\177,

line.In front

(\177,'

-\177

of him a sourceof monochromaticlight moves with c in the samedirectionand along the samestraight

3 v 2 ---\177

line.The properfrequency of light is equalto O)o.Find the frequency

light registeredby the Observer. 5.235. One of the spectrallinesof atomichydrogen has the wavelength X =656.3rim. Find the Dopplershift of that line when observed at right anglesto the beam of hydrogen atoms with kinetic energy T = 1.0MeV (the transverse Dopplereffect). of

5\177

thermodynamic

equilibriumat temperatureT. Suppose is the natural frequency of light emittedby the atoms. (a) Demonstratethat the spectraldistribution of the emitted light is defined by the formula \302\242Oo

\177,

velocity

5.38.

\177

I\177

\177

Ioe-a

(1-\177/\177Oo\1772

the spectralintensity correspondingto the frequency mc2/2kT). (b) Find the relativewidth hco/COoofa given spectralline, i.e. the width of the line between the frequenciesat which Io/2. wave propagatesin a medium 5.239. A :plane electromagnetic moving with constantvelocity V << c relativeto an inertialframe K. Find the velocity of that wave in the frame K if the refractive index of the medium is equal to n and the propagationdirectionof the

(Io= is

\302\242Oo,

-----

I\177o

wave

coincideswith

that of the

5.240. Aberration of

medium.

of stars light is the apparent displacement

attributableto the effect of the orbitalmotion of the Earth. The and the directionto a star in the ecliptic plane varies periodically, ----41\".Find within an angle star performs apparent oscillations the orbitalvelocity of the Earth. \1770

239)))

5.241. Demonstratethat the angle 0 between the propagation directionof light and the x axis transforms on transition from the referenceframe K to K' accordingto the formula cos0--\177 cos0'--t-cos0 '

\342\200\242

\177 \177

V/c

to the frame

and

K' with respect x' axesof the reference frames coincide.

is the velocity of the frame

K. The x and

5.242.Find the aperture angle of a cone in which all the stars for an observeron the Earth will be visible locatedin the semi-sphere if one moves relativeto the Earth with relativistic velocity V from the velocity of differing by 1.0% of the foregoingproblem.

light.Make use of the formula

uniformly through a medium with refractive index n emits light (the Vavilov-Cherenkov effect). Find also the directionof that

radiation. Instruction.Considerthe interferenceof oscillations induced by the particleat various momentsof time. 5.244.Find the lowest values of the kineticenergy of an electron of Cherenkov'sradiationin and a proton causing the emergence a medium with refractive index n i.60.For what particlesis --29.6MeV? this minimum value of kineticenergy equal to 5.245.Find the kinetic energy of electronsemitting light in at an angle 0 = 30 to a medium with refractive index n ----1.50 their propagation direction. --\177

T\177n\177n

\302\260

QUANTUM

RADIATION. NATURE OF

Me = where u \342\200\242

\177\"

(5.7a)

is the spacedensity of thermal radiation energy. Wien's formula and Wien's displacementlaw: o)3F (o)/T), m = b, \177-

\177 \342\200\242

is the

wavelength

Stefan-Boltzmann

corresponding to

the maximum

Me = (rT

\342\200\242

the Stefan-Boltzmann

Planck'sformula:

e Einstein'sphotoelectricequation: ha)----A

-\177

\177.

\177-

T\177

\177un

\177

\177.

\177

1.2

temperaturedecreases 2.0times. 5.252. Thereare two cavities(Fig.5.39) with smallholesof equal diametersd i.0cm and perfectly reflectingouter surfaces. The -----

\177l

of the

function

u\177.

(5.7d)

distancebetween the holesis

\177(\302\260/hT-1

c,\177

T3; (c) the radiosity Me \177 T 5.247.The temperatureof one of the two heated blackbodiesis 2500K. Find the temperatureof the other body if the wavelength correspondingto its maximumemissive capacityexceeds 0.50 the wavelength correspondingto the maximum by Ak emissivecapacityof the first blackbody. 5.248.The radiosity of a blackbody is M, 3.0W/cm Find the wavelength correspondingto the maximumemissivecapacity of that body. 5.249.The spectra], compositionof solarradiationis much the sameas that of a blackbody whose maximumemissioncorresponds to the wavelength 0.48pro. Find the mass lostby the Sun every seconddue to radiation.Evaluate the time interval during which the mass of the Sun diminishesby I per cent. 5.250. Find the temperatureof totally ionizedhydrogen plasma of density p 0.10g/cma at which the thermal radiation pressure is equal to the gas kineticpressureof the particlesof plasma. Take into accountthat the thermal radiationpressurep u/3, where u is the spacedensity of radiationenergy, and at high temperatures all substancesobey the equationof state of an idealgas. A copperball of diameterd cm was placedin an evacuatedvessel whosewalls are kept at the absolutezero temperature. The initial temperatureof the ball is To = 300K. Assuming the surface of the ball to be absolutely black, find how soon its

5.39. ----10cm. A constant temperature

Fig.

\177

:\1772c3

\177

c\177

constant.

u(\302\260

\342\200\242

o)p\177

(5.7c)

\177,

where (r

\177c

(5.7b)

T\177,

law:

(5.7f)

= Mmc is Compton'swavelength. 5.246.UsingWien's formula, demonstratethat (a) the most probableradiationfrequency T; (b) the maximum spectraldensity of thermalradiation(u\177)ma::

where

\177

u\302\242o

where

2.\177:\177

5.251.

LIGHT

Radiosity

\342\200\242

c (t -- cos0),

----

\177-

5.243.Find the conditionsunderwhich a chargedparticlemoving

5.7.THERMAL

effect: 5\177,

\177

where

Compton

\177 T\177

(5.7e)

1700K

1.Calculatethe steady-state

is maintained in cavity

temperatureinsidecavity 2. Instruction.Takeinto accountthat the cosineemissionlaw.)))

blackbody radiation obeys

A cavity 5.253.

of volume

tion at a temperatureT

----

i.0 1 is filled with

i000K. Find:

(a) the heat capacity Cv; (b) the entropy S of that radiation. 5.254. Assuming the spectraldistribution 3of thermal radiation where to energy obey Wien's formula u (co, T) = Ace exp (--aceC), 2000K the most T find for a temperature 7.64 a -\177

probable

ps.K,

---=

u\177

t\177co

\177

u\177

)\177

find the power radiatedby ---blackbody within a narrow wavelength interval radiation ---density at nm closeto the maximumof spectral ---of the K T 3000 body. a temperature of the function g (x) representing 5.258. Fig.5.40shows the plot a fraction of the total power of thermal radiation falling within

5.257.Using Planck'sformula,

|.0

determining a photons per cavity rature T in the spectralintervals (co, co-4-do))and ()\177,)\177tempe5.260. An point source emits light with wavelength -- 589nm.isotropic The radiation power of the sourceis P = :10W. Find: (a) the mean density of the flow of photons at a distancer-\177 d\177,).

\177,

A\177,

2.0m

from the source; the distancebetween the sourceand the point at which the (b) mean concentrationof photons is equal to n i00cm -3. 5.26toFrom the standpointof the corpusculartheory demonstrate that the momentum transferred by a beam of parallellight rays per unit timedoesnot depend on itsspectralcompositionbut depends only on the energy flux (I)e. 5.262.A laseremitsa light pulse of duration ----0.\1773 ms and energy E ----i0J. Find the mean pressureexertedby such a light pulse when it is focussed into a spot of diameterd i0 on a surface perpendicular to the beam and possessing a reflection coefficient p ----0.50. 5.263. A short light pulse of energy E = 7.5 J falls in the form of a narrow and almostparallelbeam on a mirrorplatewhose reflection coefficient is p 0.60.The angle of incidenceis 30 In terms of the corpuscular theory find the momentum transferred to the -\177

(a) radiation frequency; (b) radiationwavelength. 5.255. expressions UsingPlanck'sformula, derive the approximate of radiation for the spacespectraldensity << kT (Rayleigh-Jeansformula); (a) in the rangewhere (b) in the range where hco kT (Wien'sformula). Transform Planck'sformula for spacespectraldensity 5.256. of radiationfrom the variableco to the variablesv (linearfrequency) and (wavelength). a unit

area of

5.259. Making use of Planck'sformula, derive the expressions the number of i cm3 of a at

thermalradia-

-----

\177

-----

\177tm

\177-

\302\260.

plate.

5.264. A plane light wave of intensity := 0.20W/cm falls on a plane mirrorsurface with reflectioncoefficient p = The angle of incidence is 45 In terms of the corpuscular theory find the magnitude of the normal pressureexertedby light on that surface. 5.265. A plane light wave of intensity = 0.70W/cm illuminates a spherewith idealmirrorsurface.The radius of the sphereis -- 5.0cm. From the standpoint of the corpusculartheory find the force that light exertson the sphere.

\302\260

\177

0.8.

\177

/\177

An isotropic 5.266. point

Fig.

is the wavelength from 0 to x.Herex = r adiation of t o maximum the density). spectral corresponding Using this plot,find: (a) the wavelength which divides the radiationspectrum into two equal (in terms of energy) parts at the temperature3700K; (b) the fraction of the total radiationpower falling within the visible range of the spectrum (0.40-0.76 \177m) at the temperature the

spectralinterval

\177,/\177,m

(\177,m

5000K;

0.76

\177tm

242

risesfrom will increaseif the temperature

3000to 5000K.

sourceof radiation power P is located

on the axisof an idealmirrorplate.The distancebetween the source and the plateexceedsthe radius of the plate \177l-fold. In termsof the corpuscular theory find the force that light exertson the plate. 5.267.In a referenceframe K a photon of frequency co falls normally on a mirrorapproachingit with relativisticvelocity V. Find the momentum imparted to the mirrorduring the reflection of the

photon

frame fixed to the mirror; (a) in the reference (b) in the frame K. 5.268. A smallidealmirrorof mass m ----:10 is suspendedby a weightlessthread of length l t0 cm. Findmgthe angle through which the thread will be deflected when a short laser pulse with ---J is shot in the horizontaldirectionat right energy E angles to the mirror.Where doesthe mirrorget itskineticenergy? A photon of frequency coo is emittedfrom the surface of 5.269. a star whose mass is M and radius R. Find the gravitationalshift \177

\1773

1\177\"

243)))

of frequency Aa)/a) 0 of the photon at a very great distancefrom tile

star.

an X-ray tube being increased ---times,the short-wave limit of an X-ray continuousspectrum shifts by A\177,----26 pro. Find the initial voltage applied to the

5.270.A voltage appliedto =

1.5

\177l

tube.

5.27t.A narrow X-ray beam falls on a NaC1single crystal.The least angle of incidenceat which the mirror reflectionfrom the system of crystallographicplanes is stillobserved is equal to a = 4.1 The interplanardistanceis d = 0.28nm. How high is the voltageappliedto the X-ray tube? 5.272.Find the wavelength of the short-wave limit of an X-ray continuousspectrum if electronsapproachthe anticathode of the tube with velocity v 0.85c, where c is the velocity of light. thresholdfor zinc and the maximum 5.273.Find the photoelectric liberatedfrom its surface by electromagvelocity of photoelectrons netic radiationwith wavelength 250 nm. 5.274.Illuminating the surface of a certainmetal alternately \302\260.

--\177

----0.54 it was light of wavelengths ----0.35 and of photoelectrons velocities found that the correspondingmaximum with

)\1771

\177tm

\177tm,

)\177z

2.0.

= Find the work function of that metal. differ by a factor 5.275.Up to what maximumpotentialwill a copperball,remote be chargedwhen irradiatedby electromagnetic from all other bodies, radiationof wavelength = 140nm? liberat5.276.Find the maximumkineticenergy of photoelectrons radiationwhose ed from the surface of lithium by electromagnetic electriccomponentvaries with time as E = a (1 4- cosa)t) cosa)ot, and a)o = 3-60\"1015 where a is a constant, a) = 6.0-t01\177s of 5.277.Electromagneticradiation wavelength =-0.30 falls on a photocelloperatingin the saturationmode.The correspond4.8mA/W. Find the ing spectralsensitivity of the photocellis J of number produced photoelectrons yield of photoelectrons, the \177l

\177,

-1

s-I

\342\200\242

)\177

i.e.

\177

by each incident photon. 5.278,There is a vacuum photocellwhose one electrodeis made of cesiumand the other of copper.Find the maximumvelocity of

approachingthe copperelectrodewhen the cesium photoelectrons radiationof wavelength electrodeis subjectedto electromagnetic the cell. 0.22 and the electrodesare shortedoutside the circuit of a vain c urrent A 5.279. photoelectric emerging to electromagnetic is subjected when its zinc electrode cuum photocell radiationof wavelength 262 nm is cancelledif an externaldecelerat\177m

Find the magnitudeand polarity of ing voltage i.5V is applied. the outer contactpotentialdifference of the given photocell. 5.280. Composethe expressionfor a quantity whose dimension islength,usingvelocity of light c, mass of a particlem, and Planck's constanth. What is that quantity? 5.28|. Using the conservationlaws, demonstratethat a free electroncannot absorb a photon completely. 244

5.282. Explainthe

light by matter:

following features of Compton

scatteringof

is independentof the nature of (a) the increasein wavelength the scatteringsubstance; (b) the intensity of the displacedcomponentof scatteredlight grows with the increasingangle of scatteringand with the diminishing atomicnumber of the substance; (c) the presenceof a non-displacedcomponentin the scattered A)\177

radiation.

5.283. A narrow monochromatic X-ray beam falls on a scattering substance.The wavelengths of radiationscatteredat angles01= 60 = 120differ by a factor = 2.0.Assumingthe free electrons and to be responsible for the scattering,find the incidentradiationwavelength. 5.284.A photon with energy ha) = 1.00MeV is scatteredby a stationary free electron.Find the kinetic energy of a Compton electronif the photon's wavelength changedby = 25% due to scattering. = 6.0pm is scatteredat right A photon of wavelength 5.285. a free electron. F ind: by angles stationary (a) the frequency of the scatteredphoton; (b) the kineticenergy of the Compton electron. A photon with energy ha) = 250 keV is scatteredat an 5.286. angle 0 = t20 by a stationary free electron.Find the energy of the scatteredphoton. \302\260

\302\260

0\177

\1771

\177l

\177,

\302\260

5.287.A photon with momentum MeV/c, where c is the velocity of light, is scatteredby a stationary free electron, MeV/c. changingin the processitsmomentum to the value ----0.255 At what angle is the photon scattered? 5.288. A photon is scatteredat an angle0 = i20 by a stationary free electron.As a result, the electronacquiresa kinetic energy T -- 0.45MeV. Find the energy that the photon had priorto scat-

p----i.02 p' \302\260

tering. 5.289. Find the wavelength of X-ray radiationif the maximum kineticenergy of Compton electronsis T,\177 = 0.i9MeV. A photon with energy ha)= 0.i5MeV is scatteredby 5.290. ----3.0pm. a stationary free electronchangingitswavelength by Find the angle at which the Compton electronmoves. 5.29t.A photon with energy exceeding = 2.0times the rest with a staa head-on collision energy of an electronexperienced tionary free electron.Find the curvature radius of the trajectoryof the Compton electronin a magneticfield B ----0.t2T. The Compton electronis assumed to move at right anglesto the directionof the A)\177

\177l

field.

5.292.Having collided with a relativisticelectron,a photon is deflectedthrough an angle 0 = 60 while the electronstops.Find the Compton displacement of the wavelength of the scatteredphoton.))) \302\260

(a) the least curvature radius of its trajectory; (b) the minimum approachdistancebetween the particleand the

PART SIX A I

nucleus.

OM!CAND NUCI,EARPItYSIC\177 :\177

\177

6.t.SCATTERING

OF PARTICLES. RUTHERFORD-BOHR ATOM

0 at which a charged particle is deflected by the Coulomb field atomic nucleus is defined by the formula: 0 qlq', tan (6.ta) 2 2bT where ql and q2 are the chargesof the particle and the nucleus, b is the aiming parameter, T is the kinetic energy of a strikAngle stationary

ing

particle.

Rutherford formula. The relative number of particles scattered into an elementary solid angle dQat an angle 0 to their initial pro\342\200\242

pagation

{

qlq\177

, 2

d\177

sin 4 (0/2)

\177

\177-\177-\177--]

(6.tb)

where n is the number of nuclei of the foil unit area of its surface,dfl sin 0 dO tiT. GeneralizedBalmer formula (Fig.

\342\200\242

where o) is the transition tum numbers nl and n\177,

hydrogen-like

ion.

frequency (in is the Rydberg

6.t): Fig.

6.t.

energy levels with quanconstant, Z is the serial number of a

6.t.Employing Thomson'smodel,calculatethe radiusof a hydrogenatom and the wavelength of emittedlight if the ionizationenergy of the atom is known to be equal to E = i3.6eV. 6.2.An

alpha particlewith kineticenergy 0.27MeV is deflected

through an angle of 60 by a golden foil.Find the corresponding value of the aiming parameter. To what minimum distancewill an particle with kineticenergy T = 0.40MeV approachin thealpha case of a head-on collision to (a) a stationary Pb nucleus; (b) a stationary free Li7 nucleus? 6.4.An alpha particlewith kinetic energy T = 0.50MeV is deflected through an angle of 0 = 90 by the Coulomb field of a \302\260

6.3.

\302\260

stationary Hg nucleus.Find: * All the formulas

246

in this

Part are given

in the

</3,

/\177

per

s -I ) between

r>R,

0 for --Uofor r

where r is the distancefrom the centreof the well.Find the relationship between the aimingparameterb of the particleand the angle0 through which it deflects from the initial motion direction. 6.8.A stationary ball of radius is irradiatedby a parallel of streamof particleswhose radius is r. Assuming the collision a particleand the ball to be elastic,find: (a) the deflection angle 0 of a particleas a function of its aiming

direction: dN N

1\177

\177

\342\200\242

of a

6.5.A proton with kineticenergy T and aimingparameterb was deflected by the Coulombfield of a stationary Au nucleus.Find the momentum imparted to the given nucleusas a result of scattering. 6.6.A proton with kineticenergy T MeV flies past a stationary free electronat a distanceb i0 pm. Find the energy acquired by the electron,assumingthe proton's trajectoryto be as the proton rectilinearand the electronto be practicallymotionless flies by. 6.7.A particlewith kineticenergy T is deflected by a spherical potentialwell of radius R and depth U0, i.e.by the field in which the potentialenergy of the particletakesthe form

Gaussiansystem of units.

parameterb; with the ball (b) the fraction of particleswhich after a collision arescatteredinto the angularinterval between 0 and 0 4- dO; (c) the probability of a particleto be deflected,after a collision with the

ball,into

the front hemisphere(0 <\177).MeV with kineticenergy A narrow beam of alpha particles scattered The thick. falls normally on a platinum foil particlesare observed at an angle of (10 tottmthe incidentbeam direction cm locatedat by meansof a counterwith a circularinlet area the distance10cm from the scatteringsectionof the foil.What fraction of scatteredalpha particlesreachesthe counterinlet? A narrow beam of particleswith kineticenergy T = = 0.50MeV and intensityalpha I= particlesper secondfalls normally on a goldenfoil.Find the thicknessof the foil if at a distance r = 15cm from a scatteringsectionof that foil the flux density of scatteredparticlesat the angle 0 = 60 to the incident beam is equal to J = 40 particles/(cm\177-s). A narrow beam of alpha particlesfalls normally on a silver foil behind which a counteris set to registerthe scatteredparticles. On substitutionof platinum foil of the samemass thicknessfor the silver foil,the number of alpha particlesregisteredper unit time increased = t.52times. Find the atomicnumber of platinum,

6.9.

1.0

1.0 \302\260

1.0

6.t0.

\177

5.0-10 \177

\302\260

6.1t.

\1771

247)))

assumingthe atomicnumber of silver and the atomicmassesof both platinum and silver to be known.

6.t2.A

narrow beam of alpha particleswith kineticenergy T ---falls normally on a golden foil whose mass thickness is pd ---mg/cm The beam intensity is I 0 ---- t05 particles of alpha particlesscatteredby the foil per second.Find the number during a time interval x = 30min into the angularinterval: (a) (b) over 0o ----60 A narrow beam of protons with velocity v = 6-106 m/s falls normally on a silver foil of thicknessd = 9m. Find the probability of the protons to be scatteredinto the rear hemisphere (0 6.t4.A narrow beam of alpha particleswith kineticenergy T = 600keV falls normally on a golden foil incorporating n ---l019nuclei/cm Find the fraction of alpha particlesscattered through the angles 0 \177 0o = 20 A narrow beam of protons with kineticenergy T = 1.4MeV falls normally on a brass foil whosemass thicknesspd= mg/cme. The weight ratio of copperand zinc in the foil is equal to 7 3 respectively. Find the fraction of the protons scatteredthrough the 0o = 30 anglesexceeding Find the effective crosssectionof a uranium nucleus correspondingto the scatteringof alpha particleswith kineticenergy T ---MeV through the angles exceeding 0o 60 The effective crosssectionof a gold nuclenscorresponding to the scatteringof monoenergetic alpha particleswithin the angular interval from 90 to 180is equal to Ao = 0.50kb. Find\" (a) the energy of alpha particles; (kb/sr) cor(b) the differential crosssectionof scattering = to 0 60% the angle responding an electron In accordancewith classicalelectrodynamics w losesits energy due to radiationas moving with acceleration

0.50MeV

----

i.5

5.0.

\177.

\302\260

59-6i\302\260;

6.13.

1.0

90\302\260).

t.i.

\177.

\302\260

6.t5.

t.5

6.16. 1.5

\302\260.

--\177

6.|7.

\302\260.

\302\260

d(\177/d\177

6.t8.

\177

-\177=--\177

w\177,

where e is the electroncharge,c is the velocity of light.Estimatethe time during which the energy of an electronperforming almost with frequency o) --:-5.101\177s-\177 will decrease harmonicoscillations

= t0 times.

6.t9.Making use of the formula of the foregoingproblem,estimate the timeduringwhich an electronmoving in a hydrogen atom along a circularorbit of radius r = 50 pm would have fallen onto the nucleus.Forthe sake of simplicityassumethe vectorw to be perma\1771

nently directedtoward the centreof the atom. 6.20.Demonstratethat the frequency o) of a photon emerging when an electronjumps between neighbouringcircularorbits of a hydrogen-likeion satisfies the inequality o), \177 o) \177 (o\177+1, where o)\177

and

248

o)\177

+\177

arethe frequenciesof revolutionof that electronaround

Make sure that as n the the nucleusalong the circularorbits. of o) the frequency photon A particleof mass rn moves alonga circularorbit in a centrosymmetrical potentialfield U (r) -= kr\177/2. Using the Bohr quantizaorbitalradii and energy levels tion condition,find the permissible of that particle. 6.22.Calculatefor a hydrogen atom and a He+ ion: (a) the radiusof the first Bohrorbitand the velocity of an electron moving along it; (b) the kineticenergy and the binding energy of an electronin the ground state; (c) the ionization potential,the first excitationpotential and n = the wavelength of the resonanceline (n' = 2 6.23.Calculatethe angular+frequency of an electronoccupying the secondBohr orbit of He ion. 6.24.For hydrogen-likesystems find the magnetic moment correspondingto the motion of an electronalong the n-th orbit and the ratio of the magnetic and mechanicalmoments Calculatethe magneticmoment of an electronoccupying the first --\177

6.2t.

\302\242\177

--\177

l).

\177t,\177/M,\177.

Bohr orbit.

6.25.Calculatethe magnetic field induction at the centre of a hydrogen atom causedby an electronmoving alongthe first Bohr orbit.

6.26.Calculateand

draw on the wavelength scale the spectral

intervalsin which the Lyman, Balmer,and Paschenseriesfor atomic hydrogen are confined. Show the visible portion of the spectrum. 6.27.To what seriesdoesthe spectralline of atomichydrogen belongif its wave number is equalto the difference betweenthe wave 486.tand numbers of the following two linesof the Balmerseries: 4i0.2nm? What is the wavelength of that line?

6.28.For

the caseof atomichydrogen find: of the first three lines of the Balmerseries; the wavelengths (a) of a spectralinstrument (b) the minimum resolvingpower series. of the Balmer the first 20 lines of capable resolving 6.29.Radiationof atomichydrogenfalls normally on a diffraction grating of width l = 6.6mm. The 50th line of the Balmerseries in the observedspectrumis closeto resolutionat a diffraction angle0 with Rayleigh'scriterion).Find that angle. (in accordance 6.30.What elementhas a hydrogen-likespectrum whose lines have wavelengths four timesshorterthan thoseof atomichydrogen? How many spectrallines are emittedby atomichydrogen excitedto the n-th energy level? 6.32.What lines of atomichydrogen absorptionspectrum fall within the wavelength range from 94.5to i30.0nm? Find the quantum number n correspondingto the excited state of He+ ion if on transition to the ground state that ion emits with wavelengths t08.5and 30.4nm. two photons in succession \177,/5\177,

6.3i.

6.33.

249)))

6.34.Calculatethe Rydberg constant /t if He+ ions are known to have the wavelength difference between the first (of the longest =wavelength) lines of the Balmerand Lyman seriesequal to A\177.

= 133.7nm.

6.35.What

hydrogen-likeion has the wavelength difference between the first linesof the BalmerandLyman seriesequal to 59.3nm? 6.36.Find the wavelength of the first line of the He+ ion series whose interval between the extreme lines isspectral Ao)=

= 5.t8.t015 s-l. 6.37.Find the binding

energy of an electronin the ground state of hydrogen-likeions in whos\177 spectrumthe third line of the Balmer seriesis equal to t08.5nm. 6.38.The bindingenergy of an electronin the ground state of He atom is equal to E o 24.6eV. Find the energy requiredto remove both electronsfrom the atom. 6.39.Find the velocity of photoelectrons liberatedby electromagnetic radiationof wavelength 18.0nm from stationary He+ ions in the ground state. 6.40.At what minimum kineticenergy must a hydrogen atom move for its inelastichead-oncollision with another, stationary, hydrogen atom to make one of them capableof emitting a photon? Both atoms are supposed to be in the ground state prior to the colli\177-

\177,--\177

sion.

6.41.A stationary hydrogen atom

to the first line of the Lyman

acquire?

emitsa

series. What

photon corresponding

velocity doesthe atom

6.42.

From the conditionsof the foregoingproblem find how much (in percent)the energy of the emittedphoton differs from the energy of the correspondingtransition in a hydrogen atom.

6.43.A

stationary He+ ion emitteda photon correspondingto the first line of the Lyman series. That photon liberateda photoelectron from a stationary hydrogen atom in the ground state.Find the velocity of the photoelectron. 6.44.l\177ind the velocity of the excitedhydrogen atoms if the first line of the Lyman seriesis displacedby = 0.20nm when their radiationis observed at an angle0 45 to their motiondirection. 6.45.According to the Bohr-Sommerfeldpostdlatethe periodic motion of a particlein a potentialfield must satisfy the following

where a is (c) in a unidimensionalpotentialfield U----axeC, a positive constant; (d) along a round orbit in a centralfield, where the potential energy of the particleis equal to U ---(a is a positive con-

--air

stant).

6.46.Takinginto accountthe motionof the nucleusof a hydrogen

for the electron's atom,find the expressions binding energy in the ground state and for the Rydberg constant.How much (in per cent)

do the bindingenergy and the Rydberg constant, obtainedwithout

taking into accountthe motion of the nucleus,differ from the more accuratecorrespondingvalues of these quantities? 6.47.For atoms of light and heavy hydrogen (H and D) find the difference (a) between the binding energiesof their electronsin the ground\177

state;

(b) between the wavelengths of first lines of the Lyman series. 6.48.Calculatethe separationbetweenthe particlesof a system and

the ground state, the correspondingbinding energy, if line of such a of the the first series, system is wavelength Lyman (a) a mesonic hydrogen atom whosenucleusis a proton (in a mesonic atom an electronis replacedby a mesonwhose charge is the sameand mass is 207 that of an electron); (b) a positronium consistingof an electronand a positronrevolving in the

6.2.WAVE

PROPERTIESOF PARTICLES. SCHRODINGER EQUATION The de Broglie wavelength of a particle with momentum p: \342\200\242

2\177/i \177,__

\342\200\242

principle:

Uncertainty

\302\260

\342\200\242

(6.2a)

Ax Apx

A\177,

\177

masses.

around their common centre of

and

SchrSdinger time-dependent

\177

(6.2b)

time-independent equations:

quantization rule:

\177

p dq

2n\177n,

where q and p are generalizedcoordinateand momentum of the particle, n are integers.Makinguse of this rule, find the permitted valuesof energy for a particleof mass m moving (a) in a unidimensionalrectangular potentialwell of width l with infinitely 25O

V\177,-}---\177

where W

the

total wave

operator,E and U are the aI coordinates:

high walls;

(b) alonga circleof radius r;

(6.2e)

- (E--U)

. 2

V\177__ \"\177V-P-'-\177-

function, total and

-\177'r

\177r\177

\177p-\177O,

is its coordinate.part, V is the Laplace potential energieso[the particle. In spheric\177

\177p

O i sin 0 00 (sin0\1770)_+ sin r\177

\1770

0q\177

(6.2d) 251)))

\342\200\242

Coefficient of transparency

of a potential barrier

next maximummirrorreflectionis known to be observedwhen acceleratingvoltage is increased = 2.25times. electronsfalls normally 6.61.A narrow beam of monoenergetic on the surface of a Ni single crystal.The reflectionmaximumof fourth orderis observed in the directionforming an angle 0 = 55 with the normal to the surface at the energy of the electronsequal the corresponding value of the interplanar to T = 180eV.Calculate distance. 6.62.A narrow stream of electronswith kinetic energy T = = i0 keV passesthrough a polycrystalline aluminium foil, forming the interplanar a system of diffraction fringes on a screen.Calculate distance\177orresponding to the reflectionof third orderfrom a certain for a diffraction ring of system of crystal planes if it is responsible diameterD = 3.20cm.The distancebetween the foil and the screen is l = t0.0cm. 6.63.A stream of electronsacceleratedby a potentialdifference V ----i5 V. falls on the surface of a metalwhose inner potentialis if the

(x):

the

(6.2e)

\177l

\302\260

where

x1 and x2 are the coordinatesof the points between which

> E.

6.49.Calculatethe

de Brogliewavelengthsof an electron,proton, atom,all having the same kineticenergy i00eV. 6.50.What amount of energy should be added to an electronto reduceits de Brogliewavelength from t00to 50 pm? 6.51.A neutron with kineticenergy T 25 eV strikes a stationary deuteron (heavy hydrogen nucleus). Find the de Broglie wavelengthsof both particlesin the frame of their centreof inertia. 6.52.Two identical non-relativisticparticles move at right anglesto each other, possessingde Brogliewavelengths kl and Find the de Brogliewavelength of each particlein the frame of their centreof inertia. 6.53.Find the de Brogliewavelength of hydrogen molecules, to their most probablevelocity at room temperawhich corresponds ture. 6.54.Calculatethe most probablede Brogliewavelength of hydrogen molecules being in thermodynamic equilibrium at room temperature. for a de Broglie 6.55.Derivethe expression wavelength k of a relativistic particlemoving with kineticenergy T. At what valuesof T formula doesthe error in determiningk using the non-relativistic not exceedt % for an electronand a proton? 6.56.At what value of kineticenergy is the de Brogliewavelength of an electronequal to its Compton wavelength? 6.57.Find the de Brogliewavelength of relativisticelectrons and uranium

-----

reachingthe anticathodeof an X-ray tube if the short wavelength ---- 10.0 limit of the continuousX-ray spectrumis equal to pm? ks\177

6.58.A parallelstreamof monoenergetic electronsfalls normally on a diaphragm with narrow square slit of width b t.0

: : : :

\177tm.

Find the velocity of the electronsif the width of the centraldiffrac-

50 cm tion maximum formed on a screenlocatedat a distancel from the slitis equal to Ax 0.36mm. 6.59.A parallelstream of electronsacceleratedby a potential difference V 25 V falls normally on a diaphragm with two narrow slitsseparatedby a distance d----50\177tm. Calculatethe distance between neighbouringmaximaof the diffraction pattern on a screen locatedat a distancel t00cm from the slits. 6.60.A narrow stream of monoenergetic electronsfalls at an angle of incidence 30 on the natural facet of an aluminium singlecrystal.The distancebetween the neighbouringcrystal planes parallelto that facet is equal to d 0.20nm. The maximummirror reflectionis observed at a certainaccelerating voltage V0. Find Vo -----

\302\260

\177}

252

V\177

Find:

(a) the refractive index of the metalfor the electronsaccelerated

i50V; by a potentialdifference V (b) the values of the ratio V/V\177 at which the refractive indexdiffers ----t.0%. from unity by not more than 6.64.A particleof mass m is locatedin a unidimensionalsquare potentialwell with infinitely high walls.The width of the well is equal to Find the permittedvalues of energy of the particletaking motion are into account that only those states of the particle's realizedfor which the whole number of de Brogliehalf-waves are -----

\177l

\177.

fitted within the given well. 6.65.Describethe Bohrquantum conditionsin termsof the wave theory: demonstratethat an electronin a hydrogen atom can move a whole number only along those round orbitswhich accommodate of de Brogliewaves. 6.66.Estimatethe minimum errorsin determiningthe velocity of an electron,a proton, and a ballof mass of t mg if the coordinates of the particlesand of the centreof the ball are known with uncer-

tainly

\177m.

6.67.Employing the

uncertainty principle,evaluate the indeterminancy of the velocity of an electronin a hydrogen atom if the sizeof the atom is assumedto be l ----0.i0nm.Comparethe obtained magnitudewith the velocity of an electronin the first Bohr orbit

of the given

atom.

6.68.Show that

for the particlewhose coordinate uncertainty is the de where is its .=)\177/2n, Brogliewavelength, velocity uncertainis of the same order of velocity itself. ty magnitudeas the particle's 6.69.A free electronwas initially confined within a regionwith nm. Using the uncertainty principle, linear dimensionsl-----0.t0 evaluate the time over which the width of the correspondingtrain of waves becomes ----l0 timesas large. Ax

\177.

\177l

253)))

6.70.Employing the uncertainty principle,estimatethe minikineticenergy of an electronconfined within a region whose sizeis l 0.20nm. mum

\177

6.71.An

electronwith kinetic energy T 4 eV is confined a region whose linear dimensionis l--\177 9m. Using the uncertainty principle,evaluate the relativeuncertainty of its velo\177

within

city.

6.72.An electronis locatedin a unidimensionalsquare potential well with infinitely high walls.The width of the well is l. From the uncertainty principleestimatethe force with which the electron possessingthe minimum permittedenergy actson the walls of the well. 6.73.A particleof mass rn moves in a unidimensionalpotential field U ----kx2/2 (harmonicoscillator). Using the uncertainty principle,evaluate the minimum permittedenergy of the particlein that

field.

6.74.Making use of the

uncertainty principle,evaluatethe minipermittedenergy of an electronin a hydrogen atom and its correspondingapparent distancefrom the nucleus. 6.75.A parallelstreamof hydrogen atoms with velocity v = = 600m/s falls normally on a diaphragm with a narrow slitbehind which a screenis placedat a distancel i.0m. Usingthe uncerwidth of e valuate the the slit 5 at which the width tainty principle, of its imageon the screenis minimum. 6.76.Find a particularsolutionof t-he time-dependent SchrSdinger equation for a freely moving particleof mass ra. 6.77.A particlein the ground state is locatedin a unidimensional walls squarepotentialwell of length l with absolutely impenetrable Find the x of the within < < l). (0 probability particlestaying 2 a region x \177l. 6.78.A particleis locatedin a unidimensionalsquare potential well with infinitely high walls.The width of the well is l. Find the normalizedwave functions of the stationary statesof the particle, taking the midpoint of the well for the origin of the x coordinate. 6.79.Demonstrate that the wave functions of the stationary states of a particleconfined in a unidimensionalpotentialwell with infimum

\177

-\177l

nitely l

high

dx I\177n\177,\177\"

\177

(a) the particle's permittedenergy values if the sidesof the well are and 12; (b) the energy values of the particleat the first four levelsif the well has the shape of a square with side 6.82.A particleis locatedin a two-dimensionalsquare potential well with absolutely impenetrable walls (0 x a, 0 y b). Find the probability of the particlewith the lowest energy to be locatedwithin a region0 x a/3. A particleof mass ra is locatedin a three-dimensional cubic walls.The sideof the potentialwell with absolutely impenetrable cube is equal to Find: (a) the propervalues of energy of the particle; (b) the energy difference between the third and fourth levels; (c) the energy of the sixth level and the number of states (the degreeof degeneracy)correspondingto that level. 6.84.Using the SchrSdingerequation,demonstratethat at the point where the potentialenergy U (x) of a particlehas a finite discontinuity,the wave function remainssmooth, its first derivative with respectto the coordinateis continuous. A particleof mass ra is locatedin a unidimensional potential field U (x) whose shape is shown in Fig.6.2,where U (0)= oo. l\177

< <

6.83.

i.e.

6.85.

Find:

\177

walls are orthogonal, i.e.they satisfy the condition

0 if n':/=n. Here l

is the width of the well, n are

Fig.

(a) the equation defining the possiblevalues of energy of the the region E Uo; reducethat equation to the form

particlein

\177

sin kl -----4-kl \177 h2/2ml2Uo,

integers.

6.80.An electronis locatedin

a unidimensionalsquare potential well with infinitely high walls.The width of the well equal to l is such that the energy levelsare very dense.Find the density of energy levels dN/dE, i.e.their number per unit energy interval, as a func-

E. CalculatedN/dE for E = t.0eV if l----t.0 cm. 6.81.A particleof mass m is locatedin a two-dimensionalsquare

tion of

walls.Find: potentialwell with absolutely impenetrable

6.2.

k = V2mE/h. Solving this equation by graphical means, demonstratethat the possiblevalues of energy of the particleform a discontinuousspectrum;

where

(b) the minimum value of the quantity 12Uo at which the first energy level appearsin the regionE < Uo. At what minimum value of 12Uodoesthe

nth

level appear?

255)))

6.86.Making use of the solutionof the foregoingproblem,determine the probability of the particlewith energy E Uo/2 to be locatedin the region if l\177Uo---\177

z>l,

\177

6.87.Find the possiblevalues of energy of a particleof mass m

locatedin

a sphericallysymmetrical potentialwell U (r) -----0for r\177roand U(r)----- for r =ro, in the casewhen the motion of the particleis described by a wave function (r) dependingonly on r. Instruction.When solving the Schr6dingerequation,make the \177

6.94.Particlesof mass m

and energy E move from the left to the

potentialbarriershown in Fig.6.3.Find: (a) the reflectioncoefficient R of the barrierfor E Uo; (b) the effective penetrationdepth of the particlesinto the region x 0 for E Uo, i.e.the distancefrom the barrierboundary to the point at which the probability of finding a particle decreases \177

\177

e-fold.

\177

substitution

6.88.From

\177

(r) ----

\177

(r)/r.

the conditionsof the foregoingproblemfind: (a) normalizedeigenfunetionsof the particle in the states for which (r) dependsonly on r; (b) the most probablevalue rvr for the ground stateof the particle and the probability of the particleto be in the regionr \177 rvr. 6.89.A particleof mass m is locatedin a sphericallysymmetrical potentialwell U(r) -----0for r\177roand U(r) ---- Uo for r (a) By meansof the substitution (r) = (r)/r find the equation defining the propervalues of energy E of the particlefor E \177 Uo, when its motion is describedby a wave function (r) depending only on r. Reducethat equation to the form \177

\177

sin kr o =

\302\261

kr o ]/h2/2mr\177Uo,

6.95.Employing Eq.(6.2e), find

the probability D of an electron energy E tunnelling through a potentialbarrierof width l and height U0 provided the barrieris shaped as shown: with

(a) in

where k----V 2mE/h.

6.3.

Fig.

\177

(b) in

Fig.6.4; Fig.6.5.

(b) Calculatethe value of the quantity roUo at which the first

level appears.

6.90.The wave function of a particleof mass m

in a

unidimension-

potential field U (x) kx\177/2 has in the ground state the form Ae -a\177, where A is a normalizationfactor and a is a positive (x) constant.Making use of the SchrSdingerequation,find the constanta and the energy E of the particlein this state. 6.91.Find the energy of an electronof a hydrogen atom in a stationary state for which the wave function takes the form xp (r) --A (i + ar)e-at, where A, a, and a are constants. 6.92.The wave function of an electronof a hydrogen atom in the ground state takesthe form xp (r) ----Ae-r/r\177, wh'ere A is a certain constant, is the first Bohr radius.Find: (a) the most probabledistance between the electronand the al

\177

-L

\177

Fig.

6.4.

Fig.

6.5.

0 Fig.

6.6.

nucleus;

6.96.Using Eq. (6.2e), find the probability D of a particleof m and energy E tunnelling through the potential barrier shown in Fig.6.6,where U (x) ---- Uo (i

electron;

6.3.PROPERTIESOF ATOMS.SPECTRA

r\177

(b) the mean value of modulus of the Coulombforce acting on the

field of the

nucleus.

6.93.Find

mass

--x\177/l\177).

\342\200\242

the mean electrostaticpotential produced by an electronin the centreof a hydrogen atom if the electronis in the ground statefor which the wave function is xp (r) ----Ae -r/r\177, where A is a certainconstant, is the first Bohr radius.

Spectrallabelling of terms: S, ? are quantum numbers,

city, L,

L\177

X(L)\177,

i, 2,

where

\303\227

.... ... ----2S q- I

is the

multipli-

3, 4, 5, 6

(L): S, P, D, F, G, H, I,

r\177

17--9451

257)))

\342\200\242

Terms of alkali metal atoms: T=

(6.3a)

(n+\177)=

is the

B Fig.

where

Angular

with similar Hund

ML =

\177/L

rules:

(L + i),

correctionconstant

is equal to

which

unity

factor:

for light

elements.

(d+ t) +S(S+I)--L(L+t) (6.3g)

of spectral lines in a weak magnetic field: Am = (mlg1 (6.3h) \177tBBIti. With radiation directed along the magnetic field, the Zeeman components caused by the transition rn x ----mi are absent. \342\200\242

(6.35)

Zeeman splitting

m\177g\177,)

\342\200\242

(t) For a certain electronicconfiguration, the terms of the largest S value are the lowest in energy, and among the terms of Smax that of the largest L

lies lowest;

usually

the

Magnetic moment of an atom and Land6 g

M].

Ms and

expressionsfor

\342\200\242

constant, is the Rydberg correction. the diagram of a lithium atom terms. momenta of an atom:

Rydberg

6.7illu\177trates

\342\200\242

o is

where

\177'

6.97.The bindingenergy of a valenceelectronin a Li atom in the states 2S and 2P is equal to 5.39and 3.54eV respectively. Find the Rydberg corrections for S and P terms of the atom. 6.98.Find the Rydberg correctionfor the 3Pterm of a Na atom whose first excitationpotentialis 2.i0V and whose valenceelectron in the normal 3Sstate has the binding energy 5.i4eV. 6.99.Find the bindingenergy of a valenceelectronin the ground state of a Li atom if the wavelength of the first line of the sharp seriesis known to be equal to Xl = 8i3nm and the short-wave = 350nm. cut-off wavelength of that seriesto 6.100. Determinethe wavelengths of spectrallines appearing on transition of excitedLi atoms from the state 3S down to the for the S and P terms ground state 2S.The Rydberg corrections X\177

--0.4iand --0.04. 6.t01.The wavelengths of the

are

Fig. (2) for hal\177filled, \342\200\242

the and

J\"

6.7.

Fig.

basic (normal) term d = IL -- SI if L + S in the remaining cases.

the

6.8.

subshell is

less than

\177--

Boltzmann's formula:

where gl and g2

levels.

e-(E,-E,)/hT,

_____

N\177

are the statistical weights (degeneracies)of

(6.3c) the

corresponding

Probabilitiesof atomic transitions per unit time between level 1 and a higher level 2 for the casesof spontaneous radiation, in'duced radiation, and absorption: pabs__ B u pind (6.3d) radiaof the is Einstein are where A coefficients, spectraldensity B\177, of transition between the given levels. tion corresponding to frequency Relation between Einstein coefficients: \342\200\242

. ..

\177,

u\177

B\177

\177

Diagram showing

Morley'slaw

for

formation

lines: \177K\177

258

= \177 \177203

g,B,\177=g\177B\177,,

B\177

of 3

=\177

X-ray

A\177,.

(6.3e)

spectra (Fig. 6.8L

(Z\177o)

(6.3f)

yellow doublet componentsof the

resonanceNa line causedby the transition 3P--\177 3S are equaI to 589.00 and 589.56 nm. Find the splitting of the 3P term in eV units. The first line of the sharp seriesof atomiccesiumis a doub6.102. let with wavelengths i358.8 and i469.5nm. Find the frequency intervals (in rad/s units) between the componentsof the sequent linesof that series. 6.t03.Write the spectraldesignationsof the termsof the hydrogen atom whose electronis in the state with principalquantum number

\1773.

6.t04.How many

can an atom

and which values, of the quantum number d the state with quantum numbers S and L

possessin

equal respectivelyto (a) 2 and 3; (b) 3 and 3; (c) 5/2 and 2? Find the possiblevalues of total angular momenta of 6.105. atoms in the statesaP and aD. 6.t06.Find the greatestpossible total angularmomentum and the correspondingspectraldesignationof the term the principal (a) of a Na atom whose valenceelectronpossesses

quantum number n-----4; (b) of an atom with electronic ts22p3d. 6.t07.It is known that in F andconfiguration D statesthe numberof possible values of the quantum number J is the sameand equal to five. Find the spin angular momentum in these states. 17\",

259)))

6.t08.An

atom is in the

total angularmomentum

state whose multiplicityis three and the

is ]i y\177-0. What can the corresponding quantum number L be equal to? 6.t09.Find the possiblemultiplicities of the terms of the \303\227

types

(a) xD2; (b)

6.it0.A

'\177P\177/2;

possibletotal shells,and is in a state with the greatest moment for a given configuration.In the corresponding mechanical vector modelof the atom find the angle betweenthe spin momentum and the total angularmomentum of the given atom. /i ]/-6 An atom possessingthe total angular momentum is in the statewith spin quantum numberS = t. In the corresponding vector modelthe angle betweenthe spin momentum and the total is 0 =-73.2 Write the spectralsymbol for angular momentum the term of that state. 6.t12.Write the spectralsymbols for the termsof a two-electron system consistingof one p electronand one d electron. 6.i13.A system comprisesan atom in *Pa/2 stateand a d electron. Find the possible spectraltermsof that system. 6.i15.Find out which of the following transitions are forbidden 3Fa__). by the selectionrules: --)-4Ds/\177. 6.t15.Determinethe overall degeneracyof a 3D state of a Li atom.What is the physical meaningof that value? aF possessing 6.tt6.Find the degeneracyof the states aD, andmomentum. values of the total angular the greatestpossible 6.t17.Write the spectraldesignationof the termwhosedegeneracy is equal to seven and the quantum numbers L and S are interrelated as L = 3S. 6.118. What elementhas the atom whose K, L, and M shells and 4p subshellis half-filled? and 4s subshellare filled completely 6.tt9.Usingthe Hund rules,find the basicterm of the atom whose to filled

6.tii.

\302\260.

\177Da/\177--\177

\177Pl/\177,

\177P\177--)-

\177S\177/\177,

ap\177,

4F7/\177

\177P,

partially filled subshellcontains (a) three p electrons;(b) four p electrons. 6.t20.Using the Hund rules,find the total ,angular momentum of the atom in the ground state whose partially filled subshell

contains (a) three d electrons;(b) seven d electrons. 6.t2t. Making use of the Hund rules,find the numberof electrons in the only partially filled subshellof the atom whose basicterm is (c) (a) aft; (b) 6.122. Using the Hund rules, write the spectralsymbol of the basicterm of the atom whose only partially filled subshell = t; (a) is filled by 1/3,and S = S 3/2. and filled is 70%, by (b) The only partially filled subshellof a certainatom contains 6.t23. three electrons,the basic term of the atom having L = 3. Using \177P3/\177;

26O

8S\177/\177.

the Hund the given

spectralsymbol of the ground state of Hund rules, find the magneticmoment of the

rules,write atom.

6.t24.Using the

the

atom whoseopen subshellis half-filled with five

state of the electrons. ground

atoms is in the state with the 6.t25.What fraction of hydrogen = 2 K? n at a temperatureT = 3000 number principalquantum Find the ratio of the number of atoms of gaseoussodium 6.126. =

state 3/) to that in the ground state3Sat a temperatureT = 2400 K. The spectralline correspondingto the transition 3/)--\177 3Sis known to have the wavelength = 589 nm. 6.t27.Calculatethe mean lifetimeof excitedatoms if it is known that the intensity of the spectralline appearingdue to transition to the ground state diminishesby a factor = 25 over a distance 2.5mm along the stream of atoms whose velocity is v = l = 600m/s. 6.t28.Rarefied Hg gas whose atoms are practicallyall in the ground state was lighted by a mercury lamp emitting a resonance rim. As a result,the radiation power line of wavelength = 253.65 of Hg gas at that wavelength turned out to be P = 35 mW. Find the number of atoms in the state of resonanceexcitationwhose mean lifetimeis z = 0.t5 6.t29.Atomic lithium of concentrationn = 3.6.10cm -a is at K. In this casethe power emittedat the a temperatureT = 1500 resonantline'swavelength = 67t nm (2P 2S) per unit volume W/cma. Find the mean lifetimeof Li of gas is equal to P = 0.30 atoms in the resonanceexcitationstate. in the -\177

\177

\177,

\177ts.

\1778

--\177

\177,

6.130. Atomic hydrogen is in thermodynamic equilibriumwith its radiation.Find: (a) the ratio of probabilitiesof inducedand spontaneousradiaK; tions of the atoms from the level 2P at a temperatureT = 3000 (b) the temperatureat which these probabilitiesbecomeequal. 6.131. A beam of light of frequency o), equal to the resonant frequency of transition of atoms of gas, passesthrough that gas T. In this caselie) >> kT.Takinginto account heated to temperature

induced radiation,demonstratethat the absorption coefficient of where is the absorption the gas variesas = (t coefficient for T--\177 0. The wavelength of a resonant mercury line is = = 253.65 nm. The mean lifetimeof mercury atoms in the state of Evaluate the ratio of the resonanceexcitationis z = 0.t5 linewidth at a gas temperathe natural to line Doppler broadening ture T =300K. Find the wavelength of the K\177 line in copper(Z = 29) if the wavelength of the K\177 line in iron (Z = 26) is known to be equal to 193pm. law find: from Moseley's Proceeding in aluminium and cobalt: line the of the K\177 wavelength (a) \303\227

\303\227

\303\227o

e--\177\302\260\177/\177T),

6.t32.

\303\227o

\177,

\177ts.

6.133. 6.134.

26t)))

(b) the difference in binding energiesof K and L electronsin

vanadium.

6.i35.How many elementsare there in a row between those whose wavelengths of Ks linesare equal to 250and i79 pro? 6.i36.Find the voltage appliedto an X-ray tube with nickel anticathodeif the wavelength difference between the Ks line and the short-wave cut-off of the continuousX-ray spectrum is equal

to 84

pro. 6.|37. At

a certainvoltage appliedto an X-ray tube with alumianticathodethe short-wave cut-off wavelength of the continuous X-ray spectrumis equal to 0.50 nm. Will the K seriesof the characteristicspectrum whose excitationpotential is equal to i.56kV be also observed in this case? 6.i38.When the voltageappliedto an X-ray tube increasedfrom = t0 kV to V 2 20kV, the wavelength interval between the Ks line and the short-wave cut-off of the continuous X-ray spectrumincreasesby a factor n 3.0.Find the atomicnumberof the elementof which the tube'santicathodeis made. 6.i39.What metalhas in its absorptionspectrumthe difference between the frequenciesof X-ray K and L absorptionedgesequal is = nium

--\177

V\177

-\177

A\302\242o

6.85.i0s-'?

6.140. Calculatethe bindingenergy of a K electronin vanadium whose L absorptionedge has the wavelength 2.4nm. 6.i4i.Find the binding energy of an L electronin titanium if the wavelength difference between the first line of the K seriesand its short-wave cut-off is 26 6.i42.Find the kineticenergy andpm. the velocity of the photoelectrons liberatedby Ksradiationof zincfrom the K shellof iron whose K band absorptionedge wavelength is = t74 6.i43.Calculatethe Land\177 g factor for atoms pm. (a) in S states; in singlet states. 6.i44.Calculate(b) the Land\177 g factor for the following terms: (a) 6F1/2; (b) aDI/2; (c) aF2; (d) 5P1;(e)3P 0` \177

\177'L

A\177,

_\177

\177,\177

6.i45.Calculatethe

magnetons) (a) in 1F state;

magnetic moment of an atom

(in

Bohr

\177Da/\177

-\177

_\177

D\177

\302\260

262

excitedatom has the electronicconfiguration ts\1772s22p3d statewith the greatestpossible total angularmomentum. Find the magneticmoment of the atom in that state. 6.150. Find the total angularmomentum of an atom in the state with S = 3/2 and L = 2 if its magneticmoment is known to be equal to zero. 6.i5t.A certainatom is in the state in which S = 2, the total angularmomentum M = Y\177]i, and the magneticmoment is equal to zero. Write the spectral symbol of the corresponding term. 6.152. An atom in the state is locatedin the externalmagnetic field of inductionB ----t.0kG.In termsof the vectormodelfind the angular precession velocity of the total angularmomentum of that atom. 6.i53.An atom in the state is locatedon the axisof a loop of radius r = 5 cm carrying a current I = t0 A. The distancebebeingin the

\177P3/\177

\177P\177/\177

tween the atom and the centreof the loopis equal to the radius of the latter. How great may be the maximumforce that the magnetic

field of that current exertson the atom? A hydrogen atom in the normalstate is locatedat a distance r 2.5cm from a long straight conductor carrying a current ----t0 A. Find the force acting on the atom. A narrow streamof vanadium atoms in the ground state is passed through a transversestrongly inhomogeneous magnet5.0cm as in the Stern-Gerlach ic field of length experiment. The beam splitting is observed on a screenlocatedat a distance t5 cm from the magnet.The kineticenergy of the atomsis 12 T 22 MeV. At what value of the gradient of the magneticfield induction B is the distance between the extremecomponentsof the splitbeam on the screenequal to ----2.0mm? Into what number of sublevels are the following terms splitin a weak magneticfield: (c) tD.1f\177? (a) 3Po; (b) An atom is locatedin a magneticfield of inductionB 2.50kG. Find the value of the total splitting of the following

6.i54.

\177

6.155.

*F,\177/\177

--\177

l\177

\177

\177i

6.i56.

6.157.

SF\177/\177;

\177--

\177-

state; (c) in the statein which S t, L 2,and Land\177 factor g ----4/3. 6.146. Determinethe spin angularmomentum of an atom in the state if the maximumvalue of the magnetic moment projectionin that state is equal to four Bohr magnetons. 6.i47.An atom in the state with quantum numbers L-\177 2, S ----t .islocatedin a weak magneticfield.Find itsmagneticmoment if the least possibleangle between the angularmomentum and the field directionis known to be equal to 30. 6.i48.A valenceelectronin a sodium atom is in the state with principalquantum number n ----3,with the total angularmomentum What is itsmagneticmoment in that state? beingthe greatestpossible, (b) in

6.i49.An

in eV units): terms (expressed (b) aFa. (a) 6.158. What kind of Zeeman effect, normal or anomalous,is observed in a weak magneticfield in the caseof spectrallinescaused by the following transitions: ---)-alia? (c) aD1 --)-3Po; (d) (a) xP ---)-1S; (b) \177Ds/2 ---)6.159. Determinethe spectralsymbol of an atomicsinglet term if the total splitting of that term in a weak magnetic field of induction B = 3.0kG amounts to AE=t04\177eV. 6.160. It is known that a spectralline = 6t2nm of an atom is causedby a transition between singletterms.Calculatethe interbetween the extremecomponentsof that line in the magnetic t0.0kG. field with inductionB \177D;

aI\177

\177Pa/\177;

\"\177al

A\177,

\177,

263)))

6.t61.

Find the minimum magnitude of the magneticfield induction B at which a spectralinstrument with resolvingpower = is capableof resolvingthe componentsof the spectral line = 536 nm causedby a transition between singlet terms.The observation line is at right angles to the magneticfield direction. A spectralline causedby the transition 3D1--\177 3Po experiencesthe Zeemansplittingin a weak magneticfield.When observed at right anglesto the magneticfield direction,the interval between = the neighbouring s-a componentsof the splitline is Find the magneticfield inductionB at the point where the source

of a

Mean energy

\342\200\242

i.0.i05

h\302\242\302\260/\177T-

6.162.

t.32.

located. wavelengths of the Na yellow doublet (2p 2S) are nm. Find: and 589.00 equal to 589.59 is

6.t63.The

\1771

2S\177/2

\177

>\177

\"-\177

--\177

\"-\177

6.4.MOLECULESAND Rotational

CRYSTALS

energy

of a diatomic molecule:

Ej =

1 is the

where

J (J l),

(6.4a}

\177-

-\177-

molecule'smoment of inertia. energy of a diatomic molecule:

Vibrational

\342\200\242

(6.4b) where

264

\177

for molar vibrational

the natural

frequency

of oscillationsof

the

moleeuleo

(6.4c)

energy

of a crystal:

O/T

where

the

Debye temperature,

0= hO,}max/k.

._\177

(a) the ratio of the intervals between neighbouringsublevelsof the Zeemansplitting of the terms 2Pal2 and 2Pal2 in a weak magnetic field; (b) the magneticfield inductionB at which the interval between neighbouringsublevelsof the Zeeman splitting of the term is = 50 timessmallerthan the natural splitting of the term 6.164.Draw a diagram of permitted = transitionsbetweenthe terms 2P\177/2 and = in a weak magneticfield.Find the displacements(in rad/sunits)of Zeeman components of that line in a magneticfield B = 4.5kG. Fig. 6.9. 6.165. The same spectralline undergoing anomalousZeeman splitting is observed in direction1 and, after reflectionfrom the mirrorM (Fig.6.9),in direction2. How many Zeeman componentsare observed in both directionsif the spectralline is causedby the transition (a) 2P3/2 2S1/2;(b) 3P2 \177 3S17 6.i66. Calculatethe total splitting Aco of the spectralline aDa aP2 in a weak magneticfield with inductionB = 3.4kG.

\342\200\242

Debyeformula

\342\200\242

t01\302\260

\177,

A\302\242o

oscillator at a temperature

harmonic

quantum

Molar vibrational

\342\200\242

Distribution

\342\200\242

zero:

of free electronsin metal

dn= da is

where

val

the

m3/2

]/\177'\177

n2ha

Fermi

]/\177

level at T =

the

concentration

\177<

in the vicinity

\177:

of the absolute

dE,

energy falls within the the bottom of the conduction

EF =\"\177m (3n2n)2/3 where n

for

concentration of electronswhose

E, E -}-dE. The energy E is counted off \342\200\242

(6.4e)

heat capacity of a crystal

(6.4g) inter-

band. (6.4h)

free electronsin metal.

6.167.Determinethe angularrotation velocity of an $2molecule promoted to the first excitedrotationallevel if the distancebetween its nuclei is d = t89 pm. For an HC1moleculefind the rotationalquantum numbers 6.168. of two neighbouringlevels whose energiesdiffer by 7.86meV.The nuclei of the moleculeare separatedby the distanceof t27.5pm. 6.169. Find the angularmomentum of an oxygen molecule whose rotationalenergy is E = 2.t6meV and the distancebetween the nuclei is d = t2t pm. 6.i70.Show that the frequency intervals between the neighbour-

spectrallinesof a true rotationalspectrumof a diatomicmolecule equal.Find the moment of inertiaand the distancebetween the nucleiof a CH moleculeif the intervals between the neighbouring are equal to linesof the true rotationalspectrumof these molecules S-1. Ao) = 5.47-1012 6.t71.For an HF moleculefind the number of rotationallevels locatedbetween the zeroth and first excitedvibrationallevelsassuming rotationalstates to be independentof vibrational ones.The natural vibration frequency of this molecule is equal to 7.79.t01\177 rad/s, and the distancebetween the nuclei is 9t.7 pm. ing

are

265)))

6.i72.Evaluate how many lines there are in a true rotational is spectrum of laCO moleculeswhose natural vibration frequency co =4.09.t0 s-1 and moment of inertia I = 1.44. t0-39 g.cm2. 6.173.Find the number of rotationallevelsper unit energy inter-

val, dN/dE, for a diatomicmoleculeas a function of rotational energy E.Calculatethat magnitude for an iodine moleculein the statewith rotationalquantum number J t0.The distancebetween is equal to 267 pm. the nuclei of that molecule 6.i74.Find the ratio of energiesrequired to excitea diatomic \177

moleculeto the first vibrational and to the first rotationallevel. Calculatethat ratio for the following molecules: Molecule

-1 8.3

lO s 1\177

\302\242o,

(a) (b) HI H\177

d, pm 74

4.35

i60

0 40

267

6.175. The natural vibration frequency of a hydrogen molecule is equal to 8.25. t014 s-1 , the distancebetween the nuclei is 74 pm.

Find the ratio of the number of these molecules at the first excited vibrational level (v = t) to the number of moleculesat the first excitedrotationallevel (J t) at a temperatureT 875 K. It should be remembered that the degeneracyof rotationallevels is

equal to

\177

Derive Eq. (6.4c), 6.176. making use of the

Boltzmanh distribu-

tion.From Eq. (6.4c) obtain the expressionfor molarvibration of diatomicgas. Calculate for C12 gas heat capacity 300K. The natural vibration frequency of these at the temperature moleculesis equal to 1.05410 6.177.In the middleof the rotation-vibrationband of emission C\177

\177

C\177

\342\200\242

\177

s-\177

spectrumof HC1 molecule,where the \"zeroth\" line is forbidden by the selection rules,the interval between neighbouringlinesis Ace la s = 0.79.t0 Calculatethe distancebetween the nuclei of an \177

-\177.

HC1 molecule.

6.i78.Calculatethe wavelengths of the red anal violet satellites, closestto the fixed line,in the vibration spectrumof Raman scatterif the incidentlight wavelength is equal to molecules ing by = 404.7 nm and the natural vibration frequency of the molecule -\177

F\177

is co

\177

\177.o

2.15.10s-a. \1774

Find the natural vibration frequency and the quasielastic 6.179. force coefficient of an moleculeif the wavelengths of the red and violet satellites, closestto the fixed line, in the vibration spectrum 'ofl\177aman scatteringare equal to 346.6and 330.0nm. Find the ratioof intensitiesof the violet and red satellites, closestto the fixed line,in the vibration spectrumof Bamanscattermoleculesat a temperatureT = 300K if the natural ing by S\177

6.180. CI\177

266

\177

6.|81.Considerthe

linearmolecules:

possiblevibration modesin

(O--C--O);

-\177

the following

(b) (H--C--C--H). 6.182. Find the number of naturaltransversevibrations of a string (a)

CO\177

C2H\177

of length l in the frequency interval from co to co -t- dco if the propagationvelocity of vibrations is equalto v. All vibrations are supposed to occur in one plane. Thereis a squaremembraneof areaS.Find the number of 6.183. natural vibrations perpendicular to its planein the frequencyinterval from co to co -4-dco if the propagationvelocity of vibrations is equal to v. Find the number of natural transversevibrations of a right6.184. of volume V in the frequency interval from angled parallelepiped co to co dco if the propagationvelocity of vibrations is equal to v. 6.185. Assuming the propagationvelocitiesof longitudinal and transversevibrations to be the sameand equal to v, find the Debye \177-

Hereco is the natural vibration frequency of a molecule,d is the distancebetween nuclei.

\177

vibration frequency of these moleculesis co i.06.t0 s is doubled? By what factor will this ratio changeif the temperature

temperature

(a) for a unidimensionalcrystal, i.e.a chain of identicalatoms, incorporatingn o atoms per unit length; crystal, i.e.a planesquaregrid consist(b) for a two-dimensional of identical a toms, ing containingn o atoms per unit area; (c) for a simplecubic lattice consistingof identicalatoms,containing no atoms per unit volume. Calculatethe Debye temperaturefor iron in which the 6.186. propagationvelocitiesof longitudinaland transversevibrations are equal to 5.85and 3.23km/s respectively.. 6.181. Evaluate the propagationvelocity of acousticvibrations in aluminium whose Debye temperatureis O 396 K. 6.|88.Derive the formula expressingmolar heat capacity of a unidimensionalcrystal,a chain of identicalatoms,as a function of temperature T if the Debye temperature of the chain is equalto for the caseT Simplify the obtainedexpression 6.i89.In a chain of identicalatomsthe vibration frequency co dependson wave number k as co = co,\177x sin (ka/2), where is the maximumvibration frequency, k = 2n/\177, is the wave number correspondingto frequency co, a is the distancebetween neighbouring atoms.Makinguse of this dispersion relation,find the dependence of the number of longitudinalvibrations per unit frequency interval on co, i.e.dN/dco, if the length of the chain is l. Having obtained dN/dco,find the total numberN of possiblelongitudinalvibrations \177

\177)

of the

\177.

chain.

6.t90.Calculatethe zero-pointenergy per one gram of copper whose Debye temperatureis O = 330K. 6.t91.Fig.6.t0shows heat capacityof a crystal vs temperature terms of the Debye theory. Here O is the Debye temperature. Usingthis in

C\177g

is classicalheat capacity,

plot,find:

267)))

for silverif at a temperature T = 65 K (a) the Debye temperature its molarheat capacity is equal to 15:l/(mol. K); K if at (b) the molar heat capacity of aluminium at T----80 T = 250 K it is equal to 22.4:l/(mol. K); (c) the maximumvibration frequency for copperwhose heat capacity at T-\177 t25 K differs from the classicalvalue by 25%.

has one to heat classicalelectronic 6.200. Up to what temperature of mean to make the energy its electronsequal to that of free gas

electronsin copperat T----07Only one free electronis supposedto to each copperatom. correspond between neighbouring 6.20t.Calculatethe interval (in eV units) levelsof free electronsin a metal at T = 0 near the Fermi level, cm -3 and the 2.0.t022 if the concentrationof free electronsis n volume of the metal is V = 1.0cm 6.202.Making use of Eq. (6.4g),find at T = 0: (a) the velocity distribution of free electrons; (b) the ratio of the mean velocity of free electronsto their maxi-\177

\177.

mum

I \177#

L\1772

Idl

\177

]111/111\177 I.'II I/I I

]

\177

velocity.

On the basisof Eq.(6.4g)find the number of free electrons 6.203. in a metalat T ----0 as a function of de Brogliewavelengths. 6.204. Calculatethe electronicgas pressurein metallicsodium, at T 0, in which the concentration of free electronsis n---cm -a. Use the equation for the pressureof ideal gas. ----2.5.t02\177 tube of a cathodein electronic The increasein temperature 6.205. K results in the increase by AT = t.0 K from the value T = 2000 = t.4%.Find the work function of of saturation current by electronfor the materialof the cathode. Find the refractive index of metallicsodium for electrons 6.206. \177l

_.1,,\"]

]

Fig.

6.i0.

6.192. Demonstratethat molar heat capacity of a crystal at temperatureT << O, where O is the Debye temperature,is defined by Eq. (6.4f). 6.193. Can one considerthe temperatures 20and 30K as low for a crystal whose heat capacitiesat these temperaturesare equal to 0.226and 0.760J/(mol. K)? 6.194. Calculatethe mean zero-pointenergy per one oscillator of a crystal in termsof the Debye theory if the Debye temperature a

of the crystal is equal to O. Draw the vibration energy of a crystal as a function of frequency (neglectingthe zero-point vibrations).Considertwo cases: T ----0/2 and T = 0/4,where O is the Debye temperature. Evaluate the maximumvalues of energy and momentum of a phonon (acoustic quantum) in copperwhose Debye temperature is equal to 330K. 6.197.Employing Eq. (6.4g), find at T = 0: (a) the maximumkineticenergy of free electronsin a metal if their concentrationis equal to n; (b) the mean kineticenergy of free electronsif their maximum

6.195.

t35 eV. Only one free electronis assumed to correspondto each sodium atom. 6.207.Find the minimum energy of electron-hole pair formation whose electric conductance in an impurity-free semiconductor increases = 5.0timeswhen the temperatureincreasesfrom T1---=300Kto =400K. the photoelectric thresholdshort 6.208. At very low temperatures is to t.7 \177tm. in an wavelength impurity-free germanium equal Find the temperaturecoefficient of resistanceof this germanium sampleat room temperature.

with kineticenergy T

-----

\177l

T\177

\177,th

\177

6.209. Fig.6.ti illustrateslogarithmicelectricconductanceas

a function of

reciprocaltemperature(T in kK units)

for

some

6.196.

kineticenergy Tmax is known. 6.t98.What fraction (in percent)offree electronsin a metalat T = 0 has a kineticenergy exceeding half the maximum energy? 6.i99. Find the number of free electronsper one sodium atom at T = 0 if the Fermilevelis equal to EF 3.07eV and the density \177

of sodium is 268

0.97g/cm3.

2 Fig.

6.ii.

z\177

T-1 269)))

was 2.66timesgreater.Find the mean registeredbeta-particles lifetimeof the given nuclei. 6.218. The activityof a certainpreparationdecreases 2.5times

semiconductor. Usingthis plot,find the width of the forbidden band of the semiconductor and the activation energy of donor

The resistivity of an impurity-free semiconductor at room 6.210. temperatureis p = 50 \177.cm. It becomesequal to Pl ----40= \177.cm 8 ms when the semiconductor is illuminated with light, and t after switchingoff the light sourcethe resistivity becomes equal to Find the mean lifetimeof conductionelectronsand P2 = 45 g].cm.

after

\177-type

levels.

holes.

6.2ti.In Hall effect measurementsa plate of width h

i0 mm

was placed made of p-type semiconductor in a magneticfield with inductionB = 5.0kG.A potentialdifference V ----i0 V was appliedacrossthe edgesof the plate.In this case the ttallfield is VH 50 mV and resistivity p = 2.5g].cm. Find the concentrationof holes and hole mobility. and length l

--= 50 mm

-----

6.212.In Hall effect measurementsin a magnetic field with inductionB = 5.0kG the transverseelectricfield strength in an impurity-free germaniumturned out to be = t0 timeslessthan the longitudinalelectricfield strength. Find the difference in the mobilitiesof conductionelectronsand holes in the given semicon\1771

ductor.

6.213.

semi= 2.0times

The Hall effect turned out to be not observablein a

conductorwhose conductionelectronmobility was that of the hole mobility.Find the ratio of hole and conduction electronconcentrationsin that semiconductor. \1771

7.0

itshalf-life.

Find

6.2i9.Atdays. the initialmoment the activity of a certainradionuclide

totalled650particlesper minute. What will be the activity of the preparationafter half its half-life period? 6.220. Find the decay constant and the mean lifetimeof Co55 radionuclideif its activity is known to decrease4.0% per hour. The decay product is nonradioactive. A U23s preparation of mass t.0 g emits t.24. t04 alpha6.221. particlespersecond.Find the half-life of this nuclideand the activity of the preparation. 6.222. Determinethe age of ancient wooden itemsif it is known

that the specificactivity of C14 nuclide in them amounts to 3/5 of that in lately felled trees.The half-life of C14 nuclei is 5570years. 6.223. In a uranium ore the ratio of U23s nuclei to Pb nuclei is Evaluatethe age of the ore,assumingall the leadPb to be a final decay product of the uranium series. The half-life of 2\302\2606

\177--

\1771

2.8.

U238 nuclei is

2\302\2606

4.5-t0years. \177

6.224. Calculatethe specific activitiesof Na and U nuclides s whose half-lifes are i5 hours and 7.1.t0 years respectively. A small amount of solution containingNa 6.225. radionuclide with activity A ----2.0.t03 was injected disintegrationsper second in the bloodstream of a man. The activity of t cm3 of blood sample taken t 5.0hours later turned out to be A' t6disintegrations \177

\1774

\177

--\177

6.5.RADIOACTIVITY \342\200\242

\342\200\242

Fundamental

law

Relation between

half-life T:

of radioactivedecay: N = Noe-kt. decay constant

the

\177,,

i \177

\342\200\242

Specificactivity is

the mean lifetime

6.2i4.Knowing the

\177,

and the

(in

(6.5b)

of a

unit

mass Of

radioisotope.

decay constant of a nucleus,find: (a) the probability of decay of the nucleusduring the timefrom 0

to t;

the mean

\177,

lifetime

of the nucleus.

(b) cobaltnucleiwhose half6.2i5. What fraction of the radioactive life is 7i.3days decays during a month? hour by 6.216. How beta-particlesare emittedduring i.0 of Na many radionuclidewhose half-life is i5 hours? \"\177

o\177e

\177a

a counTo investigatethe beta-decay of Mg radionuclide, 6.217. ter was activated at the moment t 0.It registered beta-partithe number clesby a moment \1772.0s,and by a moment te \177g

\177a

\177

N\177

\177

t\177

270

is T = t5 hours. per minute percm3. The half-life of the radionuclide Find the volume of the man'sblood. 6.226. The specific activity of a preparationconsistingof radioactive Coas and nonradioactiveCoao is equal to 2.2.10dis/(s.g). The half-life of Coas is 7i.3days. Find the ratioof the mass of radioactive cobaltin that preparationto the total mass of the preparation \177

(6.5a)

=-7-----\177---

the activity

\17735

3t\177

per cent).

6.227.A certainpreparationincludestwo beta-activecomponents

half-lifes.The measurementsresultedin the following of dependence the natural logarithm of preparationactivity on

with different

time t expressedin hours: t In A

4.t0 3.60 3.t0 2.60 2.06 t.82 t.60 i.32 0.90

Find the half-lifes of both componentsand the ratio of radioactive nuclei of these componentsat the moment t ----0. A with half-life T -= i4.3 6.228. radionuclide days is produced in a reactorat a constantrate q ----2.7.\1770 nuclei per second.How will its soon after the beginningof productionof that radionuclide dis/s? activity be equal to A 6.229.A radionuclide with decay constant transforms into a radionuclide with decay constant Assuming that at the p3\177

\177

\177.0.\1770

\177

A\177

\177,\177

\177,\177.

271)))

initial moment

preparationcontainedonly

the

find:

the

radionuclide A1,

(a) the equationdescribingaccumulationof the radionuclide A2

time; reachesthe maximumvalue. with

(b) the time interval after which the activity of radionuclide

A\177.

6.230. Solvethe

foregoingproblem 6.23t.A radionuclide A1 goes through if

\177x

\177,2

\177,.

the transformation chain

decay constants and A1-+A\177-+A 3 (stable)with respective Assuming that at the initial moment the preparationcontained only the radionuclide Ax equal in quantity to Na0 nuclei,find the \177,\177

\177,\177.

equationdescribingaccumulationof the stableisotopeA3. A Bi 6.232. radionuclidedecays via the chain \177\302\260

6.238. Find the energy Q liberatedin K-captureif the massesof the parent atom the daughteratom and an electronm are known. 6.239. Taking the values of atomicmassesfrom the tables,find the maximum kinetic energy of beta-particlesemitted by BOo nuclei and the correspondingkineticenergy of recoilingdaughter nucleiformed in the M\177,

M\177

ground state.

directly 6.240. Evaluate the

amount of heat producedduring a day by a [\177--active Na preparationof mass m ---rag. The beta-particles are assumed to possessan averagekineticenergy equal to t/3 of the of the given decay.The half-life of Na is highestpossible energy T = 15hours. 6.241. Takingthe values of atomicmassesfrom the tables,calculate the kineticenergiesof a positronand a neutrinoemitted by C nucleus for the case when the daughter nucleus does not

t.0

\177

'\177

\177xo

\177x\302\260

\177

2\302\260G

-\177

(stable),

-6 s -x, ----t.60.t0 decay constants are -s s -1. ----5.80.10 Calculatealpha- and beta-activitiesof the Bi preparationof mass i.00mg a month after its manufacture. 6.233. (a) What isotopeis producedfrom the alpha-radioactive Ra as a result of five alpha-disintegrations and four \177--disintegra-

where

the

\177

\177,2

\177.1

\302\260-x\302\260

\177'\1776

tions?

before does U experience (b) How many alpha- and B--decays turning finally into the stablePb isotope? 6.234. A stationary Pb nucleus emits an alpha-particlewith kineticenergy 5.77MeV.Find the recoilvelocity of a daughter nucleus. What fraction of the total energy liberatedin this decay is accountedfor by the recoilenergy of the daughternucleus? Find the amount of heat generatedby i.00mg of a Po 6.235. preparationduring the mean lifetimeperiodof these nuclei if the emittedalpha-particlesare known to possessthe kinetic energy 5.3MeV and practicallyall daughter nuclei are formed directly in \1773s

2\302\260\177

\177\302\260\302\260

T\177

\177

2x\302\260

the ground state. 6.236. The alpha-decay of Po nuclei (in the ground state) is with by emissionof two groups of alpha-particles accompanied kineticenergies5.30and 4.50MeV. Followingthe emissionof these particlesthe daughternuclei are found in the ground and excited states.Find the energy of gamma-quanta emittedby the excited \177\302\260

nuclei.

6.237.The mean

path length of alpha-particlesin air under standard conditionsis defined by the formula R ----0.98.i0 v3o cm, where v o (cm/s)is the initial velocity of an alpha-particle. Using this formula, find for an alpha-particle with initial kineticenergy 7.0 MeV: (a) its mean path length; (b) the averagenumber of ion pairs formed by the given alphaparticleover the whole path R as well as over its first half, assuming the ion pair formation energy to be equal to 34 eV. -\177

272

recoil. 6.242. Find the kineticenergy of the recoilnucleusin the positron-

ic decay of a N nucleusfor the casewhen the energy of positrons is maximum. 6.243. From the tablesof atomicmassesdeterminethe velocity of a nucleusappearingas a resultof K-capture in a Be atom provided \1773

\177

the daughternucleusturns out to be in the ground state. 6.244.Passing down to the state, excitedAg nuclei emit either gamma quanta withground energy 87 keV or K conversion electronswhose bindingenergy is 26 keV.Find the velocity of these x0\177

electrons. 6.245. A free stationary Ir nucleus with excitation E = t29keV passesto the ground state, emitting a gamma energy quantum. Calculatethe fractionalchangeof gamma quanta energy due to recoilof the nucleus. 6.246.What must be the relative of a source and an absorberconsistingof free IrTM nucleito velocity observethe maximum absorption of gamma quanta with energy e = 129keV? 6.247.A sourceof gamma quanta is placedat a heighth = 20m above an absorber. With what velocity should the sourcebe displaced upward to counterbalancecompletelythe gravitational variation of gamma quanta energy due to the Earth's gravity at the point where the absorberis located? 6.248. What is the minimum height to which a gamma quanta sourcecontainingexcitedZn nuclei has to be raisedfor the gravitationaldisplacement of the MSssbauerline to exceedthe line width itself,when registeredon the Earth'ssurface? The registeredgamma quanta are known to have an energy e = 93 keV and appearon transition of Zn nucleito the ground state, and the mean lifetime of the excitedstate is x\177x

s\177

18--9451

273)))

6.6.NUCLEAR \342\200\242

Binding

(d, (b) 017 2a

REACTIONS energy of

NC\302\260;

a nucleus:

E b = Zm H

-\177-

-- Z) m n -- M,

(6.6a)

is the chargeof the nucleus (in units of e), A is the mass number, mH, M are the massesof a hydrogen atom, a neutron, and an atom corresto the given nucleus. In calculationsthe following formula is more convenient to use: E b ZA H (6.6b) A, (A Z)A n

where Z ran, and ponding

-----

\177-

where AH, An and A are the mass surpluses of a hydrogen and an atom correspondingto the given nucleus. Energy diagram of a nuclear reaction

atom, a

neutron,

q- M --',-M*

+ M'

\177

-\177-

\177'

\342\200\242

m.q--M

(6.6d) becomes possible;here m and M are the masses ol the incoming particle and the target nucleus. Tth

IQI

that of A1e7 nucleus.

6.256. Making use of the tablesof atomicmasses,find: (a) the mean binding energ), per one nucleon in

016nucleus;

in (b) the binding energy of a neutron and an alpha-particle a B nucleus; (c) the energy requiredfor separationof an 0 nucleusinto four

identicalparticles.

6.257.Find the difference in bindingenergiesof a neutron and a proton in a B11 nucleus.Explainwhy there is the difference. 6.258. Find the energy requiredfor separationof a NC nucleus into two alpha-particles and a C nucleusif it is known that the one nucleonin NC Hea, and C nuclei are binding energiesper equal to 8.03,7.07,and 7.68MeV respectively. 6.259. in atomicmass units the mass of Calculate s Li a atom whose nucleushas the binding energy MeV; (a) (b) a C nucleuswhose binding energy per nucleonis equal to \302\260

1\177

\177e

\302\260,

4i.3

1\302\260

6.04MeV.

6.260. The

--)-An

\177

A\177

nuclei involved in the nuclear reaction have the bindingenergies E\177, Ea, and Ea.Find the

E:,

A\177

\177

A\177--\177

energy of this reaction. 6.26i.Assumingthat the splitting of a U nucleusliberatesthe energy of 200MeV, find: (a) the energy liberatedin the fission of one kilogram of U and the mass of coalwith calorific value of 30 kJ/g which isotope, is equivalent to that for one kg of of the atomic (b) the mass of U2asisotopesplitduring the explosion bomb with 30kt trotyl equivalent if the calorificvalue of trotyl is 4.t kJ/g. 6.262. What amount of heat is liberatedduring the formation of one gram of He from deuteriumHe? What mass of coalwith calorific value of 30 kJ/g is thermally equivalent to the magnitude \177aa

Fig.

6.i2.

is 6.249.An alpha-particlewith kineticenergy 6 = 7.0MeVFind Li nucleus. T\177

scatteredelasticallyby an initially stationary the kineticenergy of the recoilnucleusif the angle of divergence of the two particlesis O = 60 A neutron collides elasticallywith an initially stationary 6.250. deuteron.Find the fraction of the kineticenergy lost by the neutron \302\260.

(a) in a head-on collision; (b) in scatteringat right angles.

timessmallerthan

\1776

(6.6e) Q is illustrated in Fig. 6A2, where m+ M and m'+M' are the sums of rest ma\177es are the total kinetic enerof particles before and after the reaction, and and after the r\177action gies of particles before 3 (in the frame of the centre of inertia), E* is the excitation energy of the transitional nucleus, Q is the energy of the reaction, E and E' are the binding energiesof the particles m and m' in the transitional nucleus, 2, 3 are the energy levels of the transitional nucleus. Threshold (minimum) kinetic energy of an incoming particle at which an endoergic nuclear reaction \177

numbers of protons and neutrons and having the radius one and a half

1\177

\342\200\242

n)z;

(c) Na (p, x) (d) x (p, n) Ar aT. 6.254.Demonstratethat the binding energy of a nucleus with mass number A and chargeZ can be found from Eq.(6.6b). Find the binding energy of a nucleus consistingof equal 6.255.

U\177a\177;

\177

nuclear reactions: (a) B (x, (z) BeS;

obtained? 6.263. Taking the values of atomicmassesfrom the tables,calculate the energy pernucleonwhich is liberatedin the nuclearreaction Li + H 2HeLComparethe obtainedmagnitudewith the energy per nucleonliberatedin the fission of U nucleus. 6.264. Find the energy of the reactionLiT+p--\1772He if the binding energiesper nucleonin Li7 and He nuclei are known to be equal to 5.60and 7.06MeV respectively. 6.265. Find the energy of the reactionN ((z, p) O17 if the kinetic of the incomingalpha-particleis T= ----4.0MeV and the energy

274

is*

Find the greatestpossible 6.251. angle through with an collision

is scatteredas a resultof elastic

proton.

which a deuteron

initially stationary

6.252. Assuming the radius of a nucleus to be = 0.t3 pro, where A is its mass number,evaluatethe density of nucleiand the number of nucleonsperunit volume of the nucleus. Write missing symbols, denoted by x, in the following 6.253. s\302\242\177-

1\302\260

equal to R

\177

-\177

'\177

\177

\177a

275)))

proton outgoingat an angle 0 = 60 to the motion directionof the alpha-particlehas a kineticenergy Tp = 2.09MeV. 6.266. Making use of the tablesof atomictnasses, determinethe o f the following reactions: energies (a) Li7 (p, n) BeT; \302\260

{b) Be9 (n, (c) Li ((z, n) (d) 016(d, N 6.267.Makinguse of the tablesof atomicmasses,find the velocity with which the products of the reactionB (n, (z) Li comeapart; the reactionproceedsvia interaction of very slow neutrons with stationary boron nuclei. Protons striking a stationary lithium target activate 6.268. a reactionLi (p, n)Be At what value of the proton's kinetic energy can the resultingneutron be stationary? 6.269.An alpha particle with kinetic energy T-\177 5.3MeV initiatesa nuclear reactionBe9 ((z, n)Cla with energy yield Q---\177-5.7 MeV. Find the kineticenergy of the neutron outgoingat \177)

Be1\302\260;

\177

B\177\302\260;

\177)

\1774.

\177\302\260

\177

\177.

\177

anglesto the motion directionof the

alpha-particle. 6.270.Protons with kineticenergy T 1.0MeV striking a lith-

right

\177

ium target induce a nuclear reactionp Li7 --)-2He 4. Find the kineticenergy of eachalpha-particle and the angleof their divergence provided their motion directionsare symmetrical with respect to that of incomingprotons. 6.271. A particleof mass m strikesa stationary nucleusof mass M that the threshold and activatesan endoergic reaction.Demonstrate (minimal)kineticenergy requiredto initiatethis reactionis defined by Eq.(6.6d). 6.272.What kineticenergymust a proton possessto splita deuter-\177

2.2MeV? on H whose binding energy is 6.273.The irradiationof lithium and beryllium targets by a monoergicstreamof protons revealsthat the reactionLi\177(p, n)Be MeV MeV is initiatedwhereasthe reactionBe\177(p, n)B doesnot take place.Find the possiblevalues of kineticenergy of the protons. 6.274.To activate the reaction(n, (z) with stationary B nuclei, 4.0MeV. neutrons must have the thresholdkineticenerg\177 T\177 Find the energy of this reaction. 6.275.Calculatethe thresholdkineticenergiesof protons required to activate the reactions(p, n) and (p, d) with Li nuclei. 6.276.Usingthe tabular values of atomicmasses,find the threshold kinetic energy of an alpha particlerequired to activate the nuclear reaction Li ((z, n)B What is the velocity of the B \177

E\177

\177

-- i.65

--i.85 \177

\177

\177\302\260

\177

\177\302\260.

nucleus in this case?

276

\177

i.5

\177-

\177

-----

\177a

i.55, i.80

T\177

6.28|.

\177!----

\177

-\177

(\177

(\177,

(\177a

-\177

(\177

\177

I 1.35.i(i \177-

\177

i.0.i0

\177

\177tA)

1\177

-\177

\177\302\260

\177

Find:

(a) the irradiationtime after which the number of Au nuclei decreasesby \177l----i.0%; (b) the maximum number of Au nuclei that can be formed during protractedirradiation. is irradiatedby thermal A thin foil of certainstableisotope 6.286. neutrons falling normally on its surface. Due to the capture of neutronsa radionuclidewith decay constant appears.Find the law \177

xg\177

6.277.A neutron with kinetic energy T i0 MeV activates a nuclearreactionC (n, (z) Be whosethresholdis T\177 ----6.i7MeV. Find the kineticenergy of the alpha-particles outgoing at right anglesto the incomingneutrons' direction. \177

6.278.How much, in per cent, does the thresholdenergy of gamthe bindingenergy of a deuteron(E\177 2.2MeV) ma quantum exceed Ha --\177n in the reaction p? MeV is captured with kineticenergy T 6.279.A proton the formed nucleus. of the a. Find excitation H deuteron a energy by The yield of the nuclearreactionCla(d, n)N has maximum 6.280. magnitudesat the following values of kineticenergy of bombardMeV. Making use of the 0.60,0.90, and ing deuterons: table of atomicmasses,find the correspondingenergy levelsof the transitionalnucleusthrough which this reactionproceeds. A narrow beam of thermal neutrons is attenuated ----360times after passing through a cadmium plate of thickness of interaction d 0.50ram. Determinethe effective cross-section of theseneutronswith cadmium nuclei. how many timesthe intensity of a narrow beam Determine 6.282. of thermal neutrons will decreaseafter passingthrough the heavy of water layer of thicknessd ----5.0cm. The effective cross-sections interactionof deuterium and oxygen nuclei with thermal neutrons 4.2b respectively. 7.0 b and are equal to A narrow beam of thermalneutronspassesthrough a plate 6.283. are of iron whose absorption and scatteringeffective cross-sections ---fraction the Find b 2.5b and respectively. equal to of neutronsquitting the beam due to scatteringif the thicknessof the plate is d ----0.50cm. The yield of a nuclear reactionproducingradionuclides 6.284. in two ways: eitherby the ratio w of the number described be may of nuclear reactionsto the number of bombarding particles,or by the quantity k, the ratio of the activity of the formed radionucIide to the number of bombarding particles,Find: assuming w and k (a) the half-life of the formed radionuclide, to be known: (b) the yield w of the reactionLiT(p, \177)Be if after irradiationof 2.0hours and with a lithium target by a beam of protons (over t of Be became i0 the equal to A ---beam current activity 6 dis/s and its half-life to T ----53 days. Thermal neutrons fall normally on the surface of a thin 6.285. goldfoil consistingof stableAu nuclide.The neutronflux density l0 mg. foil is m is J part./(s.cma).The mass of the Au \177 nucleiwith half-life The neutroncaptureproducesbeta-active T 2.7 days. The effective capture cross-sectionis (\177----98b.

\177

\177,

277)))

describingaccumulationof that radionuclideN (t) per unit area foil'ssurface.The neutron flux density is J, the number of nucleiper unit areaof the foil'ssurface is n, and the effective crosssectionof formation of activenucleiis 6.287.A gold foil of mass m = 0.20g was irradiatedduring t = 6.0hours by a thermal neutron flux falling normally on its surface.Following t2 hours after the completion of irradiation the activity of the foil becameequal to A = t.9.t0dis/s.Find the neutron flax density if the effective cross-section of formation of a radioactivenucleus is = 96 b, and the half-life is equal

\342\200\242

of the

\177

-\177

\177

to T

= 2.7 days.

(\177

6.288. How many

if the

neutronsare there in the hundredth generation neutrons and takes No-----t000

fission processstarts with

placein

a medium with multiplication constant k---= 1.057 6.289. Find the number of neutronsgeneratedper unit time in a uranium reactorwhose thermal power is P---=100MW if the averagenumber of neutrons liberatedin each nuclear splitting is v = Each splitting is assumed to releasean energy E

2.5.

= 200MeV.

6.290.In a thermal reactorthe mean lifetimeof one generation of thermal neutrons is x-----0.i0 s. Assuming the multiplication constant to be equal to k = i.0i0,find: (a) how many timesthe number of neutronsin the reactor,and consequentlyits power, will increaseover t = 1.0min; the T of the reactor,i.e.the time periodover which its(b)power period increasese-fold. Total energy

and momentum

T is the kinetic

----

of a relativistic particle:

moc2q-T, pc =

T (T q-- 2mock),

(6.7a)

energy

\342\200\242

2--

\177

p\177c

ra\177c

\177,

\342\200\242

.... Tth

(ral-\177-m\177-\177-

\177-

-\177

ra\177

\177-

\"\\177--(ra-\177-M)\177

c\177,

where m, M, ml, m 2 are the rest massesof the respectiveparticles. Quantum numbers classifying elementary particles:

Q, electric charge, L, lepton charge, B, baryon charge, T, isotopicspin, its projection, = 2(Q) -- B, S, strangeness,STz, Y,

particles:

interacting

(6.7d)

Q=Tzq_._\177.=Tz_ \177

B-\177-S2

Interactionsof particlesobey the laws of conservation of the Q, L and B charges.In strong interactions the laws of conservation of S (or Y), T, and its projectionTz are also valid. \342\200\242

momenta 6.29i.Calculatethe kineticenergiesof protons whose light. 0.i0,i.0,and i0 GeV/c, where c is the velocity ofwhose kinetic Find the mean path travelled by pions 6.292. ---The mean lifetime times. i.2 rest t heir exceeds energy energy are

of very slow pions is T0 =

25.5ns. \177l

6.293. Negative pions with kineticenergy T

\177

i00MeV

travel

m from their origin to decay.Find the an averagedistancel these of lifetime pions. proper 6.294.There is a narrow beam of negative pions with kinetic energy T equal to the rest energy of theseparticles.Find the ratio of fluxes at the sectionsof the beam separatedby a distance l -= 25.5ns. ----20m. The propermean lifetimeof these pions is % rouen and into a A 6.295. stationary positive pion disintegrated a neutrino. Find the kineticenergy of the rouen and the energy of the neutrino. Find the kineticenergy of a neutron emergingas a result 6.296. of the decay of a stationary Y,- hyperon (\177- --)-nq6.297.A stationary positive rouen disintegratedinto a positron kineticenergy of the and two neutrinos.Find the greatestpossible \177

---=

z\177-).

hypercharge,

\177-

neutral particledisintegratedinto a proton MeV and a negative pion.Find the kineticenergy T----5.3 mass of that particle.What is itsname? 6.299.A negativepion with kineticenergy T = 50 MeV disintegrated during its flight into a rouen and a neutrino. Find the energy of the neutrinooutgoingat right anglesto the pion'smotion direction. + A 6.300. hyperon with kineticenergy Tz = 320MeV disintegrated during its flight into a neutral particleand a positive pion outgoingwith kineticenergy T= ----42 MeV at right angles to the hyperon'smotion direction.Find the restmass of the neutralparticle

with

of the particle. When examining collisionsof particles it pays to use the iavariant: E = (6.7b) where E and p are the total energy and the total momentum of the system prior to collision, m0 is the rest mass of the formed particle. Threshold (minimal) kinetic energy of a m striking a stationary particle M and activating the endoergicreactionparticle ra M -+ ra 1

\342\200\242

numbers of strongly

A stationary 6.298.

PARTIELES E

where

quantum

positron.

6.7.ELEMENTARY \342\200\242

Relation between

... :

(6.7c)

(in MeV

units).

disintegrated during its flight into two with gamma quanta equal energies.The angle of divergenceof = is (9 60 Find the kineticenergy of the pion and of gamma quanta A neutral pion 6.301. \302\260.

eachgamma quantum. with 6.302. A relativisticparticle with rest mass m collides a stationary particleof mass M and activatesa reactionleadingto . where the M -)-rn\177 + formation of new particles:m restmassesof newly formed particlesare written on the right-hand demside.Making use of the invarianceof the quantity E -\177-

rn\177

..,

\177-

\177

p\177c

\177,

278 279)))

onstratethat the thresholdkineticenergy of the particlem required reactionis defined by Eq.(6.7c). 6.303. A positronwith kineticenergy T = 750 keV strikesa stationary free electron.As a result of annihilation,two gamma quanta with equal energiesappear.Find the angle of divergencebetween them. 6.304.Find the threshold energy of gamma quantum required to form (a) an electron-positron pair in the field of a stationary electron; (b) a pair of pions of oppositesigns in the field of a stationary proton. 6.305. Protonswith kineticenergy T strikea stationary hydrogen target. Find the thresholdvalues of T for the following reactions: for this

p---).p p p.+p ; (b) p.+p_).p.+ 6.306. A hydrogen target is bombarded by pions.Calculatethe

+ + (a) p + thresholdvalues of kineticenergiesof these pions making possible p_\302\242_\177o.

the following

reactions:

+ + Y,-; (b) +p_).K+ + A +p--\177K 6.307.Find the strangenessS and the hypercharge Y of a neutral elementaryparticlewhose isotopic spin projectionis Tz---- +t/2 and baryon chargeB = +t. What particleis this? 6.308. Which of the following processes are forbidden by the law of conservationof lepton charge: (1) n--\177p +e-+v; (4) p+e--)-n +v; + (2) \177+--+\177+ + e- + e+; + v -+-\177; (5) (3) -+ + v; (6) K- --)-\177t- + 6.309.Which of the following processes are forbidden by the law of conservationof strangeness: (t) +p--\177Y,-+K+; (4) n +p-)-A+ Y,+; (2) + p -)-X+ + K-; (5) n- + n ---)-g-+ K + + K-; (3) n- + P--+K+ + K- + n; (6) K- + p-\177-+ K+ + KO?

(a)

\177-

\1770

\177.

ANSWERS AND SOLUTIONS

= 1.1. t.2. (v)= v

3.0

= km per 2Vo(V, +v2)/(2v 4 (v)/w\177 = V t

l/2\177

hour.

o+vx+v2).

1.3.At-15s. t.4.(a)-t0 cm/s;(b) 25 cm/s;(c) t o = t6 s. -- = (vi --- I. t.5. t.6.v'=Vv\177)+v2+2VoVCOS\177 \17740 km per hour, q)'=t9 i.7. =3.0km per hour. u=(l_v\177/v,i)_i/2_l = t.8. t = t.8. t.9.O= arcsin (t/n) + n/2 = t20 1.t0.l = rot V 2 (t sin O) = 22 m. 1.tt.l = (VI+v2)Vvlv\177/g=2.5m. t.t2.t = 2a/3v. t.13.It is seen from Fig.ta that the points A and B convergo with velocity v -- u cosa, where the angle a varies with time.The \342\200\242

(r\177

r\177)/lr\177

\"\177)/l'\177l

r\177l

v\177

\302\260.

\177al\177

\177IlV

\1772_

\302\260.

\177+---\177e

\177-

\177?

\177t-

\177\"

\302\260

\177-

6.3t0.Indicatethe

bidden:

reasonswhy

the following

(t) x- A o + n-; (4) n (2) +p--+K++ K-; (5) (3) K- + n-+Q-+ K + + KO; (6) ._\177

\177-

are forprocesses

+ p -->-'X + ++ + e +\177e-; A\302\260;

Fig.

--)-\177t-

points mergeprovided the following two conditionsare met: f

(v--uCosa)dt=l,f

vcosaat=u\177,

where is the sought time.It follows from these two equationsthat. +\177

1.14.xt --

with

velocity

x\177

= \177

.\177

w+\177

--__

4.0m/s.

vl/(v

+ +/2) = 0.24km. Toward

the train

t.t5.(a) 0.7s; (b) 0.7and i.3m respectively. 281)))

1.41.R =aS/2bs,to=aV l + (4bs\177/aS)% 1.42.(a) w=2at\177,R = w = bvUa R = t.43.v = 2Rto 0.40m/s, w =(b)4Rco 0.32m/s t.44.to = (v/t) V t + = 0.7 m/s t.45.co = 2nnv/l = 2.0. rad/s. \177/\177.a;

1.18.SeeFig.tb. = nR/-c = 50 cm/s; (b) I(v)[ = 2R/T = 32cm/s; t.19.(a) (v) = = t0 cm/s 2nR/x (c) I<w>l v 1.20.,(a)=a(t--2at),w =--2aa= const; (b) At= t/a, s = a/2a. t.21.(a) X=vot (t--t/2x),z=0.24,0. and -4.0m; (l--t/2Z)vot for t s= 9 and 1t 1.1, (b) s; (c) for t_\177. [l+(l--t/z)2]Vot/2 24 and 34 cm respectively. 1.22.(a) v = a2t/2, w= aV2; (b) (v) = a ]/\177-/2. 1.23.(a) s = (V3a) (b) t = 2 V vo/a. t.24.(a) y-\177--x2b/a2; (b) v=ai--2btj,w=-.2bj, v= I<v\177l-=

b\177t2.

a\177o-c:

1.27.vo = ]/(1 a w/2b. 1.28. (a) r = vet gt2/2; (b) (v)t= -- vo gt/2, (v) = vo--g(vog)/g 1.29.(a) = 2 (vo/g) sin a; sin a, l = (b) h = 2a, \303\267

g\177_____

\177)

\303\267

\177'co\177

ag\"

\177-.

t0\177

t.46.(a) (co)=2a/3=4 rad/s, (\177)=V3ab=6tad/s2; (b) 2 \177= t2 rad/s% t.47.t = \177/(4/a) tan a = 7 (co) = \177o/3. (a) = (t e-at) COoC;(b) = COo e-\177t.

s. 1.48. -1.49. t.50. --= V-2fio sin see Fig.3. 1.51.(a)y=.u\177/fix (hyperbola);(b) y= t.52. (a) wa =vVR = 2.0m/s the \177

CO\177

vector

q\177,

is permanently directedto the centreof the wheel;(b) s = 8R = 4.0m. 1.53.(a) va=2wt=lO.Ocm/s, vs= =V\177wt=7.tcm/s, \177o=0; (b) w.,= = 2w \1771 + (wt\177/2R) = 5.6 cm/s ws = =w ]/t+ (i--tot\177/Rf=2.5cm/s'-, We= = to2t2/R = 2.5cm/s2.

\177.

\177-,

t.54'.Ra = @,

R\177

-= 2

]fl\177

,\302\242stn

[1=

CO\177COz

\177

(b) [1= (v/R) z tan

x\177;

cosa, B1= t.30.Svo/g eeFig.2. Fig. 2. t.31.l = 8h sin a. t.32.0.4t or 0.71rain later, dependingon the initial angle. sin (01-02) = 11s. t.33.At = g cos01 q- cos02 t.34.(a) x= (a/2Vo) y2; (b) w = aVo, = a2y/V t (ay/vo)% = avo/V 1 (ay/Vo) 1.35.(a) = (b/2a)x2; (b) R = (a/b) [1+ (xb/a)2]3/2. 1.36.v V 2ax. 2 --0.8 1.37.w = a V 1 m/s2. = 1.38.(a) v Vo/(l -J- Vot/R ) = we-s/n; (b) w= = V-\177vR. 1.39.tan a = 2s/R. 1.40.(a) w o = = 2.6m/s2, = \177

B\177

2v.__q_o

w,\177

282

= t2 rad/sz. = a 1/t+ (2bt/a)2 =

\177

(d)

=a\177o\177

Fig.

cos a) = (v\177/g) cos a. -y sin

(g/2v\177

(parabola).

w\177

(v\177/g)sin

V-2\177x/\177o

\177,

\177

(v\177/2g)

[\177=

\302\242p

\177.

\177

a\177/b.

-----

\177

4a\177.t

\177.

2, w=2b; (c) tan a=a/2bt;(d) (v)=ai--btj, =Va2+4b2t = V a2+ 1.25.(a) y = x--x2a/a;(b) v = a V 1 =const;(c) to= l/a. 1.26.(a) s = (b) n/2. ._.

\177

\177.

\177

v\177/2\";

\177,

---=

w\177

\177.

\303\267

vZ/wn

\303\267

(4\177n)

a%\1772/R

w\177

a\302\242o

(R/2a)\177= 2.5m/s lm= 4- a V t \177,

R2/2a\177= rk

0.37m.

mtm2g cosa klml q- keme t.6t. (a) F = (kl--kz) ; (b) tan n = taxi_m2 2 i.62.k = -+ t)l tan a = 0.t6. 1.63.(a) > sin a + k cosa; (b) < sin a --k cosa; (e) sin --kcosa <m\177/mx <sina+kcosa. 1.64. + t) =0.05g. i.65.When=g(\177--sina--kcosa)/(\177 = at/(mx + t to, the accelerations wt = \177Zmi

.\177_m

\177

[(\177

t)/(\177

m\177/mx

mz/m\177

\177z

w\177

= kgm\177/mx, = (at -Here t o = (rex + m\177)/am. SeeFig.4. t.66.tan 2a = --t/k,a = = t.0s. i.67.tan = k; a k =mg (sin + cosa)/\177 {+ k z.

when t \177

\177

w\177

km\177g)/m\177.

w\177

kgm\177

v=--

49\302\260;

tm\177n

T\177i\177

I cos 1.68.(a) mg2asinaa ; 1.69.v = V (2g/3a)sin \177

a.)))

s cos a

(b) s= 6a sinaa m\177g \177

m\177);

k---(ln \177lo)/n; (b) w = g (q-@ cos a. where r is the radius vector of the particle --m\177oZr, relativeto the origin of coordinates; F = mcoa V xa yL 1.96.(a) Ap = mgt; (b) lap I = --_ --2m(vog)/g. i.97. (a) p = aza/6; (b) s = ing

this

we obtain

equation, 1.94. F= 1.95.F=

\1771o).

\177lo)/(\177l

\177

(mv\177o/R)

\177-

_\177

az\177/12m.

i.98. s = (cot --sin cot) Fo/mcoL 1.99. t n/co; s = 2Fo/mcoZ; Fig. 6. Fo/mco. 1.100. t --+ (a) (b) v vo --sr/m, 1 . t= h(vo--v) 1.101. In -s 1.102. tan a, sin o:tan a

seeFig.6. Vmax

-----

V=Voe-t\177/\"\177,

mvo

roy

(vol\177)

Vm\177x=

Fig. 4.

Fig. 5.

i.77.w A = g/(t + rIcot w B g/(tano:q- \177lcoto:). i.78.w= ). = g (t --k)/(t + k). 1.79.wmin gV\177/(2+k+Mim t.80. = + k cota)/(cot a k). t.8t.w = g sing a(t cos a/(sin\177 + mg sin a i.82. M+2m(t\177cosa) i.83. 1.84.2.1,0.7 and t.5kN. 1.85.(a) w=gVt\177-3cos z0, T=3mgcosO; . (b) T=mgV'\177; (c) cosO=t/V\177,0--54.7 1.86. 53 1.87.0 = arccos(2/3) 48 v = V 2gR/3. z -1.88.e = i/(u/m(o Is independent of the rotation dit). roction. =- V \177 1.89.r = R/2, = 1.90.s i/zR V (kg/w\177)a-- t = 60m. 1.91.v o: V kg/a. t.92.T ----(cot 0 + mg/2n. 1.93.(a) Letus examinea smallelement of the thread in contact \177o:),

\177

w\177x

mx/m\177).

\302\260

\177

\302\260.

\177

\302\260,

v,\177a\177

\177/\177

\177

with the -\177

dFlr

284

(o\177R/g)

pulley (Fig. 5). Sincethe elementis weightless,dT k dFn and dFn----T do:.Hence,dT/T = k do:.Integrat\177

o\177;

Stota

\177

\177-\177

Instruction.To reducethe equationto the form which is convenient to integrate,the accelerationmust be representedas dv/dt and then a change of variables made accordingto the formula dt dx/v. t.103, X/o a (t --to)a/m, where t o = kmg/a is the moment of timeat which the motion starts. At t to the distanceis s = 0. 1.104. v' =Vo/I/l + kv\177/mg. 1.105. (a) v = (2F/mco) sin (cot/2)l; (b) As=8F/mco (v) ---

\177

= 4F/nmco. t.106,v = Vo/(t q- cos Instruction.Here = thereforev =--vx -b const. From the initial condition it that const = v o. Besides, = vcos(p. I

\177,

q\177).

w\177

--w\177,

and

where

\177I=

follows

t.107,w = [l --cos(l/R)]Rg/l. v\177

1.108. (a) v=V2gR/3;(b) cOS0o=2-t-\177l/5q-9\177lz ' 3 = Wo/ g , 0o \17717 . 1.109. Forn <1,includingnegative values. (1\177-\1772)

\302\260

l.ll0.When cozR g, there are two steady equilibriumposi= arccos 0 and \177

tions: =

When (g/co\177R). <g, there is only one equilibriumposition: = 0. As long as there is only one lower equilibriumposition, it is steady.Whenever the secondequilibrium position appears(which is permanentlysteady) the lower one becomesunsteady. 1.111. h (cosZ/v) sin (p=-7 cm, where (o is the angular veloc-

0\177

o,)\177R

0\177

\177

ity of the

Earth'srotation.

z 1.t12.F=mV gz N. q-(2v'co)z=8 z = 1.113. Feor 2mcoZr V 1 (Vo/cor) = 2.8N. q-co\177r

-\177

285)))

1.1t4.(a) tv'=\177o2R: Fin=mto2rV(2R/r)2--t. t.115. = mco2R ]/ (____b) 5/9 = 8 N, Foot= ]/5-]-8g/3\177o2R\177 =t7 N. 1.t16.(a) F = 2mwo sin = 3.8kN, on the right rail; (b) 2/\177m\177o\177R

Fc\177

along the parallelfrom the east to the west with the velocity v = =-2- cos 420 km per hour. Here is the angularrotation velocity of the Earth about itsaxis,R is itsradius. 1.t17.Will deviate to the east by the distance x\177 2 cm. Here is the angular velocity of the ,\177\302\242ohl/-\177/g-=-24 Earth's rotation about itsaxis. 1.118. ,4 = F (r2 -- rl) = --t7 q\177

\177oR

q\177

\177

\177o

1.1t9.A = ma4t2/8. 1.120. F = 2as -+- (s/R)2. t.t21.A = mg ]/t (h + kl). --k cot t.122.A = --kmgl/(t t.t23. = (ml + kg. 1.124. A = --(t -wngl/2 t.t25.(P) = 0, P = mg(gt--t\177osina). 1.t26.P = mRat, (P) = mRat/2. 1.127. (a) (P) = --kmgvo/2 --r[)= 1.t28. A = 1.129. = where k = + 1.t30.A = 3mg/4a, AU = = mg/2a. 1.t31.(a) = steady; (b) F\177= ba/27a see Fig.7. t.t32.(a) No; (b) ellipses F,\177n

-\177/2rn\177oV

\302\242zg.

m\177)\177/(mx

\177-

(M_\177_m)\177

(2M 3m)

(2) v2----- (M--\177m)(M.-I-2m)u,

t.t58,hp=mV2gh(\177IA-t)/(\177I--t)=0.2kg.m/s. mM t.159.(a) 1= M+m 1'; (b) F= M+m dt t.t60.1 = ml'/2M. t.t6t. z = (p cosa--M V-2gl sin a)/Mgsin a. t.162.(a) v=(2M/m)V\177sin(0/2);(b) t--m/M. t.t63,h = + m). = 1.t64.(t) A = (M where M); (2) Yes. t.166,v = t.0i-J- 2.0j--4.0k,v mM/(m 4.6m/s. -t.t67.AT = where = t.t68.(a) = (b) = = 1/3;(b) = t -J- 2 cosO =+2.0. t.169.(a) t.170. = l/2cos a = 0.25. t.t7t,v,\177=v(t V2 (\1771-t))=t.0 km per second. \303\267

\177t

(v\177

v\177)\177/2,

2rnl/(rn\177

\303\267

m\177m\177/(m\177

\177t

m\177);

4rnvn\177/(rn\177

\1771

\303\267

rn\177).

rn\177)

\177.

m\177/rn\177

\177

t.t72.Will

\303\267

continue moving

in the same direction,although this time with the velocity v' = (t--Vt v/2. For << 1 the \177

\177

Vlorm=Vo-

rn\177/rn\177

\177

(m\177

= t + m/2(M+ m) > t. M\177-2m

= 8ragC.

\177mg/8\303\227

\303\2275l/mg)

v\177

\1771

\177

v\177=

\177t

t\177/2voR.

\177.

\1771

\177

\177)\177,

l,,a\177

+ rn,z), lmin l o. 1.153. (a) 5l> 3ragC;(b) h = (t + 1.t54. = --mv/(M --m), = Mv/(M --m). (ml

\177

tan

t.136. t.t37. t.t38.t = t.t39.Al = (t + t + 2kl/mg)mg/k. = t.7 m/s. t.t40. = V t9glo/32 k mglo t cos0 \177 = = 0.09J. t.141.A -- where t.t42.A = + t.143.wc = g (t -t.145.r= (g/\177) tan = 0.8+mO cm, T=mg/cosO=5N. \177)/2(t

-\177-

2mlF/\303\227

rn\177;

m\177).

--\177tgh,

f\177/a;

xl\177

m\177v\177,

rn\177).

\303\227rn:/(mi

rn\177).

\177t

\"J-

\1771\177

whose ratioof semiaxesis a/b =V also ellipses, but with Fig. 7. a/b = t.133.The latter field is potential. 1.t34.s ---- (sin a k cosa), ,4 = --mv\177k/2(k 1.t35.h -= H/2;S,\177a: = H. \177

\303\267

(v\177

\177t

Mv\177/2g

V\177o/2g

\177t

\\\177b'(\177\\\\\\\\

\177,

/\177

v\177/vi

\177,

k\177).

\302\242o

m\177);

t.t56.(t)

(r\177

\177/2k(Al)

(\302\242oZl/g)

rn\177).

\177

A,\177=

\303\267

t.t55.Vrear=Vo--M_\177_mU ;

m\177/2)

k\177k\177/(k\177

= mg [sina

cosa] = 6 N. (b) = < V1.t47. g (k--tan a)/l (1+ k tan a) 2 rad/s. (a) V =(mava + rn:v\177)/(rna + (b) T = (va ---v\177)V2, where\177 = rnarn: (rna + t.t48.E = E + mV\177/2. t.t49. = where = rnarn\177/(rnx + 1.t50.p = Po mgt,v\177)/2, where Po = mva + m = rnx + rc = vot + gt\177/2, where vo = (mava + m\177v\177)/(ml + t.151.Vc = xV + 1.152. = lo F/x, (a) lma x = l o lmfn = /o; (b) u F\177

v\177

\1771)

i/2rn\177o

1.146. (a)

m\177/\177.

a).

velocity v' \177

t.173.AT/T t.t74.

= 5 cm/s.

= (1

\177lv/2

\303\267

2\1771)

reC)tan

= mim2/(m (a) m2). t.t75,siin+Ornax t.176,v' = --v (2 --

p=\177t]/v,\303\267v\177;

\177

(b)

0 4- m/M

-- t

T=\177/2\177t(v\177\303\267v2).

\1771

= --40%. Here

\177---

rn\177/rn\177.

equal,or

greaterthan

\177 \1771\177)/(6

\1771\177).

Bespectivelyat smaller

\1771,

l/\177-.

1.t78.Supposethat at a certainmoment t the rockethas the massm and the velocity v relativeto the reference frame employed. Considerthe inertialreferenceframe moving with the samevelocity as the rockethas at a given moment.In this reference frame the momentum incrementthat the system \"rocket-ejected portionof gas\" 287)))

acquiresduring the timedt is equal to dp = follows is evident.

What

1.179. v = --u In (meC). 1.i80.m = 1.i81.a = (U/Vo) In (mo/m).

m dv

+ dt.u-= F dr. \177t

the Sun.

moe-\177t/\177.

182.v

mo--[tt ' m\302\260

1.|83. v = Ft/rno(l + \177t

t.209,rm -= \177[4..-I- V --(2 --

4..

\1771)

me--itt w F/tno(i \342\200\242

\177tt/rno),

I.t84.v = ]/2ghIn (l/h).

: =

[

mgl/\302\242o.

mcov\177t

\177/

\177.

: : : \177

i.e.

: t\177

\303\267

--'\177

(raSM/2l)sin 0 dO.

(*)

According to the cosinetheorem in the triangle OAP the differential of this expres-4- r 2pr cos0:Havingdetermined sion,we can reduceEq.(*)to the form that is convenientfor integration. After integrating over the whole layer we obtain 5U = --\177trn 5M/r.And finally, integratingover all layers of the sphere, we obtain U = --\177trnM/r; (b) Fr = --OU/Or= --\177traM/r \177

\177.

Fig.

-\177

\303\267

/\177

1.205. /

where

(\177ms/V\177o)

2.2:i0

i.203. \177

=1,

mvo/\303\227l

frame of the centreof inertia. T 1.200. 225 days. 2n'fM/v 3 m/s2. (a) 5.2times;(b) kin/s, = T n]/(r+ R)3/2.fM. It is sufficient to considerthe motion alongthe circlewhoseradius is equal to the majorsemi-axis of the given ellipse, (r with Kepler's R)/2, sincein accordance laws the period of revolution is the same. as the Falling of the body on the Sun can be' considered motion along a very elongated(in the limit, degenerated)ellipse whose majorsemi-axisis practicallyequal to the radius of the Earth's orbit. Then from Kepler'slaws, (2\177:/T)2 = [(I\177/2)/RP, where is the falling time(the timeneededto completehalf a revolution along the elongatedellipse), T is the periodof the Earth's revolution around the Sun. Hence,\177---- T/4\177= 65 days. i.204.Will not change. l (T/2\177)2. t.206.(a) U \177M =--'fm\177m2/r; (b) U=--'f(mM/l) ln(t+I/a); F = \177mM/a (a l ).

i.201. i.202.

sin\177

is the mass of

t.2t0, n= IVt +(lv\177o/\177tms) \177.-t], where ms is the massof the Sun. t.2tt. (a) Firstlet us considera thin sphericallayer of radius p and mass 5M.The energy of interactionof the particlewith an elementary belt5S of that layer is equal to (Fig.8) rm\177

(rnv\177/2g)

\177l

is the mass of

ms being the mass of the Sun.

=-roV\177o/'Vns,

-+- \177t/mo) 2.

N = 2b V a/b. 1.185. 1.186. M = i/2rngvot+ cosa; M sin2 a cos = = 37 kg.m2/s. 1.187. (a) Relative to all points of the straight line drawn at right anglesto the wall through the point O; (b) AM = 2 mvl cosa. 1.i88.Relative to the centreof the circle. AM 2 Vi--(g/o)2l)2 ] 1.189. AM ----hmV. 1.190. M= 1.191. rn ----2krl/v 2. 1.192.= y 2gl/cosO. 1.193. F = mcooro/r. 1.194. M+ --= Rmgt. 1.195. M Rmgt sin a. Will not change. 1.196. M'-=M- [r0p].In the case when p----0,i.e.in the [rameof the centreof inertia. i.i98. = i/3 lmvo. 2 2 1.199. e,\177 o. The problem is easierto solve in the I

1.207. M=mV2\"fmsr\177re/(r\177+r2) , where m s t.208.E = T + U = --'ffnms/2a,where ms

the Sun.

Fig.9.

1.212. First let us considera thin sphericallayer of substance (Fig.9). Construct a conewith a smallangleof taperand the vertex at the point A. The ratioof the areascut out by the conein the layer --rl : The massesof the cut volumes are proportionis : dS2 -al to their areas.T\177erefore thesevolumeswill attractthe particleA with forcesequal in magnitude and opposite in direction.What \177

r\177.

dS\177

follows is obvious.

i.2i3.A --= for t.214.G = --(,1M/R3)r --(,1M/r3) r for --3/\177.\177tmM/R.

= t.215.G =

r\177>

--'lM/rThe t.2t6,p = 3/s (i -\177

--\177/sn?pl.

for r\177R, for r\177>R.

SeeFig.t0.

field inside the cavity is uniform. About atmospheres.

r2/R\177)\177tM\177/\177R

19---9451

.--\177/\177.(i--r2/3R\177)TM/R

t.8.t0 \177

\177.

289)))

sphericallayer into smallelet.217.(a) Let us subdivide thecase of interactionof the

energy ments, each of mass 5m. In this = eachelementwith all othersis 5U --?rn5m/R.Summingover all

1.232. A

,\177

?ra

are the massand

t.233,

A-

(M\177/R\177

a kJ, = t.3.t0

M\177/R\177)

v a,\177V2v[+(V\177--t) ?M\177/R, M\177 and R are the mass

\177V\177

t7 km/s.

\177

Here

v \177=

and the radius of the Earth; ?MsC,Ms is the mass of the Sun, r is the radius of the

Earth's orbit. = t.0m. l = 1.234. t.235.N= 2aF\177/mw (aB--bA)k, where k l= aB--bA axis; V\177

where M and

the radius of the Earth and the Moon.

is the

unit vector of the

\177

t.236,l= l aA--bBI/V + B t.237. -=-2F.This force is parallelto the diagonalAC and isappliedat the midpointof the sideBC. t .238. (a) I = (b) I ----l/am (a -4t.239.(a) I= l/anpbR = 2.8 (b) I---a/lomR 1.240. I = l/amRL i.24i.I ----(37/72)mR = 0.t5kgomL i.242.I = 2/a 1.243. to = gt/R (t -t- M/2m); (b) T = mg\177tV2(l d- M/2m). i.244.(a) T = l/2mg, w o ---A\177

\177.

F\177s

t0.

Fig.

\177

Uarnl\177;

b\177).

\177

g.m\177;

elementsand taking into accountthat eachpair of interactingelements appearstwice in the result, we obtain U :----\177m\177/2R; (b) U

\177

--37m\177/5R.

4.5

r = [ 0.84hour ---1.218. At 3Ar/2r-\177 2). ----t : 0.0034 : 0.0006.(6 1.219. wx : w2 : ws t.220.32 kin; 2650km. h = R/(2gR/\177 -- t). 1.221. h = R (gR/v -- 1). 1.222. 2 ----4.2 km, whereM and T are the 1.223. r--\177//?M(T/2n) of its and period revolution about its own mass of the Earth axis respectively;3.t kinC, 0.22m/sz.= where T is 6-t0 a/\177

2\177

days

(5\177---0),

\177,

\177

\177r\177-\177-

\177

\"t0\177

M= 1.224.

(4n\177RS/?T

T/z)

\177

\1774

kg,

periodof revolutionzof the Earth about its own axis. t X2\177R v'= 1.225. + V \177tMR-- 7.0 km/s, w' ( +---\177 =4.9m/s HereM is tho mass of the Earth, T is \177 periodof revolution about its own axis. i.226,t.27 times. over the E of the satellite t.227.The decreasein the total energy v as and E Fv dt. --dE to is Representing time interval dt equal and the centreof the functions of the distancer between the satellite Moon,we can reducethis equationto the form convenientfor integration. Finally, we get (V'\177-t) mlaV-\177R i.228,vx = t.67km/s, = 2.37km/s. the

\177)

\177-

=--R-\177-\177M

2\177R

i\177s

\177.

\303\227

\177

.\177.

v\177

kin/s, where M and R i.229.5v---V';\177M/R(t--]/'\177)=--0.70 Moon. the of radius and the are the mass 3.27 kin/s, where g is the stant.230.Av t)-is R the radius of the Earth. free-fallacceleration, dard -kin. t0 ---3.8. r nR/(t + 1.231, \177

V-\177)

290

\177.

\177

rnR\177.

grnr\177/I.

1.245. to = V6Fsin (p/ml. Ig i.246.= at- rn/2) R ' I m\177--

m\177

T\177

m\177

(md--4m\177)

T\177

rn\177

(m-\177-4ral)

\177

t.247.A = -- md- 2 t.248,n = (1 + k 1.249. t = = l/ato o. t.250.(to) 3/atoR/kg. (rn\177-J-m\177

(m\177--km\177)

\177

km\177g\177t

(mxd-m\177)

\177)

to\177oR/8nk

q- t)

t.251.[I = 2ragx/Rl(M + 2ra). 1.252. tan (a) k (b) T = 1.253. rng= t3 N, (a) T= >/2/\177

\177;

\177/s

g. rng\177t

a/\177

[\177

2/a

sin

\177

g/R =5.t0 rad/s\177; \177

t.254,w' = V3 (g --wo), F = x/a m (g --wo). w = g sin 1.255. -t- I/mr = t.6 m/sL = 2kg/(2--3k). t.256. = 3kmg/(2--3k); F (cosa--r/R) (cosco--r/R) t.257.(a) wx----- m(t+V) ; (b) A= 2m(t-l-V) t.258.T = rag. \177/a

mg\177t.

\177/(t

\177)

Fma\177

w,,a\177

F\177t

\177

1.259. w = 3g (M + 3m)/(M+ 9ra + di.260.(a) _ F d(b) T:- 2m,(m,-4-m\177)\" i,26i. i.262.(a) t = l/s (b) A = i.263.to V t0g(R + r)/t 7r i.264. = V t/3gR (7 cosa --4) = t.0 m/s. i.265. = i.266.voT Vrrw8gR. i.267.T = (t + \177/\177o

w--\177, (3m\177

2m\177).

I/R\177).

F\177t

\177

(3m\177

2m\177)

\302\242OoR/kg;

-----

\177.

\177

\177.

Vlio

19.

rnv\177

2/7r\177//\177).

29t)))

!.269. N l]2amo}212sin 20. 1.270. cos0 3/2 g/\177l. t.271.Ax ks. .272. = i 3m/M. =

T = 1.296.

\177

! F 1.273.

-\177

1/\177

v\"

\177 v'= 3m 3m\177v;

(a)

8My

\177

\177 \177

i.278.(a)

\177=

;

(b) A=

\177

\177.o\177

\177

i/i\177m\177l\177

\303\227

\302\242o'

mgl/I\302\242o

m\302\242o'\177l

\177

\177-

\303\227

\177

\303\227

\302\242o'

=0.8rad/s. The vector to' forms the angle 0 = arctan (w/g)= 6 with the vertical. t.286.F' = 2/5 mR2\177o(o'/l = 0.30kN. l T :O.09 1.287. = kN. N = 2nnlv/R = 6 kN.m. t.288.Fraax t.289.Faaa = 2nnlv/Rl = t.4 kN. The g\177.

w\177./I\302\242o

force

li.

exertedon

the

-----

am Ar/r = 20 atm; (b) p 2am Ar/r 40 atm. glassstrength. t.292,n= l/r2-\177/p/nl=0.8, tO rps, where am is the tensile -----

\177.

the

\177

strength, and

copper. of lead.

is the density of

i.293.= ]/r'\177/p/2nR ----23 rps, where n

strength, and

the density p l \177/mg/2n d\177E

= 2.5cm i.294.x t.295.e = I/2Fo/ES. \177

292

is the tensile

[\177

i.30i.(a) HereN is independentof x and equal to No.Integrat-

twice the initial equationwith regard to the boundary conditions dy/dx (0) = 0 and y (0) 0, we obtain y = (No/2EI)x This is the equation of a parabola.The bendingdeflection is Nol\177/2EI, where I = aVt2. In this caseN (x) = F (l --x) and y = (F/2EI)(l --x/3)xa; =(b)FI3/3EI, where I is of the samemagnitudeas in (a). = ing

\177.

\177----

i.302. F13/48EI. i.303.(a) Herep is the 3/2 (b) =- a/2 density of steel. i.304. = where p is the density of steel. 3 t.305.(a) = (l/2nr ArG).N; (2l/nr\177G).N. -- G(p/32l (b)0.5----kN.m. 1.306. N= i.307.P == -- t7 kW.-|.308.N 1/z U= 1.309. 0.04k J, where p is the density of steel. = t.310.(a) Ul/2mEeZ/p (b) U = 2/3 (hl/l) Herep \177

\177,

\177

\177/5\177pl\177/Eh

\177,

\177

(d\177

\177/\177.

d\177)

ar\177Gq\177\302\242o

\177m

is the density of steel.

q\177

r\177).

\177rZlE

\177/snrZl3p\177g\177/E;

t.3tl.

\177.

--\177

r\177)/(r\177

(r\177

pgl\177/Eh

\177

pgl\177/Eh\177;

q\177

\177.

0.08kJ.

= 7 J. 1.3t2.U== 1/s t.313,u = 1/2 = 23.5kJ/ma, where is the compressi1.314. u bility. t.315, > < The density of streamlinesgrows on transition from point 1to point 2. t,316.Q --S\177S2W 2gAh/(S -t.3t7.Q-SV2gAhpo/p. t.318,v=]/2g(h,+h2p\177./p,)=3 m/s, where Pl and P2 are the densitiesof water and kerosene. t.3t9,h = 25 cm;_ = 50 cm. = 20 cm. 1.320. h = t.32t,p Po v=/gpgh (tho -where <r < R=, Po is the atmosphericpressure. 1.322. A ---, where p is the density of water. t.323.= S/s. t.324,v = cob V 2l/h--I. = = A

p\177

outsiderail increasesby this value while that exertedon the inside one decreases by the samevalue. t.290,p = sEAT 2.2.t03 atm, where a is the thermalexpansion coefficient. ,\177

2\177)

steel.

n\177h83E/l

\177

1/\177.\177

Fig.

where p is

pco\17713/E,

\177

-----

\177r\177G\177/l

\177.

Gq\177r\177/l

amr2q\177m(o/

t.29t.(a) p

(t-

\177/\177

\302\260

Heream is

= 1/3

copper. = 1.6mm a, where is Poisson's t.297.AV'= 2\177 Fl/E ratio for copper. t.298.(a) Al = pgl\177/E; (b) AV/V ----(t -- Al/l, where p is Poisson's ratio for copper. isthe density, and 9---= 3 -1.299. --3(t --2ix) p/E; (b) (a) AV/V t.300.R = l/s Eh\177/pgl\177: 0.t2kin, where p is(tthe 2\177t)/E. densityof

-----

2(!\177t\177--\177a)\" i.279,v' = v (4 -and 4. = i.\1770.(a) = I\177/(I Io). i.\1778i. \177=6.0tad/s; F = mglo/l 25 N. I.\1772. (a) M = = sin 20; (c) sin 20. = 0.7 rad/s; i.283.(a) = sin 0 = t0raN. SeeFig. (b) F = 1.284. (o = (g w) I/nnR = 3 t0 rad/s. = ral l/ + = t.285. I\177I\177

r\177/l\177),

1/\177

(b) F= l (t\17713m) t t.275.(a) u (M/m) sin (a/2); sin (a/2); (c) x \177 =/sl. \177 = M = 1.276. (a) (1+2mx 2m/M)\177o; (b) A =l/\177m\177R=(1 +2m/\177. 1.277.(a) \177= 2mx+m\177 \177'; (b) Nz= mxm\177R du'dt * 2m\177Wma i.27\177.

(\177)

\177

= 9 N.

\302\242Ool1V

\302\260/\1773\177/ml

l/\177.m\302\242o\177l

the density of

-\177

(pgh)\177

p\177,

\177

v\177.

v\177

\177,

S\177).

l\177=\177

1/\177.

\303\267

R\177/r\177),

R\177

l/2pV3/s\177t\177

\342\200\242

\1772--\177

t.326.F 2pgSAh 0.50N. 1.327. F = pgbl (2h -- l) = 5 N.

293)))

a -= 0.7 N.m. i.328.N -= pIQ2/nr--=- 6 N t.329.F = pgh (S of i.330.(a) The paraboloid revolution:

--ts= (1--Vt --(v/c) lo/v or tsta=(t+Vt t.353.(a) t (B)-----lo/v, t (B')= (lo/v) Vt --(v/cy; = (lolu) 1/t--(t,lc)\",t (A') = lo/v.

(b) ta

s)\177/S

r where z is the height measuredfrom the surface of the liquid alongthe axis of the vessel,r is the distancefrom the rotation axis;(b) p = Po + P = n\177R*lh = 9 W. 1.331. \177-

\177,

(\302\242o\177/2g)

\177)

(v/c)2) lo/v.

(b) t

(A)

of hands \"in termsof K t.354.SeeFig.t2 showing the positions

clocks\".

In (rIR\177)

t.333.{a) \177=\1772\177 \177--

;

(b) N=4n\177R\177--R\177

= Va nv0\177; = T= (c) = 4\177lvo/R The additionalhead Ah = 5 cm at the left-hand end of 1.335. the tube impa\177s kineticenergy to the liquid flowing into the tube. \177om the conditionp\177/2 = pgAh we get v = Y 2g Ah = 1.0 m/s. t.336,e = 5. t.337,v2 = rxPx\177=5 \177m/s.

t.334.(a)

\177lvo;

(d)

\177)

\177/\177

nlR\177pv\177;

F\302\242\177

\177.

aa\177

v\177

r\177p\177

t.338.d =

=5 mm,

\177

sitiesof glycerin and lead.

i.339,t -=

\177(ptSRe\177t\177)pog

\177

]\177

where Po and

are the den-

Inn =-0.20s.

i.340,v----c]/\177l(2--\177l)'=0.tc, light.

t.34i.(a) P=a(t+V4-3\1772);(b) p=a(Vt\177+

Here

\177 \177

V/c.

t.357. = t;

V-\177

(a)

t.360,l =

l o (t

v\177

1.\17768.

tan

1.36\177.

ta\177

\177ere

\177s;

2 1.348. lo = vAt/\177t --(v/c) = t7 m. t.349./o=V-Ax-'-\177=6.0 m, v=c\177t--Axt/Ax2=2.2.tOSm/s. \"

\177

(,,/\177)\177

fi\177),

\177)

= 20

\177s

v\177)/(t

v\177v\177/c

where fi = v/c.

(vw2/c\177).

v\177-.\177

where (t-0 where \177i--\177 Cos0--glc O--\177'\177/\177

\177

= Vie.

\177)

\177i\177

\177

\177'

\177

v'V/c2

\177

\177'

\177

l\177/c

fi\177)/(t

--(V/c) 2.

t.365.(a) = \177(t-V/c. Let us make use of the 1.366.

\177.

21e/At t.350,v= t +(lo/cAt) 1.351. The forward particledecayed At = = v/c. where later, ts) .' t.352.(a) lo\177Xa--xs--v(ta V t --

\177

(v\177

\177

\177l

\177

(v/c)\177

V,l\177,)

v\177

v=V 1.361. +

\177

v\177

t\177

ns; (b) 4.0m. Instruction.Employ the invariance

\177-

fi\177

\177

> o. t3

59\302\260;

= SoV t cos 0 = 3.3m% Here = v/c. At &t s m/s. t.344,v=cV/'(2\177-\177=0.6.t0 1.345. lo = cat'V t (AtCt') = 4.5m. = t.346,s cat V t (AteCt) = 5 m. = t.4 t.347.(a) Ate = (l/v) Vt -(b) l' = l \177i (vie) = 0.42ktn.

fi\177-)

(t i.358. V'(\177- v)*+\177 1 t.359.(a) v = + = t.25c;(b) v =

= 0.9tc.

= t.08m, where \177=v/c. 1.342. /o= l V-it -fi\177sin20)/(t-fi\177) tan 0 Hence 0'= t.343.(a) tan 0'= (b) S =

t.355,x = (t -c/B, where =- V/c. -- > 0, then shown first that if At = t.356.It should be of the interval.

where c is the velocity of

t2.

Fig.

\1772)\177m/(t_

\177v/c)\177;

(b)

\177\"

\177

w(t--\1772).

\177

relationbetween the acceleration frame fixed to the Earth: w' and the acceleration in the reference \177

\177

\177'

= (t

\177/2

--v2/c\177)\"

dr\"

This formula is given in the solution of the foregoing problem (item (a)) where it is necessary to assume V v. Integrating the \177't/\177t given equation (for w' = coast),we obtain v ' 0.\177(\177't/c) The sought distanceis l (\177t light(w't/c) -year; (c --v)/c (c/w't) 0.\1777%. \177

\177

\177)

c\177/w

\177.

\177

\177/\177

295)))

i.367.Taking into o

l/t+(\177,,tlc)

\177

account that v--w't/Vi-4-(w't/c) we get \177-,

_--\177.rln

t.368,m/m o t/I/2(t t.369,v = (2 + \177

=3.5months.

w'\177

2.5.(a) p = atm; (b) M +va)BT/V= 2.0 = 36.7 += + --[- -}-- t) = 0.42kK.g/mol. 2.6.T

\177

: :p: : \177)/(t

-----

\1771)

\1771

(moc/p)\177]

c\177.

\177

\177/3

\177

\303\267

\177

p\177c

where \177=V/c,

=- rnoc , where mo is the \177

\177

ticle. 1.384. + T/2moc2-1)= 777 (a) = 2mocz (1/-1 \177

V-=\177/\177c.

rest mass of

the par-

MeV,

\303\267

\177

m\177)\177

m\177

c2\"

highest energy when the energy of the system of the remaining two particles and ma is the lowest, i.e.when they move as a single whole. (m/rno) Use the momentum conservationlaw t.388,v/c= t1--(rn/mo) 2u/c and the relativisticformula for (as in solving Problem transformation. velocity 2.1.m = pV \177P/Po= 30 g, where P0 is the standard atmospherm\177.

2u/\302\242

\177

\177.\17778\177

ic pressure. =

2.2.p = 2.3.

1/\177.

ml/m\177

(\177

mR T/p V.

2.4.p =

296

PO I\177T

-- Ap) = 0.i0atm. -- 1) = 0.50,where --a/M\177)/(a/Mx

(p\177T\177/T\177

(rn\177.-\177-m\177)

(rn\177/M\177-\177rn\177/M\177)

= t.5g/1.

'\177,

< t/a; (b)p =

\177

Mg/aRTo.

2.20.p =

p\177/(t

+ ah)

'\177.

Here

2.21.p\177d=pRT/M----280 atm; p = pRT/(M pb)80 atm. 2.22.(a) T = a(V--b) (t + b) = t33 K; = a/V 9.9 poeMC\302\260\177r2/2I\177T.

ap\177/M\177-=

---=

\177I)/BV(\177IV+

RT/(V-= b)--

atm.

\177--

-- ----t85 atm.P/moP,b 2.23. --Ra V-- ---- = 0.042 2.24. = V (V --b)\177/[BTV --2a 1/mol. (V -2.25.T > a/bR. 2.26.U=pV/(y--t) = t0 M:I. -2.27.AT == 2.28.T = t/2Mu2 (y + + p = + = + 2.29.AU --PoVAT/To(7 -- t) = --0.25 kJ, Q' --t) = 7 J. 2.30.Q = A7/(7 2.31.A = RAT=0.60kJ, AU=Q--RAT=t.00 kJ, 7 = Q/(Q BAT) = t.6. 2.32.Q = (1 --l/n) = 2.5kJ. --

(b)

\177

= 940 MeV/c; (b) V ----c V T/(T 2m0c2) ---2.12. t0s m/s. = = 1.385. Mo ]fi2mo(T 2moc2)/c, V c V T/(T+ 2moc2). t.386.T' = 2T (T\303\2672moc2)/moC2= t.43.t0\177 GeV. The particlemt has the 1.387. El = m\177+rn[-- 2m0(m2+ t/\177.moT

0 2.19. p =p\177(l--ah) (a) =

\177

F\302\261

\177l

\177/it--\177l)/(l-t-\177),

\177

=- V

moc\177/a.

e'=e 1.382. t.383.E --

\"\177

\177

e\302\260Mgh/RT)

\303\267

where = v/c-

-- 0.6c,where c is the

c\177

FII---m0w/(t

(\1771

2.8. = poe-et/v. 2.9.t = (V/C) In = i.0min. 2.10.AT = (rng+poAS) l/B = 0.9K. 2.tl.(a) Tm\177= 2/3 (po/R) Vpo/3a;(b) Tm\177=po/efiR. 2.12.Pmin ----2RV 2.13.dT/dh =--33mK/m. 2.t4.dT/dh ==--Mg/R --Mg(n -- t)/nR. 2.t5.0.5and 2 atm. 2.16. h = RT/Mg = 8.0km; (b) h \177IRT/Mg = 0.08kin. 2.17.m(a)= (t -PoS/g.

(C/\177l)

\1772)\177/2,

t)/\1771

In 2.7.n = In (i+ AV/V)

70, where \177=v/c.

-1]2----0.44%. t.370.(c --v)/c t -- [t + t.371,v = 1/-\177--t i/2c A = 0.42 1.372. mo instead of 0.t4mo s m/s. v 1.373. 2.6.t0 1/2cV\177 e 0.0t3. 1.374.For e << t the ratio is T/moc2 1.375. ]fit(T+2moc2)/c--t.09 GeV/c, where c is the velocity of light. 1.376. (I/ec)V T--(T 2moC2),P = T1/e. 1.377.p = 2nmv\177/(t v2/c2). 2 F2t 2, l = ]fi(moc2/F)z + c2t --moc2/F. 1.378. v: Fct/Vm2oc F 1.379. F l] v and F_L_ v; (b) = mow V 1.380. (a) In two cases: -----

\177a)

V\177

'\177

\177

(\1771

\1771

--\177l)

\1773Ma)/(\177l

\177M\177

To\177l'

velocity of light. The definition of density as the ratio of the rest mass of a body to its volume is employed here. cV\177l

(v\177+v\177

----(\177xMx

(T\177p\177

(T\177

T\177p\177)/(T\177

T\177)/(p\177

T\177)

p\177)

\177

\303\227

b)\177].

T\177T\177

(p\177V\177

(piV\177

p\177V\177)/(p\177V\177T\177

p\177V\177T\177);

V\177).

p\177V\177)/(V\177

\177RTo

2.33.7= \177 t) +\177 t) = t.33. 2.34.cw = 0.42:I/(g.K),% = 0.653/(g.K). 2.35.A = RT(n--t--lnn). 2.36.A = poVo In + t)/4\177l]. --In n) = t.4. 2.37. = t (n --t)/(Q/vRTo 2.38.See (\177--

(\1772

\177,\177(\177--t)+\177(\177--t)

[(\1771

\177

13where V

Fig'. isochore,p an isobaricline, T isan isothermal line,and S is an adiabaticline. = 0.56kK; (b) A' = 2.39.(a) T------t)/(Vt) = 5.6kJ To\177l\302\242\177-i\177/'\177

is an

BTo(\177I\302\242\177-\177/\177-

297)))

2.40.The work in the adiabatic processis n -- In i.4 timesgreater. --i)/(7 2.41.Ti)----To [01+ t)U4\177l](v-iv2. --t) M 3.3km/s. 2.42.v = V'27RT/(7 --= RAT 2.43.Q (2 7)/(7 --t) t). 2.45. = R (n --7)/(n (7 -- t); Cn \1770 for t 2.46.C = R(n--7)/(n t) (7 t) = --4.2J/(K.mol),where -\177

\177l

-----

C\177

\177

n---- In A

\177

2.47.(a) Q = R (n --7) AT/(n -- t) (7 -- t) -- t) = 0.43kJ. = --RAT/(n [\177/ln

----O.tt

kJ; (b)

2.59.(a)

--b)u/cv = const;

T (V

(b) C\177--Cv----i--2a(V--b)'./\177rV\177

\342\200\242

2.60.AT= --3.0K. = 0.33kJ. 2.6i.Q -2.62.\177-----p/kT i.10 cm-a; (l)= 0.2ram. 2.63.p (i rnRT/MV i.9 atm, where M is the mass of an mole. -= 1.6.i0 cm -a, where 2.64.n (p/kT -and are the massesof helium and nitrogenmolecules. 2.65. 2amy cos\1770----i.0 atm, where ra is the mass of a nitrogen molecule. 2.66.i = 2/(OvUp --t) ----5. 2.67.v/v,q-----I/(i-b2)/3i;(a) 0.75;(b) 0.68. for volume molecules. 2.68.(s)= (3N--3)kT kT for linearmolecules. (3N--5/2) and i/(2N--5/3) respectively. i/2(N--l) 2.69.

(\177--i)

\177aV\177

RV\177

v\177a

--__

(V\177+V\177)

(V\177

V\177)/V\177V\177

\177

-----

\177-

-\177

\177l)

N\177

\177

m\177

l\177

O/ra\177)/(i

m\177/m\177)

m\177

\177

p-\177

--(a)3)/(6N--5); --i). = (N--

(6)

----(6N

(c)

\177/\177)/(N

Fig.

--t); --t)/(7 --t); (b) A ---2.48.(a) AU (c) C = (7 t)/(7 -- i). 2.49.(a) C----\177R/( 7---i); (b) TV(v-\177)l\177----const; (c) A-------2RTo (i -i). 2.50.(a) A ----(i --O:)RAT;(b) C = R/(7--t)+ R (t--o:); C <0 for O:\1777/(7-t). 2.51. A \177

-\177

O:V\177(\177]

\177/\177

(\1771\177

\177/\177aV\177

\303\267

\177l(\177-v)/\177)/(7

= RI(7 --(a)t)

2.52.(a) C = Cv -F R/O:V; (b) C = Cv -F R/(t -F O:V). (a) C---7R/(7--t)+ aR/poV; (b) AU=po(V\177--- 2.53.-= V\177)/(7

t); A a=po(V\177-In

i) + 2.54.(a) C

-v\177)/(7 \303\267

V\177)

+aln(V\177/V\177);

7po(V\177-

(v\177/v\177).

298

7v\177)/(3v\177

(5v\177

5v\177).

-\177

\177l

\177/\177

(\177

-\177

\177

l/\177R

v\177r

V--b, t 2.57.A=RTln\177d-a ( v, der Waals constants. k\177.

-\177

\177

2.55.

2.58.(a)

-----

\177

\177l(i-\177)/ti-\177)

\177

= 3.8

-\177

\177l(i+\177)/i

RTo In (V\177/Vx). (a) Ve-aT/

const. 2.56.(a) A ----const.

volume molecul\177, 2.70.A/Q: i/(3N--2)forfor linearmolecules. i/(3N--3/2) For monoatomicmoleculesA/Q=2C. 2.7i.M = R/(cv-- cv) 32g/tool:----2/(c,/c\177--i) 5. 2.72.(a) n ----i/2 is the polytropic index. 3, where---2.73.7 -b q2.74.Increasesby Ap/p Mv\177/iRT 2.2%,where i ----5. 2.75.(a) v,q=V3RT/M=0.47 kin/s, (e)----a/\177.kT=6.0.iO a = 3 d 0.i5 m/s. (b) v,q---2.76. i ----[/2kT/n9 7.6 times. 2.77.Q --i) iraRT/M i0 kJ. 2.78.o)sq = x/2k\177/]-=6.3.10\177rad/sec. 2.79. 2.80.Decreases times,where i 5. 2.8i.Decreases 2.5times. 2.82.C = (i -F i) ----3R. 2.83. = V2p/o= 0.45kin/s, (v) = 0.5ikin/s, v,q = ----0.55km/s. = i.66%; 2.84.(a) 6N/N =

t v\177

) '

where a and

are Van

(8/V-\177)

e-\1776\177

(b) 6N/N = i2V3-\177e-a/\1776\177-- i.85%.

m (Avp 2.85.(a) K; (b) 7'-- = 340 K. T=k(V.\177_V.\177)\177.---380 -2.86. :_330K; (b) v= V 3kTo . rn\177

AU-\177a/Vl--a/V,=O.ti (b) k\177;

--b

Q=RTInv---\177-\177_b-\177

m(\177|--\177)

(a))))

2.87. 2\177

2.it2. dN= (2gno/a3/2)e-u/aT (b) Up= 2.it3.I(a)n the latter case. 2.it4. -- = 9. (b) \177l=t--ni/V-l=0.t8. 2.ti5.e(a)==\177l=t--n\177-v=0.25; (i-2.ii6. i 2Ta/(T\177 + Ts). = 60%. 2.ii7. = i ---n-(\177-i/v). 2.1t8. = t -2.tt9. = i (n y)/(i yn) 2.i20.In both cases\177=i 2.i2t.In both cases\177=i 2.i22. i alan\" 2.t\177.(a) \177=i--vn--i ., (b) \177=i 2.i24.(a) \177=i n--i\177(?-- i) nlnn '

= 0.37kK.

(i--V\177) \177

\177fUdV;

2.88.v= V BkTmIn2--m1 = t.61km/s. 2.89.T = i/amv\177/k. 2.90.dN/N = 2uv.dvz (rn\177/rnx)

\177

\177_

\177)/\177

\177

2.9i. 2.92. = kT/m. 2.93.v = (v), where (v) = W 8kT/\177m. x/\177a

2mv\177

.v\177

= akT,

(v\177)

_\177

where

e-mv\177l

nv_\177

4\177

T1/-\177-\177;

\177m.=\177-/dc\177;

no.

\177

(b) n = t

]/6-'\177\"

2.98. N

\177lnn

\177

(i+\177)

V3kT/m; (b) ep----kT. l/2 S dn (df\177/4n) v cos0 = n (2kT/nrn) sin OcosOdO.

when Tl

Q\177/Tm\177

(df\177/4n)

\177l

(M2-MO\177n/nr

\1771/\1771o

(n\177/n\177)

(m\177--

-l\302\260N.

\177l

\177,

\177s

m\177)

2.t07.(U) ----kT. Doesnot depend. 2.108. \177lBT/Ml 70 g. ,\177

M\177

\177l

\177o)

(r\177--

\177

2.i29. Accordingto ----

\177l

3/\177

300

e-\177/aT4ur 2dr;(d) Will

T\177

Then

the Carnot theorem the expressions

dT/T.Let us find

\177A/SQx

For an infinitesimalCarnot cycle (e.g.parallelogram1234shown in Fig.t4) for 6A and 6Qx. \177A

6Q\177

Tmi\177.

or \177lcarno\177.

Tm\177x

Q\177

and

= dp.dV= (Op/OT)\177dT.dV, -I--p dV = [(OU/OV)r + p] dV.

dU\177.

Fig.

V+dV

i4.

It remainsto substitutethe two latterexpressions into the former one. 2.t30.(a) AS= t9 J/(K.mol); (b) AS -\177Rlnn =25 \177/(K.mol). = 2.0. 2.i3i.n = 2.i32.AS =vBlnn=20 I/K. m 2.i33.AS= --M \"lnn=-t0 2.i34.AS = (y In a --In vB/(y -- i) = --tt J/K. eas/\177\"

= 280rad/s.

(a)dN=noe-\177/a\1774\177r\177dr;

T\177

< O. Hence

Rln----\177n

\177

2.i09. (P--2RTpIn rl) 2.1i0.(o = V (2BT/Ml2) In

is replacedby

Qx-Q[\177Tmax--Trnln

n/2

2.iOi.dv = I dn v cos0 = n (rn/2gkT)3/2 e-rnv2/2aTv dr. 2.i02.F = (kT/Ah) In ----0.9\"t0 2.t03.NA ----(6BT/ad\177Apgh) In 6.4\"t0.tool -x. 2.i04. = e t.39. 2.i05.h = kT In g 2.i06.Will not change.

= (a/ukT)

2.i28.The inquality ) \177--\177 \1770b\177omes even stronger 5Q\177

e-eo/ar.

]fl-\177o/\177kT

v,\177=

2.100.dv

(\177--i)

\177

\302\242

AN / N

2.99.(a)

\177

Then

2.1ti.

\177

i) In n

? (a--i)

2.t25. x In +(\177i)/(V--i) (x--i) Inn 2.i26.\177 V/(V--i)

The principalcontributionto the value of the integral is provided by the smallestvaluesof e, namely e eo.Theslowly varying factor ]/\177-can be taken from underthe radicalsign if ascribed the constant ]/\177o-

n--i\177(\177-\177

(nkT)3/2 I\177

value

\177

\177=

2hT = (m/2nkT)i/2n. dv:,. 2.95.(t/v)= V 2ral\177kT= (v). 2.96.\177N/N=2\177(nkT)-\177e-\177V, = 0.9%. 2.97.\177/N = 3 e-o\177

\177

(v\177)

2.94.p =

n\177-v

\177

3/2e-\177s/2\177r

\177

\177/\177.kT.

(b) rp\177=Vk-\177.,

increase

(c)tiNC=

\177._\177_\177

\177/K.

\177--i

\177)

30i)))

lna--In 2.t35.S2--St----vR 2.t36.AS = (n--i) R t) In

t.0 J/K.

)

(n--\177)

\177.

(\177--

2.t37.AS= v(\177+t)R

lna-----46J/K. \177).

\177

3a Fig.

i5.

2\177za\177/pg.

2.t50.The=pressurewill be higher after the fast expansion. In (t + i/n) = 5.t J/K. 2.t5t.AS ---- In (t + n) In In 2.t52.AS (T/T\177) ----4.4J/K, where (T/Tx) + \"t- v\177R

m\177c\177

cx and c2

heat capacitiesof copperand water.

2.t53.AS----Cv In

\177 4T\177T\177

O. log

are the specific

(t/\177)

the mean time which takes a helium atom to cover distancesof dimensions. the orderof the vessel's ----24.6%. 2.t55. ----NI/[(N/2) 252. I]\177

2.t56.Pn = n[

respectively. 2.t57.P==

-\177-

PNt\177

\177P\177/2n

' t/32,5/32,t0/32,t0/32,5/32,t/32

\342\200\242

(N--n)l2/v

NI nl(N-n)l p=(t--p)

where p=V/Vo. where no is Loschmidt'snumn-'\177,

2.t58.d=\177/6/nno\177l\177=0.4 = t.0.t0 ber; (n)= 2.t59.Will increase = (i + ATCo) \177ra,

: \177.

t/\17712

\177/\177o

times\342\200\242

-a atm. 2.t60.(a) Ap = 4a/d ----t3 atm; (b) Ap = 8a/d = t.2.i0 2.t6t.h ---4a/pgd 2t cm. -a =- lisped (t -2.162. t). 2.t63.P Po d- p -gh d- 4a/d ----2.2arm. = 5 m. 2.t64.h -- (n t) + 4a (n -- t)/dl/pg -= tt mm. 2.t65.Ah ----[po4a [ cos0 [ -\177

\177laln)l(\177l

-----

\177

(d\177

3(\1772

\177

2.t8i.A' = F + pV ln (piPe), where a. 4a/R, V 2.i82.C --Cp==+ a/sPor/a).

dl)/d\177d\177pg

\177

\177/\177nR

t/\177Rl(l 2.t84.(a) AS --2(da/dT) (b) AU = 2 (a -- T da/dT) = t.2 2.i85.A = AmRT/M 20 g, V,= t.0 1. Here 2.t86. (V --mVl)/(V\177 -v; is the specific volume of water. 2.t87. Mpo (Vo -- V)/RT 2.0g, where Po is the standard atmospheric pressure. 2.i88. ==(n -t)/(N --t); I/(N + t). 2.t89.AS mq/T = 6.0k J/K; AU = m(q -- RT/M) = 2.1MJ, A(\177;

\303\227

\177I.

-----

m\177

V\177)

,\177

m\177

\177

-a s is where \1770 2.t54.(a) P--i/2\177;(b) N----\177\17780,

f\177v\177

2l\177lR\177/ar

\177\177>

(2v-\177

T----(mxclT1-f-rn\177c\177T\177)/(mxc\177-f-m\177c\177),

\177

T\177).

v\177R

\177

\177,

(T\177/T\177);

\177I/K.

\177/spgha/a

\177

(T\177

\1771

_____

\177-

,\177

R\177

\177

(T\177/T\177)

(d\177

t/a\177d2

\342\200\242

\177n

2.142. S aTeC. 2.t43.AS---- m[a In (T\177/TI) + b -T1)]= 2.0kJ/K. 2.\17744. C = S/n; C \1770 for n \1770. 2.t45.T = Toe(S-So)/c.See Fig. 15. 2.i46.(a) C= --a/T;(b) Q=a In A d- Cv (T\177-(c) ----a in 2.t47.(a) (b) \1771----(n-(n--t)/2n; --i)/(n + i). 20 2.t48.AS =vRlnn-----1) RTo/(7--t), AS= 2.t49.AU = -----R in 2.

: :: :

2.i70.h = 2acosO/pgxScp. 2gl--4cz(n--t)/pd 0.9 2.t7i Vl cmS/s. V n I ----0.20mm. 2.t72. -2.t73.m 2aR\177a[ cos0 ](n -- t)/gh = 0.7 kg. 2.i74.F 2arn/ph = t.0 N. 2.i75.F 2aR2a/h = 0.6kN. 2.i76.F 2a\177l/pgd 13N. 2.t77.t ---a. 2.i78.Q ---2.i79.(a) F -- uad = 3 9J; (b) F ----2had =-- t0 9J. 2.t80.AF 2had (2-t/a _ 1) --1.5

2.t38.Vm-(t + 2.t39.T-=---To+!R/a)in-2.t40.AS R [(Y2 (V/Vo). b)/(V1 -- b)]. -= 2.t4t.AS= Cv In (T2/T1)+ R In [(Y2 -- b)/(V1-- b)]. \"\177po/a

2.i66.R = 2a/pgh 0.6mm. 2.i67.x = l/(l + --=h)lt.4d/4cm. cos0. 2.i68.a = [pgh +Pod/da) pol/(l 2.i69.h = 4a/pg --dl) ----6 cm.

_--

\177

\177l

where T = 373 K.

2.i90.

h\177,

(Q--rncAT)

poS(\177

-\177-

----20cm, where c is the specific = 100K, q is the specificheat of vapo-

qM /RT)

heat capacity of water, AT rizationof water, T is itsboilingtemperature.

2.i9i.A ----mc (T -- To) RT/qM = 25 where c is the specific heat capacity of water, T is the initial vapour temperature equal to the water boiling temperature,as is seen from the hypothesis, q is the specific heat of vapour condensation.

\177,

where p is the density of 2.t92.d 4aM/'qpRT 0.2 2.t93.9 = \177IPoVM/2nRT = 0.35g/(s.cm\177), where Po is the

\177

water.

\177tm,

standard atmospheric pressure. = 0.9nPa. p \177tV2\177RT/M

2.t95.Ap = a/V\177M = t.7-t0 atm. 2.t96. pq. About 2.t0 atm. = 3.6atm.l\177/mol 2.t98.a = = 0.0431/mol. = 4.7 cma/g. 2.t99. = = t.5. 2.200.(n + 3H -- t) = 2\342\200\242t94.

\177

p\177

\177

2U\177R\177T\177,./p\177.

Vg\177

3/sRT\177./Mp\302\242\177. \177)

(3\177

8\177,

\177

303)))

2.20i.(a)

= 3binC=5.01; (b) = a/27b\177 = 230arm. -----= a/bR 0.30 M/b = 0.34g/cm*. 2.202. 8/27 kK, Per = where 2.203. 0.25, p is the density of Mp\177flpRT\177 \177l----8/a etherat roomtemperature. Let us apply Eq. (2.4e)to the reversibleisothermiccycle 2.204. Vmax

P,\177a,,

T\177r

i/\177

1-2-3-4-5-3-1:

Sincethe first two integralsare equal to zero, p dV = 0 as well. The latter equality is possibleonly when areasI and II are equal. for Note that this reasoningis inapplicableto the cycle 1-2-3-1, example.It is irreversiblesinceit involves the irreversibletransition at point 3 from a single-phaseto a diphasestate. where q is the specific heat of melt2.205. = c I t I/q=0.25, \177.

ing of ice;at t = 2.206.AT ------(TAW/q) Ap ific heat of melting of ice. \177l

--80\302\260C.

= --7.5mK, where q

is the spec-

\177.

,\177

\177)

\177

\1771

\177

(T\177/T\177)

q,\177/T\177

m\177q,

m\177c\177t\177

where T=

(T'--t--ln rn\177

\177r

\177

\1771

\302\243

\1771.

\177

(b)\177

3.8

\177

\177tm.

d\177n

t\302\260

s-\177;

= \177/2]/r\177d2n2(v)=l.Ox = (v) V 8RT/nM. (b)

\177

s-\177.cm-a,

102\177

where n=_po/kTo,

2.229.(a)

2.230. (a) = const,v increases\177 times;(b) decreases n times,v increasesn times. \177

2.231. (a)

\177T-\177, v

T\177--t-In \177/(K.mol),

T\177

1=o. where

Cv ----

Ta.

\177

2.232. (a) V (C) T1/(1-n), 2.233. (a) C \177

\177

T(n+l)/\177(n-l}.

= 1/2B(i + 2) = 29 J/(K.mol). 2.234. n = nee-m,where = 4V/S (v), (v) = V 8RT/aM. 2.235. Increases(t + \177)/(1 + \177) times. = 2 times. 2.\1776.Increases aa/\177 2.237.(a) D increases n times, \177=const; (b) D increases n 3/2 times, increases\177 times. 2.\1778.D decreases 6.3times, increasesnm t.6times. 2.239. (a) n = 3; (b) n = 1;(c) n = 2.240. 0.t8nm. = t.7. 2.241. d\177r/dHe 2.242. 2a\177Ra/AR; p =.\177kT/ud\177nAR = 0.7 Pa. \177

\177

n\177/\177

\177

N\177

2.243. =

2.244.N = t/2\177aa/h. \177

2.245. N= 2.246. 2.247.T =

(t/R\177

\177/3\177aap \177a\177M

2.248. x=

2.2t8.C = Cv --qM/T = --74 = .\177/(.,,, --t). 3\1774

\177--\177-\177

m\177Tt-\177-rn\177T\177--ra\177q/c\177

qM/T\177

\177

\1771

\177

2.207. qAT/TSp = t.7 ma/kg, where q is the specificheat of vaporization,T = 373 K. 2.208. p,, Po (t + qMAT/RT = 1.04atm where q is the pressure, specificheat of vaporization,po is the standard atmospheric AT = t.i K. 2.209.hm/m = (qM/RT t) AT/T = 5%. 1 Theseassumptionsareadmis2.2t0.P = Po exp (\177o-\177) ]\" siblein the caseof a vapour narrow temperatureinterval, far below the criticaltemperature. 2.2tt. cpTAV'/q = 0.03,where c is the specificheat capacity of ice, T 273K, q is the specificheat of melting. 2.2t2.(a) 2t6K, 5.iatm; (b) 0.78,0=.57,and 0.2tkJ/g. 2.2t3.AS m [c In + q/T\177] 7.2=k:\177/K. c In 2.2t4.As + qo/T\177 8.6:l/(g.K). + --t0 :I/K,where c is the specific 2.2t5.AS = mc In (T/T\177)= heat capacity of copper,T ----273 K (undertheseconditionsonly a part of the icewill melt). 2.2t6.(a) When < not all the icewill meltand AS=m2c\177 T')=9.2J/K; \177 \177 the ice will melt completelyand (b) When rn2czt2>mtq, _ in )----t8J/K, In AS -\177+c\177(rni V'\177v

2.2t9.AS = + Cp In (T2/T\177). 2.220.(a) 0.37; 0.23. (b) 2.22t. = A//ln 2.222. (a) P = e-at;(b) 2.223. (a) X= 0.061xm,= 0.t3ns; = 6Mm, hours. 2.224. 18times. 2.225. L = (2aN\177,/3b)V'a (kTo/V\"\177upo) = 84 nm. 2.226. = ud\177PoNAV2?/MRTo = 5.5GHz. 2.227.(a) 0.7 Pa; (b) 2.10 cm -a, 0.2 2.228.(a) v = V\177n (v) -= 0.74.10

2.249.T (z)

(Z\177

2.250. AT = T 2.251.

T\177

N\177/4a\177.

uM/2RT.

P\177PI

(x\177T\177/l\177

t/R\177)

(m/Z)

x\177H\177).

Z\177/x\177).

(AT)oe-\177*, where

{I +

x\177T\177H\177)/(x\177H\177

Z\177)/(Z\177/x\177

\177

(I/C,+ --

i/C\177)

2\1779451

I]}\177/\177,

Sx/L

where z is the

tancefrom the platemaintainedat the temperatureT [(T\177/T,)\177/\177

dis305)))

2.252.q--2iR3/2 9\177t3/2/d2NA

(T3\177/2--

T\177/2) -\177-

40 W/m 2,

where

i=3, d is the

V\177.\177

diameterof helium atom. 23 mm l, consequently,thewhere gas is ultra-thin; q =p (v)(t\177-- t\177)/6T(\177-- i) = 22 W/m = VSRT/nM, T = + in \177 2.254.T = +

effective

2.253. \177

--\177

\177

\177,

1/\177

T\177).

(T\177

3.t6.E --1/3ar/eo. 3.t7.E --1/3koo/eo, where k to which the with 0

is the unit vector of the z axis respect angle is readoff. Clearly, the field inside the given sphere is uniform. 3.t8.E = --'/6aR\177/eo . 3.t9. (I) = 1/2)\177R/e o. The sign of (I) depends on how the direction of the normal to the circleis chosen. I

3.20.

T\177--T\177

: : T\177

T = 2.256.

2.257.T q/m

ln(R\177/R\177)

To + (R To + (R

The ratio 3.t. =

Fe\177/Fg

\177

0.86.t0 C/kg. N. 3.2.About 2.t015

--r --r

3.5.\177T-

\177)

3.22.E,,\177

w/4n. w/6u.

is equal to

3.23.E =

t04\177

and

I.103fl respectively;

\342\200\242

3.6.E 2.7i--3.6j,E 4.5kV/m. 3.7.E-3.8.E=\177=0.t0 kV/m. l >> r the strength 8.9.E= 4980(rg @ ) 3/2 caseof

3.t0.E--

\177or

a point charge.E\177.x= 6

E\177

for

\177'

\177s

4\1778o

3.13.(a) E= thread. (a) 306

directedtoward

4\177eoy

-----

l=r/\177.

\177f\177eor\177

\177l

\177/,

E For

(b) E

' (b) E--

\342\200\242

\177

V%\177--\177;

4\177eo

x >> R

In both cases

(\177*--a*)\"

\177

3.34. 4\177eo/\177

the

\177/\177

) .When q

where q

3.35.E --a,i.e.the field is uniform. 3.36.(a) E --2a(xi--yj); (b) E------a(yi--xj).Here i, j

are the unit vectorsof the x and case a > 0.

axes.SeeFig.16illustratingthe

3.37.E :--2(axi-\177 ayj -+- bzk), E = 2W a (x y2) -f- b2z of revolutionwith semiaxes]/f\177a and V(p/b. (b) (a) An ellipsoid In the case of >0, a single-cavityhyperboloid of revolution; when = 0, a right round cone;when q) < 0, a two-cavity hyperboloid \177

\177

\177-

The vector E is directedat the angle 45 to \302\260

\342\200\242

the

(\177=-\177o,

x3' where p

1/-----\177

r\177/\177;

2-h\177o

\342\200\242

4aeor for r\177a.

3.i4.E= k the

the vector a is

3.30.A(p ----\177( t 1/t-4-(a/Rp)\" 3.3i. --(pa = In 5 kV. 3.32.(p = oR/eo, E = 0/%. 3.33.\302\242p=-\177-o(Vt+(R/l)2--t),E_---\177o (t oR o . l-,-O,then E=-\177o,when l>>R, then (p\177

4\177eo/\177

\342\200\242

3.t2. (a) E-- 4eoR ;

4980r\177

\177/3

for

V\177/\177--\177-\177-\177

4\177eo

--\177---q

directionof thevectorE corres-

3.29.E = 1/\177aD/eo, where axisof the cavity.

3qR\177

3.tl.F--

\177 \177

the for

3.28.E----1/3a9/%.

4\177eoX4

strength E

sign of (I) depends on ).theThecircle is chosen.

3%\177r\177

in the

\177

3.24.(I)

.25.

\177

oo/eo,with = 4aRa. 1/\177

pondingto the angle

qqo 892\1770r2

q\177

----\177/aeol.

(b) E,,x= 1/9 9oR/eo for rm = 3.26.q = 2aR2a,E = 3.27.E =

3/2aV 2neomg/l.

\177

1/t.4_(/\177//) \177

[

-l\302\260

3.3.dq/dt

q(t--

]\302\242]-----\177

directionof the normal to 3.21.[\177 = \177/3apro (R --r\177)/eo.

how the \177

[

\177.

q\177

of revolution.

3q . 3.38.(a) q)o=S=soR, (b) q)=q)o( t--3--r'),

(b)

\177

20*

r\177R. 307)))

p 3.39.E=VE\177+E\177 = \177Yi+3cos 20, where Er is the radial componentof the vector E, and Ee is its componentperpendicularto Er. 3cos\1770--t P 3sin0cos0 ., 3.40.Ez= P , r r E_\177--

4\177 o

4\177eo

E A_ p at the pointslocatedon the lateralsurfaceof a conewhoseaxis is directedalongthe z axisand whose semi-vertexangle 0 is found

3.45.

]f\177P ,

280

\177,

2eo(X\177_]_R\177)3/2

E where p = \177R20/. In the formulas for the potential the plus sign correspondsto the spaceadjoining the positively chargedplate and the minus sign to the spaceadjoiningthe negatively chargedplate. then

\177

and

\177

\177eoX

2\177eoz\177,

\177

3.46.(a) F=0; (b) F--

--2.l.i0

3P*

3.47.F

\177P

, 2geora\"

(c)

\177P \342\200\242

2\177eor*

-\177

3.48.\177=--axy\177const.

3.49. = ay (@--x + eonst. a.50. == (az + bz) + eonst. \177

\177)

--\177

\177

p p p q=

\177.51.

\177.\177.

8.58. 3.54.

6%ax.

4l V

\177eokx.

3.55.A=-q\177 i6neo/

i23.5\302\260).

\342\200\242

3.56.(a) 3.57.F = (2 \177--i) 3.58. 32geo/a ql 3.59. 2a(l

these

\342\200\242

\177

(2V\177-l)q\177Saeo/\177

E:2(t

(b)

5V51 )

\177eol2

q\177

32\177eo/\177

\342\200\242

\177

3.60.(a) Fl=

\177

\177/2

\177

q\177nd:

--q.

r\177)

(b) a=\177(l

\177+z\177)\"

4\177eo\177\"

3.6t.(a) o=\177;(b) o(r)= V\177 t 3.62.(a) o= (l + R*) .' (b) E = 4aeo 4l* [t \"

\177

4\177e

q o R

3.63.

\177=

Fig.

i7.

is the projectionof

,\177

\177

3.44.

(\177

funcand

(\177l

2eo\177Z\177---\177h-\177

\177t+4(l/R)*

E\177:=

\177

olR*

2%(z

\177

Rs)3/2

SeeFig.

+\177/a

(R/l)*l

\177/\177'

\177\"

4aeo/

i/r--

tla b 3.65.qe=.--\177q\177; \177:\177 (l--b/a)rif 3.66.(a) = = o, q\177

\177 \177

\342\200\242

(\177--

308

i7.

axis. The

ql 2\177eoX-------

\177/2

\177

Fig.

the vector E on the x If JxJ >\177R, then tions are plottedin Fig.

2\177

if a

\177

r\177b.

x)/l, q, --qx/l.Instruction.If the charge (1is imaginedto be uniformly spreadover the planepassingthrough \177/\177eoh\177/d,

3.67. = --q

]\177

\177

\177/\177eoh\177/d. \177

q\177

309)))

that chargean d

parallelto the plates,the chargesqa and q2 remain, obviously, unchanged.What changesis only their distribution,and the electricfield becomeseasy to calculate. 3.68.dF/dS = V2o2/8o. = 0.5kN. 3.69.F = = 3.70.F 1/\177R2o\177/eo. 3.7t. N = (s--hopt) %E -- 3.t0 Fig. t9. where n o is the concentration of molecules.

\177 q\177

3.84.(a) E, = 2eEo/(8-4- 1), = 2Eo/(e-4- t), ---- = = = 288oEo/(8 + Eo/8,Di = D2 = %Eo. = = = 3.85.(a) =t); (b)=E1Eo, Eo, = eoEo, = 2Eo/(8+ t), Dx = 2eoEo/(8+ t), = 8D1. (b) 3.86.E = q/2n8o(e t) r 2. 3.87.9 = po8/(8--t) = t.6 g/cma, where e and 9o are the perE\177

E\177

D\177

E\177

D\177

8D\177;

D\177

E\177

D\177

\303\267

kerosene.

mittivity and density of

\177,

3.72.F= i.tR (attraction) 3.73.(a) x= (b) z= 0.29R (repulsion).SeeFig.t9. 3.74.P= e--1 -q-r' q'= e 3.76. = --q (e--t)/e,q'ot,t = q (e--l)/e. 3\177P2

\342\200\242

4\1772eo/7

R/V-'\177;

q\"

4\177r3

\177

Fig.

q\177nn

2t.

Fig. 22.

3.88. = (8 -- t) 8oE = 3.5nC/m q' = aR (8 --t) 8oE= 3.89.(a) Since the normal componentof the vector D is contin\177

(\177o\177

-----

t0 pC.

\177,

uous at the dielectricinterface,we obtain

=--ql(8---- t)/2ar (e + t), \177

-q t)/(e + t). 3.90.F = (8 (e --

(b) q' =

q\177

3.91.D =

Fig. 20.

3.77.SeeFig.20.

\303\227

lEo sin 0.

e\177,

\177RaE\177cos

\177

3.80.(a) E=

\177;

\303\227

\177--.\177_\177.te

pl/8% for

l<d, > d,

--p/\177/288o

for

l<\177d,

--(d/28--?l--d) pd/% for The plots Ex (x) and (x) areshown in Fig.2t.(b) a' = pd (8 --t)/8, p' --p(8 -- t)/8. for r< R, 3.8t.(a) E= pr/3eo8 2 pR3/3%r for r>R; (b) p' = --p (8 -- t)/8, o' = pR (e --t)/38.See Fig. 22. 3.82.E 3.83.E = --Po(t -U= \177

pd/eo for 1

q\177

\177

l>\177d.

q\177

xVd\177)/eo,

3t0

4dPo/38\177.

= q/2n%(t e) r / both in vacuum and in dielectric. and o' -+0. 3.92. = ql (8 -t); for l (8 3.93.o' = ql (8 --t)/2\177ra8 t)/2ar38. 3.94. = Ph/%d (between the plates), hid)PC, = = Phial. r\177

\303\267

--\177-0

\177-

E\177

\177o

D\177

--\1770;

\177

(\177'

(\177--

\177'

74\302\260;

\177'

t)/16n8o/\177 (8 + t). q/2n (1 + 8) r in vacuum, 8q/2n (t -4-\" 8) r 2 in dielectric;

\177

3.78.E=--\177-\177/cos2\177o + e2sina\177=5.2 V/m; -- % I) Eo cos = 64 pC/ma. tan = 8 tan hence, e = 3.79.(a) E dS = (b) \177Ddr=--% (8--t) \177

q/2\1778o

for 1 --\1770 and

E\177

=--(t-

3.95.p'=--2a,i.e.is D\177

independent of r. 3.96.(a) =E =--P/38o. 3.97.Eo E - P/3eo. 3.98.E = 3Eo/(8 + 2), P = 3%Eo (8 -- t)/(8 + 2). 3.99.E 3.t00.E=--P/28o. 2Eo/(8 t); P = 28oEo(8-t)/(8 t).

\303\267

3.101. C= 3.102. The strength ='12C$ (8 -- t)/(8 + t). soS 3.103. (a) C=

d,/e\177%-d2/e\177

3.t04.(a) C -= 8o

(8\177.--

8\177)

decreased

' (b) \"

S/d In

8oV

(82/e\177);

80/dS82. (b) 9'= --q(8z-e\177d\177

e\177d

3tt)))

3.t05.C = = ezR2Ez, 3.t06.When4neoa/ln(Rz/R,). 3.107. V = R1E1 [In (R2/R,) 3.t08.C neeIn (b/a). 3.109. C 2Zteo/ln (2b/a). 3.110. C .\177.2Zteoea. Instruction.When e,R\177E\177m

3.t35.(a) W = 3q'\177/2OaeoR;--(b) = = t/5. 3.136. W = (q2/8a%e) (t/a t/b) 27 mJ. 3.i37.A =-- (q\177/Saeo) (t/Rx -- tIRe,), t 3.t38.A-( W\177/W\177

.\177

q(q\302\260+-q/2)

.\177

4r\177o

b >> a, the

chargescan

be assumed to be distributed practicallyuniformly over the surfaces of the balls. C 4Zteoa. = C,-f- C2 -f- C3; (b) Ctot\177 l = C. 3.112. (a) 3.113. (a) C = 2eoS/3d;(b) C = 3eoS/2d. 3.114. V -+- C1/C2)= 9 kV. = t0 V. 3.t15.U = Vi$/(t(1-f-3\177q-f-\17732) = = 3.1t6. C (V\177 t)/2 0.62C.Since the chain is infinite, all the links beginningwith the secondcan be replacedby the capacitanceCx equal to the sought one.

3.ttl.

\177

Cto\177aZ

\177.<

C\177

3.117. VI =

= t0

q/C\177

3.i18.V1 = --\1771)/( t C1/C2),V2 =(\1771-- \1772)/( t C2/C1). 3.119. q = 1\1771-- CxC2/(Cx C2). = 3.t20. In the easewhen C\177/C2

(q\177.

(\1772

-\177-

\177f-

'\177,

\17721

C\177C\177--C\177C\177

\177

\302\242\177,--\177z

\342\200\242

(C\177+C\177)

=- C3/C\177.

(c\177-{-C\177)

3.121. =0.06mC, q= 3.t22. = $C2,q2 = --\177CiC2/(Ciq- Cz). = --24 3.123. qa= (C\177--Cz)/(C\177+C:) q2 = q3 = (C2-(Ci--C2)/(Ci@ C2)= 36 3.124. --Cz= --Ct$1)/(Ct+ Cz+ C3). + 3.125. %= q\177

\177Ca

\177C,

\177

\177C,

60\177t

(C2\1772

\302\242,

\177C\177--\177

\177C\177

\177=

C\177)

\177

C\177+C\177+C\177

\1773= C\177+C\177+C\177

+Cs

3.126. = Ca+ + 3.127. (a) W=(\177+4)qV4\177%a; 2C\177C\177

Ctot\177t

+C\177)

(C\177

C\177

2C\177

(b)

W=(\1774)q\177/4\177oa;

3.t28.W-- 21n2 \177 a 3.129. W = 3.130. W = 3.131. = + = --0.03mJ. = 3.132. Q $\177CCo/(2C+ Co). It is remarkablethat the result obtainedis 3.133. Q= independent W= 3.134. + + =\177(\177+\177_\177). q\177

q\177q\177./4a%l.

\177W

\177

\177/zV\177C\177C\177/(C\177

C\177)

/zC$\177.

o\177

\177.

q\177

W\177

312

W\177

q\177

\177

3.141. (a) A

(q2/8aeo)

(x2

x\177)/2eoS;

\177

x\177)/2x\177x2.

(x\177

VzCV2\177I/(t

\1771)

\1771

\177tA.

,,\177

3.t49. a = (a,-f3.t50.(a) (\177

\177tA.

(b) a

(a2 + \177lal)/(t q(a) a/sR; (b) (C) Rx = R (V\177-I). R = (1 V t -+- 4R2/R\177) R1/2= 6 D. Instruction.Since is the chain infinite, all the links beginningwith the secondcan be replacedby the resistanceequal to the sought resistanceR. Imaginethe voltage V to be appliedacrossthe points A and B.Then V = IR =IoRo,where I is the currentcarriedby the lead wires, Io is the current carriedby the conductorAB. The current Io can be represented as a superpositionof two currents. If the current flowed into point A and spreadall over the infinite wire grid, the conductorAB would carry (becauseof symmetry) the current I/4. Similarly,if the current flowed into the grid from infinity and left the grid through point B, the conductorAB would alsocarry the currentI/4.Superposingboth of thesesolutions, R = Re/2. we obtain Io = I/2.Therefore, R = (p/2a/)In (b/a). In the caseof b--\177R = pl4aa. R = p = 4aAtab/(b a) C In R = p/2\177a. (a) = 2aIV/gr\177;2 (b) R = 9/433. = (9/\177) in (1/a). (a) = lV/29r In (l/a); (b)

3.151. 3.152.

\177132)1(t

\1771);

,\177

\1771)-

3]\177/\177.

\177/t2/\177;

\303\267

(C\177+C\177)

(-t/a- l/b). =eoSV -(b) -- = t.5 = 3.142. (a) --A= t)/[e (e --t)12 = 0.8mJ. (b) VzCV2qe (e 3.i43. Ap = see(e -- t) V:/2d = 7 kPa = 0.07atm. 3.144. = (e- 1)a2/2eoepg. 3.t45.F = aReo(e -- t) VVd. 3.t46.N = (s -- t) 3.147.I = 2a%aEv= 0.5 I 23% --t) rvV/d = 0.tt 3.148. 3.t40.A =

3.153.

t/C\177q-t/Ca-{-I/C\177

\177C2

R\177

q\177q\177

3.154. 3.155. -3.156. p(b--a)14aab. 3.157. 3.158. 3.159.=]] I VC/peeo = 1.5 3.160. RC = peeo. 3.161. (r = D, = D cosa; ] ----D sin o:/eeo9. 3.162. = 5 nA. I = VS -- in 3.163. 3.165.= (r 3.166. eoV (e292 -- elPX)/(pxdl 92d2), (r = 0 3.167. q = SoI (eep2 -3.168. l). 9 -- 2eoV -R\177

\177tA.

((r\177

(r\177)/d

((r2/\177rx)

-\177-

(\177l

t)/d\177

(\177l

expx

\"+

313)))

= 3.t69. 2. (b) E = 2aaI/S 3.t70.t(a)= Rx--RC2acz/S\177; In (I -- V/Vo) = 0.6 la = =

3.17t.9 3.172. I=

3.205.(a) I = (Vo/R) e-amc;(b) Q = 3.206. e/m = lor/qR = i,8-i011 C/kg. 3.207.p = IIm/e= 3.208. s = enl (v)/]\177 t07 m, where n

--In1)2$/R]1.4.t0 e-ntmc. 3.t73. = $/01-t- 1) = 2.0V. = ($x -3.t74. = v. + = 0 in the sourceofSxcurrent 3.t75.R = -- Rx, ternal resistance -- = 0. 3.t76.(a) I = 3.177. -- =(b)($x -- RJ(Rx + = --0.5V. \177s.

\302\2421

g\177.m.

\177/eoe

[(\177

R1/(/\177x

$\177)

\302\242\177

R\177

R\177)

0.\1770

--\177

in-

R\177. \177z;

q\177

\302\2429a

\177

3.178.= =0.8 A. Ix $R\177/(RRx + \302\242a

3.t79.V =

\177

Vox/1.

\177)

R1R\177

VoRx/[RI q- Ro (l

t.2A, I2 = IxRJR2=

R\177)

R\177R)

--x) x/1];

for

R >> Ro

\177,

3.180. = RxRJ(Rx+ $= $2R\177)/(R 1 --R\177$x)/(RR I= 3.181. = 0.02A, the + RIR + current is directedfrom the left to the right (see Fig.3.44). 3.t82.(a) I = -= 0.9V. +RaRe)= 0.06A; (b) 3.183. I 3.184. = \177t.0 V. + + 3.t85.IRaRe) ($\177R\177

\177-

(R\177$\177

R\177),

\177

\177A

\177B

R\177

\177

$\177

R\177).

R\177R)

(u\177

\177

\177/z$

q\177R/ht;

V\177/4R;

\177

R\177

RxR\177/(Rx

3.20t. --AW 3.204.(a)

= (1 --

\177/ze

\177,

e-\177V\177e)q\177/2C

mJ.

\177T;

x:)Z/\177=

\177

+ a) I/R. I V = 2.0\177T.

(\177o/4a)

\177/l

3.228.(a) B--(\177o/4a) V4\177 aZI/R:O.30 (b) 3.229.(a) B between the (b) B 0 outside the

\177oi/2;

planes.

\177T;

\177oi

planes and

inside the plate, outside the plate. 3.231. In the half-spacewith the straight wire, B \177oI/2ar, is the distancefrom the wire.In the other half-spaceB 0. 3.232. The given integral is equal to \177oI. for B:\177Veto[jr] r\177R, 3.233. [jr]Re/re for r\177R. 3.234. B [jl],i.e.field inside the cavity is uniform. \177o]d

\177

3.230.B--[\177]x

0.5mJ. (e h 3.202. where 9 is the density of water. o (e-t) VVggd 3.203.(a) q = qoe-t/e.eo;(b) Q = (t/a -t/b) q\177/8aeoe.

\177/\177

V\177C$\177RJ(Rx

t) CV \177

3\1774

R\177);

\177ohI/4a\177Rr,

(\177o./4a)

3.227.B

2\177

\177

\177oR\177I/2(R

2.3[tT. B 3.220. tan (a/n)/2aR, for n B \177oI/2R B sin 3.221. 0.t0roT. 3.222. B (a -- + tan T) \177oI/2nR 28\177T. 3.223.(a) B:\177( +\177)\" (b) B:\177 +\177) ( 3.224.B where r is the distancefrom the cut. 3.225. B 3.226.(a) B (\177o/4a) (aI/R); (b) B (\177o/4a) (t + 3a/2) I/R; 4\177oI/\177d

The current flows from point C to point D. 3.t87. = r + 3r). 3.t88.RV as (r(t +--3R)/(R e-2\177mc).

3.200.(a)

\177oI/2R

n\177oI

(

\177/\177

: :: : : : : :

\177

\177d.

: :: : : : :

ms. 3.215. t=(n--t)/V2:t3 t = 4.6 days. 3.216. I = evoe 3.217. 3.2t8.j 3.219. 6.3 (b) B -(a) B eo\177U/e\177id

3.189. = (b) Q = x/z In 2.q\177R/ht. 3.t90.R(a)=Q 3Ro. 3.t92.Q = I ($ V) = 0.6W, P = -IV = -2.0W. 3.t93.I = = 1/2. P\177= = 2%. 3.t94.By V/2R; 3.t95.T -- To = (l --e-\177/e) VVkR. 3.t96. R = 3.197. + 3.t98.n 3.t99.Q = + Re) = 60 mJ.

-\177.

u\177)

= 6.t0 cm

IxR\177

0.2A.

R\177Rx)

is the concentrationof

\177ol\177/2Vo.

R\177Ra

CVoL

electrons,(v) is the mean velocity of thermal motion of an electron. 3.209.(a) t = enlS/I= 3 Ms; (b) F = enlgI= t.0 MN, where p is the resistivity of copper. E= 3.210. m/2eV =32 V/m, h\177=(I/4aeo) V m/2eV= =0.80V. (I/2\177eor)V 3.211. (a) O(x) = --\177/oeoax-2/a; (b) ] --\177/9eoaa/z V 2e/\177. 3.212. n = Id/e l0s cm VS = 2.3. + 3.213. Uo = 9 3.214. cm-a.s-1; (a) n, = I\177jeV = 6.t0 (b) n = \177i/r=

free

with

Aq\177

\177tN.s.

t/\177

\177

\177/\177o

3.235. ] (r)

\177/\177o

315)))

B= \177onI/\177t + (2Rll)e. 3.236.

B = i/2\177onI (1--x/Wx

3.237.(a)

x > 0 outside seeFig.28;(b) xo = solenoid;

x<0 inside the

the solenoidand

e\303\267

R2), where

: :

nR. 3.264.I\177 = V when R = pd/S, the power P= 3.265. P = Pmax = 2pV. 3.266.U = almost t :1. 3.267.n = ]B/eE= 2.5.t0 -a m-l; = 3.2\"t0 3.268. m\177/(V's) uo = F= 3.269.(a) F = 0; (b) F = (\177o/4a)2Ip,\177/r F\177B; 2F\177,\177/po

v\177B\177d\177R/(R

pdlS)\177;

\303\267

\342\200\242

\177/\177v\177B\177dS/p

\177/\177oI\177/n\177R\177ne

3.238. B= 3.239. 3.240. \1771

V\177'(t

(\177t\302\260I/h)

\177

(\177o/4a)

N/a =

\177

(1)

= t.0 \177tWb/m.

8.2I/r, t0

(h/2\177/\177)e=

\177.

(\177to/4\177)

3.241. (1) = (I)o/2 =

> R.

0.3mT, r< R,

\177Wb.

3.243.

30 mA. m 3.244.

(b)

= i/3nIN (a

2 (b--a)

\177

3.246.(a) B 3.247.B 3.248. 3.249.

\303\267

3.250. Fm/Fe

3.25t.(a) 0.13mN/m.

3.252. B= copper. 3.253. B= of

F\177

\177oeo

v\177

(b) .\177B

Fig.

23.

poI\177/4R

t.00.t0-%

= 0.20mN/m;

(b)

F\177

\177oI\177/al

\177N;

[(2\177 k\177.

2I\177I\177.

F\177

\177

(B\177

--B,)/2po. Theforce is direct\177

ed to the right. The current in the conductingplane is directed beyond the drawing.

3.26t.Ap = IB/a= 0.5kPa. 3.262. p ---3.263. p \177toI\17718a\177R\177.

oppositedirections.

%1; in

\177).

\303\267

+ =

\177

\177).

3.284.From

0 Fig. 24.

2\177RB/(\177oNI

p,\177a\177

\177.

circulationof

the vector H we

H:t.51--0.987H (kA/m).

\177\302\260\177d

\177\302\260NI

1.0.t0

\177

the theorem on

obtain

B\177

bBl\177tond

H = 0.06kA/m, 3.283.

TherequirB and H areinterrelatedas shown in Fig.3.76. Besides, ed values of H and B must simultaneously satisfy both relations. Solvingthis system of equationsby means of plotting, we obtain H 0.26kA/m, B t.25T, and = B/\177oH 4.t0 \177

\177

F 3.285. 3.286.(a) z,,= t/V 3.287.A 3.288. By V 8w/a. I = Bvl/(R 3.289. 3.290.(a) Aq9 ,\177,

4\177

B\17712\177o.

I\177o\177

$\177

\177;

\177oF,\177a\177

V'\177/VB\177o

= 3.6.t0

-\177.

where

R\177),

\177/2(o\177a%n/e

3.29i. E dr = -/\"

(b) Z =

i/e%VB\177/\177o.

\303\267

20 mV. c

\177.

\177

F\177

3.259.F1= 3.260. In all three cases =

Blsin 0.

2Bo\177/(t 3Bo\177/(2

meability is bB) = 3.7.t0

\177

3.256. R\177V\177o/eo (ln\177)/a=0.36 3.257. = 3.258. = \177 b In (t +b/a). F\177

(\177t--

\177

(4\177

(\177oalIo/a) In

\177)

H\302\242

\177

\303\267

3.280. = NI/l 6 kA/m. = 3.281. H =0.t0kA/m. 3.282.When b << R, the per--

\177.

q/2m.

\177

dr = (t --

3.278.B = 3.279.B =

kT, where \177 is the strength of (2pgS/I)tan0 = t0 mT, where p is the density copper. 3.254.B = Amgl/NIS= 0.4 T. -- t) = 0.40 A = 3.255. (b) (a) F = 2\177olIo/a ad\177z\177/4RI

\177/\177.

3.277.B--

pT.

(V/C)

x\177)

\177

3.275.(a) I\177 %I; (b) 3.276.See Fig.24.

\177)

\303\267

\177

\177tT;

pm/M

\177/\177R\177;

\177

\177/\1772B

+ b = t5 mA.m 29

2/3\177off\177R

p\177

\177

(\177o/4a)

ileNId\177

\177,

\177

: : :: : :: In (b/a)

\177c)

\177,

2Ipm/r F r.

3.270.F = 6nR\177Ipmx/(R 3.27t.F a/\177op\177,e\177/nl = 9 nN. 3.272.I' 2Bxa/\177oR = 0.5kA. 3.273.B'= B V sinea cose a. cos0. t)/\177o; 3.274.(a) H dS =

\177/6o

\177.

p\177olN

(\177o/4a)

\1771

=0.5A.m Pm 3.245.(a) B = \177.

\342\200\242

\177

2INhIn = = 2\177R3B/\177o =

(\177o/4\177)

t/\177IB

\177onlS/2,

where (1)o is the flux of the vector B through the cross-section of the solenoidfar from its ends.

3.242.(1) =

\177s

i/2\177oBd

\177

3.0 nV;

(b)

t0 mV.

tle\177on\177I\177.

3\1776

3t7)))

....

3.292.$t = 1/2(--t)nBagt, where n = 1, 2 is the number of the half-revolutionthat the loopperforms at the given moment t. The plot St (t) is shown in Fig.25 where t,, =

3.293.Ilnd = a/r, where a --1/2t%lvI/aR. 3.294.St = z2Inky (z + a)\" a 3.295. 1/,2 (coa3B + 2rng sin

3.318.= density of

sin

rngR B\177l

L\177

\177z

dent of

b v In \177; (b) F=-\177-\177 {

\177\302\260I\302\260v

2\177----R--

\177oIo 2\177

\177,

-\177

3.304.(a)

3.328\342\200\242

3.305. I

U\177coBo

3.306. = $\177m

3.307.St =

3.308. E= 3.309.I =

b)/p

\177/3\177a\177NcoBo \177

a/2wl/\177t

copper.

1/2\177onlaUr \177/\177tonSd]/p

-\177/\177).

(L\177+L\177) \177\302\260b

Z\177-

,L\177 /2\177---

ln (l +\177-)

b In a

2\177t ,\177,

t) O-tn/L].

(L\177+L\177)

(b)

U2\177toZtaUb;

\177/2\177tona\177l/b.

(I)\1771

2aRq/\177toN.

,\177

U2\177to\177a*/l

\177.

3.334. = \177(t --e-tn/L,).

0.5A.

L\177

\177zL\177

I\177

3.335. Q

3.336.W

> a.

3.338. (a)

3.337.W =

where p is the resistivity

L\177

2R\177

r < a,

for r = 2 mA,

(\1771--

t\177#h.\177___\177N

\342\200\242

for

1/2\177t\302\260n\177r

L\177

= t2 mY.

\177

/t:

50 A.

-2)

*L\177

3.330.z _. 3.33t.(a) LI2 3.332. Pm: 3.333.

c\177

In the round conductorthe current flows clockwise, there is no current in the connector;(b) in the outside conductor, clockwise;(c) in both round conductors,clockwise;no current in the connector, in the left-hand sideof the figure eight, clockwise. = (d) -- =

3.329.L,z=

U\177mv\177.

a---\177

gaB

3.327.I=-\177-(1 e

3.302.(a) s = vomR/l*B (b) Q = F 3.303. v= (t --e-at),where ----B212/mR.

+\1771)

3.326.I=\177-[t +

Fig. 25.

\177

-3.30t.(a) I--

\1771.

bto(ln-\177

3.300.q-= \177In b-j-a b--a'i.e.is indepen-

I= aa=B/L;= (b)2 A. =

I :3.325.

+l\177BuC/rn

\177t\177aI

3.324.I Io (t

3.297.w= t g sin 3.298.(P) -- (acoa\177B)UR. = 0.5T. 3.299.B = U\177qR/NS

=- 0.26\177tH/m.

,NZaln(t+-\177-).

\177---\177

\177,

\1771

poh/b = 25 nH/m.

=.--\177-\177

3.322. 3.323. (a)

\177z

\177t\177o

3.32i.Lx =

\177

\177/\177

copper.

L 3.320.

\177o

3.296.v=

__--0.7ms, where p is the resistivity, Po is the

/PPo

4\177

3.319.

V2---\177-\177n/g.

4\177

\177i

\177o

\177

(t+ao/R) =

UzN\177I

BHn\177a\177b

0.5J. 2.0J, where H = 1/2 NI/\177b.

W\177,/W,\177

vb/ad=3.0;(b) L

\177toSN\177

\177

3.339.W 1 = 3-t08V/m. 3.340.E = = t.t-10 = 3.34i.Wm/W B/V\177o\177o= = L/2. = 2L; (b) 3.343.(a) 3.344.Lt2-----Y 3.346.W -2 cos0. = 3.347.(a) Jd --J; (b) Id = q/%eO. current should be 3.348.The displacement addition to the conductioncurrent. 3.349. = Im/eocoS= 7 V/cm.

b+ad/\177

=0.t5H.

\177o\177,\177co\177a\177/8a.

\177

-\177.

%\177toco2aVl

\177

Ltot\177

L\177ot\177

L\177Lz.

4rRlb

\342\200\242 \177

\177\302\260\177a\177

\1772

3.3t5.l fV4\177tloL/tto=O.tOkin. 3.3t6.L= mR , where p and Po \177o

4\177

/PPo

are the resistivity and the

I\177I

taken into account

E,\177

cos(cot +a), 3.350. H:H,\177

density of copper. L

2---'-'\177--

and

where

a is determined from the formula

__ rVm

H,\177--

tan

1/O2 --

_}_

(e0eco)2

a:eoeco/a. 3t9)))

318

3.35t.

ilz\177R21r

Here

\177

----

3.368.(a) E'= l/t-- t.4 nV/m; (b) B'= B V t-fis =0.9T, a'\1775t

< R, for r > R.

for r

il\177r

]\177=

\177

sin cot.

\177

\177tonI,\177\302\242o

\302\260.

3.370.B'= B Vt --(E/cB)2 0.t5mT. the positive directionof the x axis of the reference frame K. Let us pass into the frame K'

2qv . qv 3.352. (a) ja=\177,(b) ja= \177-$ 3.353. = 0, Jd max = 4ha

\177

3.37t.Supposethe chargeq moves in

\177r

x\177

\342\200\242

\177

H= q [vr] 3.354.

3.355. (a) If

4\177r

\342\200\242

V E = --OB/Ot=/= O. The spatial derivatives of the field E, however, may not be equal to zero E =#, 0) only in the presenceof an electricfield. (V E = --OB/Ot=/=O.But in the uniform (b) If B(t),then V

B (t), then

\303\227

at whose origin of coordinates this is at rest (the axesof the two frames coincideand charge the y and y' axesare parallel). In the frame K' the field of the chargehas the simplestform: -- t q with the following componentsin the plane x, y r'a 4no0

r',

=0.

\303\227

E field V (c) It is assumed that E a/(t),where a is a vector which is independentof the coordinates,(t) is an arbitrary function of time. Then --OB/Ot= V E = 0, that is the field B doesnot with time. Generally speaking,this contradictsthe equationVvary H = = OD/Otfor in this caseitsleft-hand sidedoesnot depend on time whereas its right-hand side does.The only exceptionis the case when (t) is a linearfunction.In this casethe uniform field E can

-----

\303\227

be time-dependent. 3.356. Let us find the divergenceof the two sidesof the equation \"q H = j -4- OD/Ot.Sincethe divergenceof a rotoris always equal \303\227

zero, we get 0 = V.j+\177-(V.D).It remains to take into account that V.D = p. 3.357.Let us considerthe divergenceof the two sidesof the first equation.Since the divergenceof a rotor is always equal to zero,\177.(0B/0t) = 0 or (V.B)= 0. Hence,V.B= const which doesnot contradictthe secondequation. V to

\177t

3.358. E =-[o)B]. E' = [vB]. 3.359. 3.360. ----%vB = 0.40pC/m 3.36t.p= --2co)B= --0.08nC/m3, \303\227

(\177

3.362.B-- r3 E' = br/r where r 3.364. q[vr]

(\177=eoatoB=2 pC/m2.

Eqs.(3.6i),

is the distancefrom the z' axis. where r is the distancefrom the z' axis.

3.365. B'=- airy] ' c*r\177

3.367.(a) E'= E whence

a\1775t\302\260;

t--

\177

\177//

(b)

B'=

E== E'=,

E\177

= E;/y t --(v/c)Z.

Solvingsimultaneouslyall these equations,we obtain E = E\177i

4:\177e

\" r\177

(t--\177sin\177O)\177/\177

Note that in this case (v ----const) the vector E is collinearwith

the vector r.

3.372.v----\177/O/2ale/m = tfi km/s. 3.373.tan a = al 3.374.(a) x=2Eo/a;(b) w=qEo/m. 3.375.t= = 3.0ns. eeE 3.376.w= me (t + rn

\177

\177- \177//

\342\200\242

2\177y\177

(T+2m\302\260C\177)

= eE---A-t V t --(reC) wheree and m 0 are the charge moVo a proton;(b) vx=vo/]/i+ (t--v\177/c (eEt/mocZ)e. T/rnoc\177)\177

4\177

\177,

r'sq Y\"

x = r cosO= x'V l (v/c)z, y = r sin O= y'. Besides,in accordancewith the formulas that are reciprocalto

V\177r

\177.

E\177'=

Now let us make the reversetransitionto the initialframe K.At the moment when the chargeq passesthrough the origin of coordinates of the frame K, the x and y projections of the vector r are related to the x' and y' projections of the vectorr' as

\177

\177o

\303\227

\177

cos*

t--\177 \177Esin\177z

\177z

--9 kV/m; tana, = tan ct V\177, _t4

\177tT.

3.377.(a)

tan

and the mass of

\177

\177,

\177)

3.378.a = arcsin(dB 3.379.(a) v=reB/m=tO0km/s, T=2nm/eB=6.5 Fs; (b) v= =4.t ns. =c/\177/ t + (moc/reB)2=0.5t e, T= eB 2r\177m\302\260

V\177

\"\177-\177

320

21--9451

(v/c=)

32)))

3.380.(a) p=qrB; T =moc\177(V l +(qrB/moc)Z--l); r

3.397.(a) T=

[t -4-(moc/qrB)2]

\177lrno

3.382.AZ =

respectively. 2.0 era. ]f2mV/eBZcos

Instruction. Heres

3,383,q/m = r = 29 sin 3.384.

1,where p = \177- sin -- 2n m 1 3.385. rmax:ae where b-]

(\302\242\177/2)

\177o

\302\242\177,

\302\242\177

Since N is large,

cos

\177

q/ra =

2\177mEn 2

3.387.(a)

Y n-----

qB2

(b) tan

;

3.388.z = l tan V2q\177y;

for

\177.

rob

o\177

\177,

\177 \177

\177

2EAx

= a (1 --cos where a = y = qB/m. The trajectoryis a cycloid (Fig.26).The o\177t),

o\177t);

dp dt

r0 =

\177/z

X\177 '

............ \177x v.', o '1 . k.,/ k .......... V //

3.395. y

a\177

322

\177

\177,

....... \177..

o \"\177Zln-\177-

\177-

Fig. 27.

V. a

(sin cot--cot cos cot),

a = qEm/rn. The trajectory has the form 3.396.V 2\177v\177mrAr/e = 0.10MV. \177

w\177

\177

b\177--

\177

motion of the particleis the motion of a point locatedat the rim of a circleof radius a rolling without slipping along the x axis so 8rnE/gB\177; that its centretravels with the velocity v = E/B;(b) s = E/B. (v.) (c)

\177/z

(v\177/a\177)

Fig. 26.

3.394.

\177

2Bo/3a. = 3.\17705. dECr (to) -(\177) = O. 3.\17706. AW = 2=r\177eB/At = 0.10 keV. z- 1)mocZ; (b) s = WAt/reB. 3A07. (a) W = (V 1+ (reB/moc) (a) SeeFig.27; (b) + (z/a) = I and = --\177z. 3.\1770\177.

\302\2431.

e eE= 2at

where p is the momentum of the electron,r is the radius of the orbit, is the magneticflux acting insidethe orbit. On the other hand, dp/dt can be found after the relationp = erB for r = const.It follows from differentiating the comparisonof the expressions obtained that dBo/dt = \177/zd (B)/dt.In particular, this condition will be satisfied if Bo = (B).

2\177

\177.

\342\200\242

B\177

2-\177- (\177o

\177dn.

\177

V= 3.393.

v\177

3.403.On the one hand,

<<i this equation reducesto

a (a+2b)

rnE/qB

W\177,

\177

2--h--\177-.

\177tN,

3.392.(a) x =- a (cot --sin

\177

\177o

\177)

3.391. q/rn-

\177

= (2mE/ql\177B zz. 3,389,F = mEI/qB= 20 = 6 era. 3,390,Al= 2nraE eB tan \177

3eY

= 2nvW/eBc = 9. 3.400. = \177o/V 1 + at, where = qB/m, a = qBAW/nm'c v = '/zrqB/m,p = r/2. 3.401. 3.402. N = W/e\177 = 5.10revolutions,s = 2nrN = 8.t0 s kin.

In (b/a) \177z----

\177

3.3\177.n

\" \177

s\177

\177\"

V r\177B

\177

v\302\260lb,

3.386.u----rB InV (b/a) '

\177

leB my

(b)

=20MHz. = 0.74 kin.

V\177m

4n3V\177mr\177

where is the velocity of the particle after the nth passage across the acceleratinggap.

\342\200\242

\177i(B\177_Bx)\177

Vmin=-\177r

\177

\177s;

cz, 5 keV and 9 MeV \177-=--

2\177

-- 12MeV; (b)

3.398.(a) t= e\177=t7 N n'\177vmr

T= 3.381.

(erB)22m

of unwinding

4.2.(a) The

Where

spiral.

amplitudeis equal to a/2, and the is T seeFig.28a;(b) = 4\177x (a --x), seeFig. period 4.3.x = acos(\177t+a) =--29cm, =-- 81cm/s, where = a V xo + z, = arctan (-\177

\177/\177,

\177

v\177

(V\177o/\177)

2t*

\177

V\177o/\177Xo).

323)))

4.5.(a)

(v)

=3a/T=0.50m/s;

(b) (v) = 6alT=

i.0

4.i8.T=zVml/F=0.2 s. 4.i9.T = 2u V nt/g (\177- = t.1 s.

m/s.

: : :T:

4.20.T = 2 V/\177

1\177

4.2t.t=

(a)

(vx)

(b>

\177F\177 q

where B=wlg.

\177

--\177--n

aco;

(b)

I(v)l=

4.29.(a) x + (g/R)x = 0, where x is

aco; (c)

i--cos(wt--n\177/2)],

H\177re

of the displacement itsradius, g is the = 42 rain,

\177

4.7.s = [ a[n+s\177(\177t--n#/2)],n is odd. n is a whole number of ratio 2\177t/#. \177

the

body relativeto the centreof the Earth, R is standard free-fall acceleration;(b) z =

a\177. \177n

\342\200\242

a [n

(c\302\242/[\177)].

4.22.T V 4\177m/pgr z 2.5s. 4.23.T=2\177(t--B) s. m/u=O.13 4.24. 4.25.T:2aVm/x,where u= 4.26. V 2To/ml. 4.27. 2a Vm/Spg(1+cos0)=0.8s. 4.28.T = oo V 2likg= i.5s.

Fig. 28.

4.6.(a)

+ arcsin

[\177t/2

is even,

V\177 = 7.9 km/s. 4.30.T = 2:tVl/I g --w = V g2 ly2 __ 2gw cos

0.8s, 4.31.T=2:t/V\303\227/m--coz=0.7s, 4.32.k = 4a2algT = 0.4. 4.33.(a) 0 = 3.0 cos3.5t; = = l

th\177

\177--

\177_

V\177-\177

where

g --w l

[\177.

rad/s.

\302\242o\177V\177/m=lO

\177

\302\260

4.5 sin3.5t;(c)

0 = 5.4 cos (3.5t 1.0).Here t ].5 is expressedin seconds. ZO 4.34. F= -f- ms) g -ts t-

(b) 0

\302\260

-\177

\"a

mlaco

4.35.

29.

s x2 V a In both casesa = 7.

4.i0. 4.ii. 4.i2.47.9x and2.73aco. 52.i 4.i3.i8 or 26s Hz. s-l, i.5g. 4.14.(a) yS/b2 = i, clockwise; (b) w = --cost. 4.t5.(a) y\177: +4xs(l = a(t---xS/aS); (b) Fig.29. 4.16.T = 2n Vm\177o. 4.i7.T = 4ha V\177. Vm=

\177.5

(l-b

2x\177/aS).

= O.

Tmu\177

4.37.(xlro)2

4.38.(a) y

-----

\177r-

\302\242z

(ylVo)

w/co\177;

----t0

(b) Tma\177: ----

(b) y---- (cot--sin cot)

cm, E = m\177gS/2k 1+2h\303\227/mg, E = mgh q--

mg/k

\"-\177

30.

\177u-\177m;

(1--coscot)

Here co ----V'\177/m. 4.39.Ahrn\177 = See

Z\177

\177

2rag,

x\177/a

324

(a)

a\177\302\260----\177-\177

Fig.

l/\177

0 q-- g coscot)' see Fig. 30; cm; (c) a = Fig. (b) arnin = g/co2:8 cm. =(co 4.36.(a) y ----(i --coscot) ragC,where co = =

\177)

4.8.s=0.6m. 4.9.dP/dx=

(m\177

60 and 40

4.8

----

4.40. V 4.41.a (ragC)W 1qq- M)g. 4.42.Let us write the motion equation in projectionson a ----

(mg/\303\227)

x and

axes:

mSgS/2\303\227.

2h\303\227/(m

coy,

= --cox,where co = aim. 325)))

Integratingthese equations,with the initial conditionstaken into account, we get x = (vo/co) (l --coscot), y = (vo/co) sin cot. Hence (x --vo/co) + y2 = (vo/co)2.This is the equationof a circleof radius

a= [ --\177/2, when

\342\200\242

4.69.(a) ao=

Xo\1770,

\177

Vo/co

with the Will

4.43.

centreat the point, xo = vo/co, Yo 0. increaseV l + el5 (R/l) times. It is taken \177

into

\177

accountherethat the water (when in liquid phase)moves translationwise, and the system behaves as a mathematical pendulum.

4.44.co = V 3g

2\303\227/

4.45.(a) T = V l/3g= 1.1s; (b) E--i/emglaZ=0.05J. 4.46.q\177m=q%Vl mR q%/2kq%,2 E = 1' 4.47'.(T) = + /12ml 0o. 4.48.T = 4u/co. -- = 0.8g.m2. 4.49.I = rnl (co]-4.50.co = V (Itco + Izco\177)/(I, + 12).\" 2\177

e'\177

\177-

4.52.T =

W2h/g, -= coo V 3acoz/2

4.53.

4.54.coo = V

l\177

T 4.57.T = -----

V 3 (R V 3rn/2\303\227.

\177

4.80. = *l

(a) T =

tions, here A =

4.83.x=

m\177rnz/(m

\342\200\242

\177

2:\177V

I\177Ie/(I\177

where mo 4.6i.co\177/coi_-=Wl+2mo/mc\177l.9,

and mc are the

masses of oxygen and carbon atoms. 4.62.co--=sV2ypo/mVo,where y is the adiabatic exponent.

4.63.q = 4h V aeorng (,1

\177-

\17712

....

4.68.(a)

=-- arctan 326

(0) =

n\177

[\177(Po,

where

\177

(0) =

= 0.28s; (b) n = (xo--A)/4A= 3.5oscilla-

]/-\177-\177\303\227

krng/\303\227.

Fo/m

\177<

\177,

>\177

\177,

-----

\177

co\177ox

t x + coax = 0, x = a cos[coo(t -- +\177z], where = k/rn, a and are arbitrary constants.From the continuity of x and x at the moment t = we find the sought amplitude: \177)

\177z

co\177

\177

l) = 2.09c.

field increased = 25 times. 4.65.x = (Vo/co) sin cot, where co= lB/V mL. 4.66.x= (l coscot) g/coe, where co=lB/W-\177-\177. 4.67.(a) ao and aoco; (b) ta=-\177-(arctan--\177+nu) , where n=0, 2

4.64.The inductionof the

2\177

(coscoot--coscot). 4.84.The motion equationsand their solutions: t -kF/m, x= (l--coscoot) F/k,

4.58.coo = ]/\177t,where = m\177me/(m 4.59.(a) (b) E 2 cm. Here = + me). 4.60.T = I'/k,where I' = \177t

2RI/a\177B

\177

\177t

-----

2\177,hI/\177R\177T.

4.81. =

r)/2g. s-\177

e-\177/2)

(2\177/\177.\177)

a/\177(4uz+\177,z)

4.82\342\200\242

co=V\303\227/\177t=6

e-I\177t.

\177/\177

\303\227/(rn

2:\177

\177.\177)

\177

4.56\342\200\242

(\177/co)e

[\177o

\177l

= hi2.

+ I/Re). cosa 4.55.coo ~- W MR 2mg 2mR (i + sin a)

s-\177.

4.71.(a) v(t)=aoWco\177@\177e-\177t;(b) v(t)= [Vi+ 4.72.The answer depends on what is meant by the given question.The first oscillationattenuatesfaster in time.But if one takes the natural timescale,the period T, for each oscillation, the second oscillationattenuates faster during that period. 4.73. = n\177.o/V 1+ (1--n z) z= =3.3,n'= Wl + ----4.3times. 4.74.T = Ax/g = 0.70s. 4.75.Q = \177n/ln = 5.10\177. 4.76.s l (1+ e-\177/2)/(l 2 m. 4.77.Q = 4.78.T=V R/g=0.9s. W(4\177e\303\267

co\177)

Tm\177=2u|/\177/\177V

\177

4.70.\177=coV\177l\177-l=5

if --nl2<cz<O,

\177

\177,

x=\177/2V\177,

=/2<a<u, xo<O. if

(\177.o/2\177)

\177

4.51.

xo>Oand

\177,

2\"2

1/+mglO\177

g/l)/(co\177

with --[xolVt+([\177/co)\177, a=arctan(-[\177/co),

([\177

coe)\302\242Po;

n=O, 1,2, .

(b)

t,\177

4.85. 4\342\200\24286.

= (2F/k)lsin(coot/2)l.

' vi\177 = = 5.t.t0 t--(\177,/2\177)

\177

co\177.o=

]\177\"

corn

W (co[+co\177,)/2

4.87.(a)

\177o=

W\177;

= V\1774.88. = (t + 4.89.A = o sin (\177

\177

\177)\177/12. X\177/4a

\177

\177aF

\177)

g/X

\177,FoAl

a\177.o

(b)

2.t.

\177.

327)))

4.90.(a)Q=I/2V

--(o2) tan

(o)\177--

\177o\177)\177

tan 2 q)

-1 2.2; -=

(b)

1.0.10 2. 4.i12.Q -- 2 = = 20 mW. 4.1i3.(P) = R (IS) 4.1i4.(P) = t/2RCV\177/L = 5 roW. t t 4.115. (0 = V LC 4RzCz .' R<tV-\302\243\" 4.i16.\177+\177=\177 and \177=\177. 4.ti7.I = te_t/\177; I = V2m

--\177

\177ma\177

V\177f\177

(\302\242o\177--

= 6 mJ. HereO)o ----V \177m.

q\177

t/\177RI\177

4.91.(a) (P)--(\177--\177)\177+4[\177; (b) F\177t\177/m

\177=\177o,

(P)ma\177='F\177/4\177

= 4.92.(P)'\177':--(P) <p>\177 \177%.

\177

\177o0

5.53.(a) A= --a\177mNmsin\177;

(b)

\177

2 sin

\177V\302\260

\177

4.t18.I= 4.t19.I=

\177.se. @)

(a) T

\177.98.

\177

2\177

= 0.7 ms;

\177 C\177)

(C\177

\177

Ima\177--

\177.9\177.

at the moment

[cos(o)t --q\177)--cosq\177-e-m/s],tan = o)LIR. tan = [cos(o)t -- --cos e-t/\177c],

\302\242\177

Vm -\177

l//\177

q\177.

q\177)

q\177

t/((oC)\177

\342\200\242

o\177RC

cos the left-hand capacitor,and \177.\177.

\177

I/\177

\177

Vo, where the plus sign refers to the minus sign to the right-hand one;

\177t)

by the equation tan

2/\177c.

.... \177

\177o\177

(W\177).

4.10t.(a) t,=\177,(b) n=0, 1,2

\177n.

4.t02.Vo/Vm= 4.t03.Vc =ImV\177

t\177=w

C Wr/Wc

4np The current is ahead 4.121.

Here [arctan(--5)+\177n] \"

\342\200\242

sin

(\177t

\177

\177)

with

tan

\177

\177/fi;

Vc (0)

\177

R\177

\177.

\177=

I',

\302\242oil

328

(Vm/l\177Im)

2-1.

Fig.

31.

4.t22.(a) V'= Vo Vr, cos((ot--a), where Vm=Vo/V-t'+((oRC) a = arctan (\177RC); (b) RC = V \177z:i/\177= 22ms. 4.t23.SeeFig. 31. \177L\177

\177/\177C

\302\260

\177

V\177

\177

I\177

\177

oo/Voo

2\177

where

t 4.126.For C=\177=28\177F; V\177=Vm\177I+(\177L/R) \177=0.54kV; Vc=Vm\177L/R = 0.5tkV.

5.i27.I=Imcos(\177t+\302\242),

tan

---

(b) Q=\177 --\177; it should be taken into

\177,

\177

account that solving the problem, -I is the cur= of the is the where I charge capacitor, q dq/dt rent in the V/R). winding, I' is the leakagecurrent

When

4.125.

m\177.

4.1t0. \177ln\177=l.O

4.ill.(a)

o=Vo\177-2\1772;(b) = R/2L. = I/LC, (a)

\177

t\177

\302\260,

tan

I\177/\177C

R\177.

\177o--

\177

cp----60

o\177

the phase angle defined by the equation by

4.1\177. A; (b) tan\302\242= (a) [m=Vm/\177z@(\177L--I/\177C)2=4.5 of the voltage); is ahead , \177=--60(the current R = 0.65kV, VR + \177L = 0.50kV. (c) Vc =

\177

Woe-\177/\177

the voltage

= L/CR = 5. 4.i04. 4.i05.L =L\177 + R = + t = \177ln = 0.5s. 4.106. 4.t07.n 1 \177 4L 1 t6. \177\177U.070.. 4.108. 4.109.(a) Wo 0.t0 L\177,

\177

\177to\177Va

q\177

-\177-

\177

e-\177t

=Im

4.t20.The current lags behind the voltageby phaseangle% defined

(I'=

\177

where

= \177RC.

4.128. V c 4.129.Q=\1771/4. \1770

\177

Ira-- R

V,\177

z \303\267

(\302\242oRC)

and

L\177

(LIL\177__L\177)

4.130. Q= \177 (n--t) z 4.131. (a)

\177o=V\177,\177;

(nZ--t) \177

(b) Q=\177x\1772

329)))

4.133. --]/l (Q2 1/4)(q2 __ 2.2and 19respectively. 2 4.135. (a) (b) I----\177Io\177l.15Io; Io/I 4.134. t = i/2nt o.

\303\267

4.t36. R V\177--I=2kHz. 4.t37.The current tags behind the \302\260,

x\177

For R=coL--r=0.20 Pm,\177-4.138. 5.139. Increasedby V\177--I\17730%.

V.\177

kg\177;

For Q>>I the 4.140.

2\302\242oL

= o.16kW.

2a\177l

= 0.3rad.

\177 5.158. r = (airi -+-a2rz)/(ai a2). -\177

4.159. (a) y-= r--ro

voltage by the phase angle

V P ----\177W z2

\177

v)\177

ln(\177lr\302\260/r)

\177

\177,

2\177

Aq\177--

I=--\177Io\177,l.llIo.

v-----\177-\302\243

--arccos]/1--(XL/Z)2

4.156. See Fig.33. 4.157. In (1--*1)

1)2/\177]2,

.\177_

0.08m-i; (b) Vm--

2a\177ao.

....

= 15cm/s.

4.160. (a) SeeFig.34a.The particlesof the medium at the points \1771

solidstraight lines (y =x-4-n\177,n = 0, 1,2

lying on the

--0.11kW.

)

ratio is

Fig. 33.

oscillatewith maximum amplitude,those on the dotted lines do oscillateat all. (b) SeeFig.34b. The particlesof the medium at the points lying

Fig. 32.

4.141. P2 ----'/2 ( V2 -- --4.t42.Pl ---4.143. Z= R/V 4.144. See Fig. 32. \177.1\1775.

mA,

(a)

\177--\177

V\177

V\177)/R

not

30 W.

on the straight lines y----x\302\261 nL, ----x

W. \177--I\177)R----2.5

___

\302\261

: t:

\177C

1/2)\177,

and y----

2\177

(R\177+ \177L\177)--

\177L

\177

\177o\177L\177

(\177CR)\177T(i--\177CL)\177\"

4.149.

(n\302\261

\177\"

A. A, Ic:F\177--(\177)a:l.0 I\177F\177l.0

4.156. tan

y----x\302\261

1/5)L oscillate respectively along those lines, at

,\"

//I

\177L\177L\177I\177

(\177)

2(R\177+\177LI)

21 4.150. (VE+ V\177

(\177)

Fig. 34.

\177

= 4.151.5\177

:: : : :

4.152. k=\177 4.153. \177:a\302\242os[(i--V/\177)\177t--kx'], 4.155. 5.1.\1770\177; (b) (a) 3.2.I0 a/\177

(OUOx)\177

(OUOt)\177 isthe veloeity of the wave. (\302\242)

330

-\177,

\177

wher\177 \177

(OUOx)\177,

\177:\177/k.

II cm/s, 3.2.10-\177; where 0.34km/s \177

right anglesto them, or move alongthe circles(heren ----0, l, 2 At all other points the particles move alongthe ellipses.

4.161. (w) =

2t\177w

.... ).

1 4.t62.<*)=2\177/2Io (t ]/i+(\177//y)=20 pW. 4.t63. = P/]/1-+-(2Rlh)2=O.07W. (\177)

331)))

FeeFig.35,for (a) and 4.164. 2

5.165. (a) swp

kx.coso)t;

----1/2pare(e2 sin o)t. SeeFig.37.

cos \"kx-sin m

\303\227

(b); seeFig.36 for (c). (b)

4.175.u= \177 (\177/%)2_1) \177=0.5m/s. 4.i76.(o = \177(\1771.+(AV/Vo) --t) = 34 \177

(Vt\303\267

i/\177pam\302\242om

\177

s-\177.

4.i77. = \177o/V 1+ 2\177/v = 1.35kHz. \177

4.i78.(a) V=vol(l--\177)=5 km. kHz; (b) r=lVl+\177=0.32

4.t79.Decreasesby 2u/(v + u) = 2.0%. 4.i80. = + u) = 0.60Hz. 2\177ou/(v

\177

4.18i.\177

6.l0-3 m -t.

(\177r\177/r\177)

2 (r2_rl)

= 50dB; (b) x= 0.30 kin. 4.182. L' =L--20?xloge = 4.t83.(a) L Lo \177.20 log (ro/r) = 36 dB; (b) r> 0.63km. 4.184.(a)= ln(rs/r\177)/[\177+ r\177)/v]= 0.12s-l. 4.t85.(a) Let us considerthe motion of a plane elementof the (r\177-

\177

medium of thicknessdx and unit areaof cross-section. In accordance with Newton'ssecondlaw p d\177 =--dp,where dp is the pressure

incrementover the length dx. Recallingthe wave equation we can write the foregoingequation as

Fig. 35.

5.166. ama x = 5 mm;

\177

(O2\177/Ox\177),

32--\177

increase\177: ]/I--aT/T I+\177/\177

\177'----

= --dp.

pv \177 ax a

to the third overtone.

4.i67. = l/\177I\177+\177)I 1.4. \177.168.Will

\177

Integratingthis equation,we get

Ap ----

times.

\177

o\177 _1_

\302\242onst.

In the absenceof a deformation (a wave) the \"surplus pressureis Ap ----0. Hence,const ----0.

4.i86.

\177R

\177

(Ap)\177/2pvX

= it roW.

4.187.(a) (Ap)m ----Vpvp/2ur --5 Pa, (Ap)m/p ---- 5\"i0-\177; a/X = 5.10 (b) a = (Ap)m/2\177vpv -= 4.i88.P----4urmem++o.t0\177-----t.4 W, where L is expressed (\302\242I))

\177

%/,,,

...............

:: 2lv

4.i70.(a) vn

Fig.37.

(2n

\177-

: ....

1),six oscillations; (b) vn

n:0,l, 2

Here alsosix oscillations. 2n\177t

\177

kHz; four \177.17t.\177==\177 \177=3.8(2n+t)

frequencies2fi.g, 36.2,\1771.8, and

4.172.(a) Tmax= 4.i73.W = 1/amSp\177+a+/k. 4.i74.v = 2VoVU/(V -- u i\"

\177

m\1772a

\177

\342\200\242

ma\177,

\177

332

bels.

4.i89.A\177=(l/V\177--l)c/v=--50 m. 4.190. t -= 2 ( V\177llcIn(el/e2). 4.19t. = 2.. 4.i92.H=-\177y'e-\177o[kEm]cOs(ckt), where ]1]\177,

= 0.34km/s. \177--

\177f\177)

\177=r/\177

Fig. 36.

4.i69.v

\177 \177)

-\177.

3\177tm,

6\177.\177

(b)

kHz. =

+ i),

oscillations with

(T)=llsm\1772a\177.

2VoU/V

-\177-

the wave

1.0Hz.

\177/2\177Veeo=

coskx = --0.30ez; -cos W (ckto kx) = O.i8e\177. Hereez is the unit vector axis, H is expressedin A/re.

e\177E,\177

of the z

is the velocity of

in vacuum.

4.i93.(a) H =

(b) H =

\302\242

e\177Em Yeo/\177to

eo/\177to

4.i94.e,,= 2vvl2E,,/c = 13inV. 4.196.(S) = 4.197.(a)

i/\177eocE\177

\177lzkeoc\177EL/(o.

3.3 ]\177o

----

\177

\177tW/m

\177eovE,\177

= 0.20

mA/m\177;

(b)

(S)----

\177.

333)))

4.i98.Sincet >> T, where T is the periodof oscillations, W = 5 kJ. tlsV 4.199. B= sin kx.sin (or, where Bm 2_ Era, with Bm=Em/C. 4.200.Sx --Bm sin 2kx.sin2(or, (Sx) = 0. 4.201.WinCe = t/seo\177to(osRs = 5.0.10 -15. = 5.0.10 4.202.We/W,\177 = -la. 4.204.q)s = I S = Is V 4.205. 4.207.To the left. (I) = VI. 4.208.

(e21rn)2\17716\177.

t/aSo\177to(OSRs

4.209.((1))=

t/2VoI o cos

4.2il.The electric moment of the system is p = = (elm) Mrc, where Mdipole is the mass of the system, r c is the \177.

r\177,

4.212. (P)= l 4.213. P= (4nee) 3

4.214.AW

ega20)a

4\177o

3c \177

( 3c3

0 too. -t5 W.

s K

T ) 2. a\"

After

t is the moment of timewhen the particle a definite point x, y of the circle,and t\" is the moment when the information about that reaches the point P.Denotingthe observed values of the y coordinateat the point P by y' (see Fig.4.40), we

is at

shall write

' g'(t')=g(t).

The sought acceleration is found by means of the double differentiation of g' with respectto t': dy

\177-\177-\177\"dt

dt'

d\177y

\"\177'\177 dt

dtd [ dy' -'7 dt

\177

v\177

v/c--

\177

Rcos(ot,and

dt' ]

-\177-\"\177(t\177vy/cR)

= v/R.

flow

\177/a\177rSSo

\177/6p%)\177/\177eo

334

responseof an eye for the given wavelength.

5.3.(a) (E)=t/2Eo; (b) (E)= t-5.4.M = ---- o. sin2 0; (b) M = 5.5.(a) q) 2 =

I = 50 Ix. \177-\177-

2/3y\177L

\177L.

\177.

the planesof the given mirrors, eo, %, ea of the incident ray and the rays reflected from the first, second, and the third mirror.Then (seethe answer to the foregoingproblem): % =et--2(e\177ns) n s, ea = es 2 (esna) el = eo 2 (eon\177) the left-hand and right-hand sides of these termwise Summing it can be readily shown that ea = --e\177expressions, = arctan n 53

n\177,

. 1= 1.25. 5.16.nt/ns= l/V1]s5.17.x= [1-V (1 sin2o)/(n2- dsin0=3.1cm. 5.18.h' = (hn cosa O)/(ns --sins 0)a/s. 5.21.0 = 83 . 5.22.From 37 to 58. 5.23.a = 8.7

5.i5.

\302\260

\177-

0\177

\177-

sin\1770)]

\177

\302\260

2 sin (0/2) 5.24.Aa = ]/in \177

portionalto the squareof the

4.219.(P) = . 4.220.(w) = a/sPo/\177r2c. 4.221. P= ca.

\302\260.

radiation S is prodensity of electromagnetic of the observed accelerag projection tion of the particle. Consequently, St/Ss = (1 v/c)a/(l v/c) (b) Energy

\302\260

where the following relationsare taken into account:x = R sin

t.1V/m,

is the relative spectral

e\177,

S1/S\177

1.6mW/lm,

E,\177

4.2i8.(a) Supposethat

dy'

Here A =

\177

\302\242

hence

\177o/eoAq)/2\177rSV\177,

-+- h\177/R

l/ae\177B\177/\177eocarna.

\177--z(t)

3.0mA/m.

R/V-\177

2.5s for the electron, 1.6.10s=0.5.10 years for the proton. = 3. 4.2i7. = tan s ((elC)

t'=t-\177

Ema\177

4.215. = 2.10 AW/T = t/aeaB/eoCam2 -ts. 4.2i6.T= Toe-ut, where a= t\302\260

E\177

\177

qe2

3carn\177vb

5.2.

i.61m,

\177/s(Vl\303\267

i.6

UIoR\177/h

\177

(4\177%)

\177

\177LAS

\177e4q

\177

5.i.

E =LS/4R 401x. 5.6.h\177R, 2 = 3.10lm. a 0, (I) ---5.7.I ----Io/cos 5.8. = (9/16nV\177) pES/R2 = 0.21lx, at the distance from the ceiling. 5.9.E =----nL. 5.10.E = nL. = 7\"102lm/m 5.1t.M ----Eo (1 = 252)Ix. 5.12.Eo= nLR2/h 5.13.e' e --2(en)n. of the normalsto 5.t4.Supposen, n 2, n are the unit vectorsare the unit vectors and

--5-10

X--\177--R-

(o)\177--o\177)\177\"

3P/16uc\177OMc 0.6\177m. = V\177)(I)\177/A (a) 3 and 9 roW; (b) (I) = where A = mW/lm, V1 and V s are the values of relativespectral responseof an eye for the given wavelengths.

4.224.R =

\177/t

radius

\177

caser c = 0, P =

and in our

\177

t--1/t--(/\177//)\177

\177er\177

vector of its centre of inertia.Sincethe radiationpower P

\177

eeo/\177toESm\177R2t

t/\177eocE\177

4.222.(P)I(S) = (e2[ra) (o 4.223.(P)/(S)=-6a

\177.

sin\177(O/2)

= 0.44 \302\260.

5.27.(a) [----l\177/(l-- = 10 cm; (b) = 2.5cm. cd. 5.28.I' = pie/s([ --s) = 2.0.103 \1772)

l\177l\177s/(\177s--

\1771)

\177

is a point sourceof light and S' its image to Fermat'sprinciplethe opticalpaths of all A ccording 38). (Fig. rays originatingat S and convergingat S' are equal.Let us draw

5.29.SupposeS

335)))

circleswith

quently, the

centresat S and S' and radii SO and S'M.Conseopticalpaths (DM) and (OB) must be equal:

n.DM= n'.OB.

However, in the

,\177,

5.47.F D/do = 20. 5.48.F = 60. 5.49.(a) F = 2alo/do = 15,where lo is the distanceof the best

the

h\177/(--2s)

case of

and OC

paraxialrays DM h'\177/2R.

,\177

(*)

AO -f- OC, where

Besides,OB -=- OC --BC \177

\177,

into (.) and h'\177/2s h'\177/2RSubstituting these expressions = (n'--n)/R. n/s n'/s' we obtain h' that h, taking into account \177_,

,\177

\177<

vision (25 cm); (b) F

o. 5.50.The principalp2alo/d lanes coincidewith \177,<

the centreof the lens. air and water: = noC)-=--4-15era. Here q) = (2n--n where n ar\177dno arethe refractiveindicesof glassand water.TheI)/R, nodal points coincide and arelocatedin water at the distancex = ]' / = 3.7cm from the lens. 5.5t.See Fig. 39. 5.54. -opticalpower of the system is q) = --dq)lq)\177 (a)= The + 4 D, the focal length is 25cm.Both principalplanes The focal lengths in

\303\267

q)\177

H H

-o

\177

ca)

Fig. 38.

5.30.x = (1_Wi (In--t)t) r2 5.31.6.3era. 5.32. = d (n--l)/nR=--0.20;(b) = 42 lx.(a) 5.33.(a)q) -= = (n no)/(n t) = 2.0D,]' ---= \177

\177

arelocatedin -----

nolo=

q) 6.7D, ]= ----85era; o(2n-no-t)/(n-t) = t/O t5(b)era.f' = noC 20era.Heren and n o are the refractive indicesof glass and water. ram. 5.35.Ax Alp/(l -- = 0.5 = 20 era; 5.36.(a) f = [l (Al)2]/41 2 = 20 cm. + (b) f = l = = 3.0ram. 5.37.h = t5 lx. 5.38.E = (t --a) 5.39.(a) Isindependento=f D; (b) isproportionalto D 35 era, where no isrefractiveindex 5.50.f = noR/2(nl of water. 5.41./ = R/2(2n-- 1) = t0 era. 5.42.(a) To the right of the last lens at the distance3.3cm from it; (b) l = t7 cm. 5.43.(a) 50 and 5 cm;(b) by a distanceof 0.5cm. 5.44.F ----Did. = o.6'. 5.45. = 5.46. (F+I) ?to 1=3.t,where no is the refractive \177]

l/\177q)

,\177

\177

])\177

\177

V-\177-/(I

\177

V-\177-)

V-\177t'h\"

\177

\177LD\177/4f

\177.

\177

n\177)

F'--\177

index of water. 336

\177--_ ?t\177

\177

Fig. 39.

E=\177n\177D\177L/4d\177=

\302\242o

q)\177

(b)

n'\177-

n_\177i

\177

lens:the

front of the

front one at a distance converging t0 cm from the converging lens,and the rear one at a distanceof 10cm from the diverging lens (x = dOsC and (b) d= 5 cm; about 4/3. 5.55.The opticalpower of the given lens is q) = q)x + ---(d/n) = 5.0cm, x' = --dq)l/nq)= 2.5era, q)lq)O.,X

i.e. 5.56.

x'=-

q)\177

dq)\177/nq)

both principalplanes arelocatedoutside the lens from the side of its convex surface.

: :

/principal plane

ld\177

/\177+/2--d

. The lensshould be positionedin the i.e. at a distance of

of the system, d) from the first

front

x----

lens. 5.57.q) 2q)' --2q)'\177l/no 3.0D, where q)' ----(2n--no--l)/R , n and n o arethe refractive indicesof glassand water. 5.58.(a) d = nAR/(n -- 1) = 4.5cm; (b) d = 3.0cm. 5.59. q) = -]\177d/([1%

]\177

\177

d (n l)2/nR > 0, the (a) planesarelocated on the side of the convexsurface at aprincipal distanceof d from each other, with the front principalplanebeingremoved from the convex surface of the lensby a distanceof R/(n 1);(b) q) = (I/R\177--I/R1) X x (n \177 l)/n \177 0; both principalplanes pass through the common curvature centreof the surfaces of the lens. 5.60.d = -:--9.0cm, F (R\177-+- R\177)/(n-R1/R2 -----5.0. \177

5.6t.q) = 2(n2 -5.63. = 3.10 5.65.p1.9a. m; t/\177n

\177

= 37 D. \177)

l)/n\177R

Wnl=

1.6-\1770

-\177

m-1

22--9451 337)))

5.66.Let us representthe kth oscillationin where

a\177

the

Hence,we seethat

complexform

is the complexamplitude.Then the complex

ae\177(\177-i)\177

+...

= a [i+ etv -+-f-ei(N- = = a (et\177N- t)/(O\177--1). Multiplying A* by the complexconjugatevalue and extracting the square root, we obtain the real amplitude 1--cosNip = a sin (Nqv/2) A a ]\177\177, sin A*

aei(

\177-

\177

t)\177

etZ\177

: : -----

\177------\177s\"

\177

5.67.(a) cos0 (k-=

q\177/2\177)k/d,

U4, k = O, i, 2, -- (d/L) 5.68. sin (cot -{5.69. = 2hxAh/l (*1 -- l) -- 0.6

(b) q\177:

\177/2,

Aq0

5.71.(a)

= E(n, -

(b) E\"

amplitudeof the resulting oscillationis

k d/k 2rt [k

\303\267

c\302\242)],

=0, l, \302\261

where k ----0,

; .

+|,

l.l

\177.

\177tm.

= ram, 9 maxima; (b) the = (b/r) = 13r)/2ar shift is mm; (c) the fringe pattern is stillsharp when ?)x Ax/2, hence 15max = (l r/b)L/4a = 53 5.72.L = 2aAx ----0.64 5.73.(a) Ax ----L]/a --- 0.15 ram, i3 maxima;(b) the fringes Ax=

\177ix

L(b\303\267

\177il

\177

\177.\177

\177tm.

are stillsufficiently sharp when is the shift Ax/2, where = of the fringes from the extremeelementsof the slit,hence, = Lf\177/2ab = 37 5.74.L = 2a O(n -- l) Ax/(a b) ----0.6 5.75.Ax L/20 (n --n') = 0.20ram. 5.76.The fringes are displacedtoward the coveredslitover the distance Ax---- hl(n--l)/d = 2.0ram. 5.77. n' = n + NL/l t.000377. 5.78.(a) Let E, E',and E\" be the electric field vectors in the incident, reflected and transmitted waves. Selectthe x-,y-axesat the interface so that in direction with E and H in the incident wave. they coincide \177Sx

\177ix

\177

k=0, 1, 2

\177tm.

\303\267

The continuity yields

of the tangential componentsacrossthe interface

]/'n\177\177sin\1770

d=\177.

(r\177--

\177

at the inter-

.... \177tm,

\177

\177tln.

r\177)

\177

cos 5.84.Ax= 0x ]/'n 5.85.(a) 0 = \177/zk/nAx = 3'; (b) AL/L Ax/l = 0.014. 5.86.Ar 5.87.r' = V rZ --2RAh = i.5ram. 5.88.r= V ro + (k--t/2) =3.8ram, where k=6. 5.89. = (d\177--d\177)/B (k\177-- = 0.50\177tm, where are the numbers of the dark rings. kz 5.90.(I) = 2(n--l)(2k--l)k/d =2.4D, where k is number of \177,

O\177

2\177

\177-

sin\177

\177

\177.R

\177

k\177)

\177/\177

k\177

\177

the bright ring. (a) r

and

the

5.9t.

ram, where no is the refractive index of water. ----r/I/\177o=3.0

5.92.r=Wt/z(l+,2k)\177R/nz=l.3 mm,

\177

\177-

.... n\177

\177tm.

\177)rnax

\177tm.

E the phaseabruptly changesby the phasejump does not occur. and E\"

n\177

5.80.dmln\177-0.65 5.8t.d ='/,X(l + 2k)/V-\177, where k = 0, i, 2 15 5.82. sin 20./i0 d 5.83. 4nl (l--k)

-+-2

is.cophasal.

5.79.d=U\177L(l+2k)/V'nz--sinzOi=O.14(t+2k) where

t)q\177]

... ....

((p/2) k

> <

is at 1/2 face.If n 2 that

E are collinear,that + n2), nO/(n\177

E'\" and

5.93.k\177, =

5.94.The transition from

if the following

\177/zLd(L\177

\177,\177)

----

140.

where k=5.

one sharp pattern to another occurs

conditionis met:

where k is a certaininteger.The correspondingdisplacement Ah of the mirroris determinedfrom the equation2Ah = kL\177. From these

two equationswe get

E+E\" =E\" The minus sign beforeH\" appearsbecauseI-I\" 11H. Rewrite the secondequation taking into account that I-leonE.Solving the obtained and the first equation find: 2En\177/(n\177

+ n2).

\177= 0.3ram.

5.95.(a) The conditionfor maxima:2d cos0 = kL; hence,the orderof interferencek diminishesas the angle 0, i.e.the radius of the rings, increases(see Fig.5.18). (b) Differenting both sidesof

the foregoingequation and taking into account that on transition from one maximum to anotherthe value of k changesby unity, we obtain 60 = sin 0; this shows that the angular width of the with an increase of the angle0, with a decrease fringes decreases in the orderof interference. \177/\177L/d

E'\" =

\177 (\177.\177--\177.\177)

22*

i.e.

339)))

----k/k = kS/2d _ 5 pro. 5.96.(a) = 2d/k = t.0.105, (b) 5.97.Io=\177 I(r)rdr. o -- 2.0m. 5.98.b = --rl) (a r s) b)/2ab = 0.60 5.99. 5.t00.(a) I 4/0,I 21o;(b) I lo. 5.101. (a) I 0; (b) I lo/2. 5.102. (a) 11 9/lO lo, Is 13 i/\177Io, (i --q\177/2n)sI0; (b) lo, lo lo, lo, =1o,I (i 810. Here is the angle covered by the screen. 5.toa.(a)h = (k + 3/8)/(n-- l) = 1.2(k 3/8) (b) h ----!.2 Here k (k (c) h i.2kor i.2(k 3/4) = 0, i, 2 7/8) 5.i04.h\177.(k\1773/4)/(n--i), where k----0,i, 2, (b) I,\177 8Io. ----A\177.

k,\177ax

arS/(k\177.a

_\177

\177

-\177

\177.

\177tm.

\177-

(r\177

\177

\"\177

\177

I\177

\177

....

\177

i/\1771o,

I\177

\177

s\177/\177

q\177/2\177)

9/\177

I\177

\177

\177-

-----

hm\177n

\"\177

\177.

5/8)/(n

\303\267

_\177

\177tm.

....

2.5

where k 2. where k\177--l, 3, 5 \177tm,

5.106. f) =0.90 mm, s --- i.0m. 5.i07.b' 5.i08.(a) y' = yb/a = 9 ram; (b) 0.i0mm. 0.6m. This value 5.109. /\177ab/(a b) ]/\177

W'k\177fb/(b\177

...,

\177tm;

-----

\177

5.i05.

\1779/\177

\302\242p

\177tm,

\177-

.\177

-----

b/\177l

h,\177,\177

\177

\"\177

ab\177./D

5.i22.(a) A0-----arcsin (n sin 0) --0 ----

dition

7.3

7.9\177; (b) from the conwe obtain A0---- 0+l--0_x----

(sin0\177nsin0)

=\302\261X

5.123. s -- d/2k = 0.6 5.i25.55 . (a 5.i26.d ----2.8 5.127. k (d sin A0)/V 5--4 cosA0 = 0.54 5.128. (b)-64 (a) l --8 cm. 5.129. 2R/(n.--l) V From the condition d [n sin 0 --sin (0 5.130. 0a)] obtain 0 o ------18.5 k\177, 0+o----\177-78.5 Fig. 40. 5.i3i.ha (k --i/2)/(n --i), where k i, 2 ; a sin ---i.5km/s. 5.i32.v \177_

\302\260.

\177

: \177

\177tm.

a\177)

\302\260

\302\260.

45\302\260;

x-\177

(d/)\177)\177--

-----

\177/2. -----

... k\177we

\302\260.

\177

0\177

\177

\177-

\177-6,

0+\177----0\302\260;

\302\260,

\177tm.

Se\177

\177

k\177//Ax

5.|33.Each star producesits own

diffraction pattern in the

focal plane,with their zeroth maximabeing separated objective's

to the corresponds as focal from which there are other points point, apart principal \177

...

\303\267

well.

5.110. (a)

=0.60(2k i) l , 5.iil.(a)where and are1.7, ----0.7 the h

Herek----0, 2,

\303\267

\177

I,\177/I,,\177,\177

\177tm,

v\177

meteralongCornu'sspiral.

5.112.

\177tm;

(b) h

(b)

\177.

2.6.

\177,

the correspondingvalues of the

l) 0.30(2k\177 (Ax)S/b(v\177

\177m.

correspondingvalues of the para-

s= 0.55 5.i13. \177.----(Ah)S/2b(v s --Vl) I\177,,\177./I\177.

\177--

\177tm,

where

v\177

and v s are

parameter=along Cornu's spiral.

-- where k 0, i, 2, . . . 5.1i4.h (k i.9.3/4)/(n l), 5.115. 5.ii6.I 2.81 o. :4:7. 5.ii7.1l:Is:I\177 5.ii8.I Io. s 5.1i9.Io (sin s, where a:(ub/\177.)sin 0; b sin 0 k----i,2, 3, leadsto the transcendental 5.i20.The conditionfor a maximum = \"\177

\177.

\177

\177-

\177

I\177/I\177

...

\177

-----

\177z)/\177z

k\177.,

where a (ub/\177)sin 0.The solution of this equation tan a-\177 provides the following equation(by meansof plotting or selection) root values: 2.46\177, \177 ----3.47\177. Henceb sin 0 i.43\177, ----1.43k, 3.47L b sin 2.46\177., b sin 0a \177z,

\177

\177zl

\177z\177

5.i2i.b(sin0--sin0o) and 27 0\177

-----

angles0 are equal to 33

\302\260

34O

\177

_\177

kX; \302\260

for k

\177

\177-I

respectively.

and

k-\177--Ithe

Fig. 40.

Fig.

the angle 9 beangle (Fig.4i).As the distanced decreases in each diffraction pattern increases, neighbour\177ng maxima and when 0 becomesequal to the first deteriorationof visibility occurs:the maximaof one system of fringes coincidewith the and minima of the other system. Thus, from the condition9 = the formula sin 0 ---we obtain = \177/2d 0.06\". 5.134. (a) D k/d l --(k\177/d)a-\177 6.5ang. min/nm, where k 2; (b) D k/d V l (k\177/d sin 0o) = i3 ang. min/nm, where k ----4. by an

\177

tween the

-\177

2\177,

2\177

\177/d

\177

5.135. = (tan 5.136. A0 = W i 5.139. 0 --- 46 5.i40.(a) In the fourth order;(b) 5.i41.(a) d=0.05mm;(b) 6cm. 5.152. (b) not in the first (a) 6 and i2 dO/d\177

0)/\177,.

2\177/Nd

\177

(k\177./d)

-\177-

\302\260.

5X,\177,

l\177-

\177tm:

cond order.)))

\177

\177.S/l

-\177

pm.

order,yes in the se-

5.i43.According to Rayleigh's criterionthe maximumof the line of wavelength 3. must coincidewith the first minimum of the line of wavelength 3.+ 153..Let us write both conditionsfor the least deviation angle in terms of the optical path differences for the extremerays (seeFig.5.28):

--(DC q- CE) =0,

Hence,b6n

,\177

3..What

+ 6n)--(DC+

b (n

CE) ----3. +

63..

follows is obvious.

5.i44.(a) and 0.35.t04(b) 1.0cm. \1772bB/3.a; 1.2.104 5.i45.About 20cm. 5.t46.R = 7.10 cm. a, 5.i47.About 50 m.AYmtn\1774

5.t63.P ---- --i)/(i-- cos -- 0.8. 5.i64.(a)Let us representthe naturallightas a sum of two mutual(\177

componentswith intensitiesIo.Supposethat each perpendicular polarizertransmits in its plane the fraction of the lightwith oscillationplane parallelto the polarizer'splane,and the fraction with oscillationplane perpendicular to the polarizer'splane. The intensity of light transmittedthrough the system of two polarizersis then equal to ly

\302\242z\177

A\177b'

A\177b

(A\177b

= = D/do = 13. telescopeis 5.i49. u ----1.4 d,\177 ----0.613./sin 5.t50.Supposedmi n is the minimum separationresolvedby the

of the

F,\177in

A\177b'/A\177b

\177m.

microscope's objective, is the angle subtended by the eye at the objectover the distanceof the best visibility l 0 (25 cm), and is the minimum angularseparationresolved by the eye (A\177' = = 1.223./do). Then the sought magnification of the microscope is = = 2 (lo/do)sin ----30. Fm\177 5.i51. 26,60,84, 107and 134 5.152. a ----0.28nm, b = 0.41nm. 5.i53.Supposea, 6, and 7 are the anglesbetween the direction to the diffraction maximum and the directions of the array alongthe Then the values of theseanglescan periodsa, b, and c respectively. be found from the following conditions: a cos b cos = and c cos ---Recallingthat costa+ cos\177 + q- cos\177 = 1,we obtain A\177

A\177'

A\177'/A\177

c\302\242

\302\260.

(1- a):

k\1773.,

\177

k\1773..

\177

2kt/a

+ + (ka/c) 2 5.154. --244pm, where 3.=\177- V\177sina \177

-\177

\177\"

:\177

(k\177/b)

(k\177/a)\177

of a

NaC1 molecule.

d:2 sin 5.|55.

\177

(\177z/2)

k\177

5.i56. 3.5 5.i57.lo/4. 5.158. = 0.6mJ. 5.i59.E(a)----Io; (b) 21o. 5.i60. = %. (cosT)\177(N-I) 0A2. 5.i61.Io/I= \177 60. = P/(l P) = 0.3. 5.i62. k\177

2\302\242z

c\302\242

\177(I)o/\302\242o

-\177

\177l

,\177

\177

I\177o\177

342

/I\177,\177

2, m

is the mass

when their planes are

\302\242z

is the glancing angle

(:\177I09

parallel,and to

their planes are perpendicular; accordingto the condition, On the otherhand, the degreeof polarization producedseparately by each polarizeris Po = (\177 --\1772)/(\177 + \177)Eliminating and \177 from theseequations,we Po = V (\177-i)/(\177 + i) = 0.905. = 0.995. (b) P = F -The relative intensity variations of both beamsin the 5.165. casesA and B are

when

\177

I\177/I

\177.

\177

ge\177

: :: :

\177/\177

(AI/I)\177 Hen\177

\177

cot (\177/2).6%

4 tan (T/2)-\177%

(AI/I)\177

= (5I/I)\177/(AI/I)v cot (T/2), = ii.5% \177

\177

5.166. 90% 5.167. (a) p

-- i)2/(n2

l) =

0.074; + n2) H ere n is the refractive \1770.080. (b) P:p/(t--p)= (l+n2)\177+4n index of glass. I = Io(i --p)/n = 0.721Io,where n is the refractive 5.168. index of water. 5.169. + l)]sin2T = 0.038,where n is the p = [(n -\177/z

\177

\177-

\177

refractive index of

\177

l)/(\177

water.

: : :

p 2O(l--O) \1770.i7. 5.170. 0.087, i-P\177:P3=i, l-p 2p (i--p) each of reflection 5.17i.(a) In this case the coefficient --

pro, where ]flk\177-+-k\177--2k\177kecos(a/2)=0.28

are the orders of reflection. = cm, where r = l tan found from the condition2d sin ----k3.. and

k-\177

III ----a\177Io

3./6\177

5.i48.Suppose and are the minimum angularseparations resolved by the telescope's objectiveand the eye respectively = i.223./D, Then the sought magnification A\177b'= 1.223./do).

2\177)

\177

P\177=

surface of the plateis equal to p

Io(l --

P: ient

I\177

(b)

\1776Io\177/(\177

p)\177

\177

(a\177

\177

0.725Io;

o\177

inciden\177

and therefore

i)\177,

0.i6,where p'

of reflection for the component

at right anglesto the oscillates

\177rom

l)\177/(a\177

a\177)\177

\177--(i--P')\177--(\177+\177)*--\177* \177 l+(i--p')\177--(i\177a\177)*\17716n*

p\177=

is the coeffic-

light whose electricvector

plane.

\177), where a 2a/(l (l --a\177)/(l a is the refractive indexof glass;(b) 0.16, 0.31,0.67,and 0.92re-

5.i72.(a)

spectively.)))

\177

a\177),

----(n -- l)2/(n-t- l) = 0.040;(b) = I --- (l5.i73. --p)2N(a) p 0.34, where N is the number of lenses. 2

il\177)/l\177)

5.i75.(a) 0.83;(b) 0.044.

5.i87.(a) The light with right-handcircularpolarization(from on passingthrough the observer's planepolarized viewpoint) becomes a quarter-wave plate.In this casethe directionof oscillations of wave forms an angle of the electricvector of the electromagnetic -\17745 with the axisOO' of the crystal (Fig.43a); in the caseof lefthand polarization this anglewill be equal to --45 (Fig.43b). (b) If for any positionof the plate the rotation of the Polaroid (locatedbehind the plate) doesnot bring about any variation in the intensity of the transmitted light, the initial light is natural; \302\260

\302\260

Fig. 42.

5.i76.SeeFig.42, where o and e are the

dinary rays.

5.177.6

\177

ordinary and extraor-

For the 5.178.

\302\260.

right-handedsystem of coordinates: (l) circularanticlockwisepolarization,when observed toward the incomingwave; clockwise (2) elliptical polarization,when observed toward the incomingwave; the majoraxis of the ellipsecoincideswith the straight line g----x; (3) plane polarization,alongthe straight line g --x. 5.179. (a) 0.490ram; (b) 0.475mm. 5.180. 0.55and 0.5ipm respectively 4dAn/(2k+ l);0.58, at k = \177

15,16and 5.18i.Four. 5.i82.0.69and--0.43 = 0.25 5.t83.d = where k ----4. 5.i84.An =(k 1/2)----0.009. mm, 5.185. Let us denote the intensity of transmitted light by \177/An

\177/@Ax

I\302\261

Ii\177

\177

\177/\177I

\177

\177/\177I

h \177k\177,

.... for

HereA is the opticalpath

dinary rays,

k----0,l,

any

\302\242p

A=(k-\177t/2)

\177,

5.188. Ax = n;) ---(b) d(n;-no) O, = 0.71.10where a is the rotationalcon5.189. An = stant. 5.190. a n/Ax tan O = 21ang. deg./mm,I(x) cos (nx/Ax), where x is the distancefrom the maximum. = 3.0mm. 5.19t. = arcsin 5.i92.8.7 ram. 5.i93.[a]= 72 ang. deg./(dm.g/cma). = t0.6 kV/cm; (b) 2.2.10inter5.194. (a) E,\177 ----t/ I/ second. ruptions per 5.i95.An = where c is the velocity of light in vacuum. -= 0.015ang. min/A. V = 5.196. ((Pl -=

--no) @Sx< 0. --2(n, (a)

\177/2\177(no--

a\177,/n

-\177,

\177,

\177

V\177-\177

(1/c\302\242)

\177

4B'--\177

2cHV/\302\242o,

5.t97.If one lookstoward the

The conditionsfor the maximumand the minimum: Polaroids Ima x Imf n \302\261

cularly polarizedlight.

dm\177n

in the caseof the crossedPolaroids, and by in the caseof the Then parallelPolaroids. = o sin 2(p.sin (5/2), o[l--sin 2(p.sin (5/2)1. Ill = I\302\261

if the intensity of the transmitted light varies and drops to zero, the initial light is circularly polarized;if it varies but doesnot of natural and cirdrop to zero,then the initial light is composed

-----

\177xm.

\177

Fig. 43.

\177/z

\177p\177/4

difference for the ordinary and extraor-

\302\242O\177)/lH

transmittedbeam and countsthe VNH)l, where N is positive directionclockwise,then (p----(athe number of times the beam passesthrough the substance (in is N = Fig.5.35the number 5.i98.H,n\177n = rt/4Vl = 4.05). kA/m, where V is the Verdet constant.The directionalong which the.light is transmitted changes

opposite. t = 5.199.

to the

= 12hours.Although the effect is very small, was observedboth for visiblelight and for SHFradiation. rnc\302\242oo/\177I

345)))

5.200. a=eEo/raco2=5.tO cm, where Eo=V-2-\177oc , = ace= 1.7(a)cm/s; -li. (b) Fra/Fe -= 2.9.10 5.20t.(a) e = t -2, v = V\" t + 5.202. n = 2.4.107cm -3. no =

I = Io(1--p)2 5.217. 5.218. = V (ln I = \177(1\177p)2 5.219. 5.220. Will decreaseexp 5.221. d = 0.3ram.

-l\177

noe\177/eornco

(noe\177/4n\177eomc

\302\242

\177)

5.203. n--t=-noe\177X\177/8n2eomc\177-=-5.4.10-7, where n o is the concentrationof electronsin carbon. 5.204. (a) x = acos(cot + @, where a and are defined by (4n%\177meo/e\177)(t

Here = \177/2m,

eEo/m

= 8 ram. 5.222. d = (ln 5.223. N = (ln 2 = 5.6. 5.224.c = 2lz t0s m/s. ni) = 3.0. 5.225. First of all note that when v << c, the timerate is practic\177)/ln

2\177

for

(P)\177=\177{eE\302\260\177

5.205. Let us write the wave equationin the form A

= 2n/E.If n' = n + ix, then k = (2n/Eo)n'and real form

frames fixed to the sourceand to the ally identicalin the reference

receiver.Supposethat

= Ao cos --k'x), i.e.the light propagatesas a plane wave whose amplitudedepends on x. When x < 0, the amplitude diminishes(t\177 attenuation of the wave due to absorption).When n' = ix, then A = Ao ex'x cos This is a standing wave whose amplitude decreasesexponentially total internalreflection (if x < 0).In this casethe light experiences in the medium (without absorption). = 2.0.109 5.206. cm no = ex'\177

(\177t

\177t.

4n%omc\177/e\177X\177

a/\177

(b) u

= 2v; (c) u =

\177.

\177/\177

\177

v\177

\177o)1\177o

\177,

\177

p\177+

=Ioa (1

\177);

\177)\177/(l

\177

where a = exp(\177xd). \177)\177z(l

\177o

\177

5.214. x= \177 =0.35cm t = 0.034cm 5.215. x = \177 In (1 5.216. z = (1 + II. exp p)\177/(l

a\177p\177),

(\177/\177)

-\177.

d\177

\177d\177

-\177.

\177

p)\177

[\1771/2

(x\177

x\177)

cos0.

(v/c)

\177).

2T/mc cos0 = --26rim. = 25 days, where R is the radius of the 5.227.T = = 3.t0 kin, m = d= = 5.228. = 2.9.10kg, where is the gravitational constant. -5.229. = (t + = V/c; where o0(t + 2V/c). 5.230. v = 900km per hour. 5.231. for t' and x' (from the Lorentz Substitutingthe expressions --kx = \177't' --k'x', we transformation) into the equation A\177,

\177

4=R\177

c6\177

\177

(A\177/E)\177c\177/a

(AE/E)\177ca\177/2ay

\177

\177)/(1

\177o

\177

\177),

\177

\177/\177XAv

\177

\177t

obtain

\177=\177'

where

\177

V/c.

5.232.From = v/c = 0.26.

5.233. v=c

\177

\177/2Ioe-\177t

c/\177

5.226. = --E

\177,

v = c/n = 5.210. s m/s. 1.70.10 5.211. It is sufficient to discussthree harmoniccomponentsof the train of waves (most easilywith the help of a plot). I= 5.212. sin where = VIH. 5.213. I -- (b) I = +(l -- p\177+...) = Io(l --(a) =Io(l--p)\177(l + z\177+z'\177'+...)

where A is a constant. s m/s, u = [1+ (E/n) (dn/dE)lc/n = 1.83.10

A/\177

sourceemitsshort pulseswith the inter-

the

vals To. Then in the reference frame fixed to the receiverthe distance betvceen two successivepulses is equal to E cTo-vrTo, when measuredalongthe observationline.Here is the projectionof the sourcevelocity on the observationline (Vr = V COS The frequency = Vo/(t of receivedpulsesv = = i/To.Hence v\177/c), where v o (\177

5.208.(a) u = 5.209.e = I +

(n\177

o\177---o\177

is the mass of an electron.(b) 2

\177=k/m, m

\177

\302\242p=

\177\302\2603

\177o

\177

tan

mS(eEo/m\177

or in the

= 0.6.10times. \177

(\177d)

2)/\177

1/(\302\260\177--\302\260\177)\177+4\177

where k

e_\303\227\177_\177.

\302\242p

a= =

\177l)l\177zd.

q\177

\177)

the formulas

2;\177

--\303\227,)

(\303\2273

A\177,

\177o

k=k' (i+\177)/Vi-\177, the formula (X/X,)3

_ \303\267

\177

5.234.co = cooV 3/7. \177

atom.

+\177)/V

\177-\177,

into accountthat co' = ck'. co'=co V(I--\177)/(I+\177) we get

Hereit is taken

(\177./\177.,)\177

5.235. AE

(\177

ET/rnoc \177

\177

\177-\177

7.t.t0 km/s. \177

= 0.70nm, where

is the mass of the

5.236. co=coo/\177t--[\1772=5.0.10 s-1 ; (b) co=co0Vt-[\177= = t.8.10\177o(a) s- . Here,6 = v/c. 5.237.The chargeof an electronand the positivechargeinduced in the metalform a dipole. In the reference frame fixed to the electron the electricdipolemoment varies with a period T'= d'/v, where d'= d Vt The corresponding\"natural\" frequency (v/c)\177 \177o

\177

34\177 347)))

is v' ----old'.Due to the Dopplereffect the observedfrequency is

i -- (v/e) cos0 ----t --(vie)cos

0\"

The correspondingwavelength is = c/v = d (c/v cos0).When 0= and v c the wavelength is 0.6Fro. 5.238. (a) Let v= be the projectionof the velocity vector of the radiating atom on the observationdirection.The numberof atoms with projections falling within the interval v=, \177

t = (\177a_ l)cpd/18(\177T\177 = 3 hours, where c is the specific heat capacity of copper,p is itsdensity. 5 252.T2=T\177d-\177=0.4 kK. 5.253. (a) Cv (OU/OT)v=i6\177TaV/c\1773 nJ/K, where U

5.251.

.-----

5.254. (a)

\302\260

\1775

\177

exp(--m\177/2kT).dv=. The frequency of light emittedby the atoms moving with velocity is = (l v\177/c). From the expressionthe frequency distribution of atomscan be found: n (\177)d\177 = n (vx)dye. And finally it shouldbe taken into accountthat the spectralradiationintensity n (v=) dv=

\177

v\177

I\177

\177

\177o

= 2V(21n2) kT/\177 \177n(\177). (b) 5.239. u=\177.If V(<c,then = km/s. 5.240. v = 5.242.0' = 8% a\177/\177o

medium.

5.257.AP =

(b)

\177z;

u\177

e2nav/hT

--1

4n\177c\177hTaAX/b

a) \177a

16\177c\177-\177

\177a

= (h/n\177c

e2ahc/hTk--I

2n\177c/\177b\177

0.31W/cm

is the constantin Wien's displacement law. = 5.258. (l (a) (c) (T\177/T\177) \177m; (b) 0.37;

i.l

= 4.9.

\177

P\177/P\177

5.259. = 1 5.260. (a) (])=

\177d\177

d\177

\177

eh\177/\177T__

\177

nz dX

8n\177

-a

\177,

where

--g\177)/(i--g

d\177

\177

6.10

e2nhc/\177T\177__

i\"

\177

5.262.(p) =4(i+p) E/nd\177cz\177 50 atm. 5.263. p = (E/c) + + 2pcos20 = 35 nN.s. 5.264.p = (I/c)(l+ p) cos =0.6 nN/cm F = nR\177I/c = 0.18 5.265. 5.266. F = P/2c(i + \177i

\1770

\177.

\177N.

\177).

5.267.(a) Ap= =V/c. It is evident that 2h\177

\1771--\177 l--\177

\177

' (b) Ap=

2h\177

\342\200\242

i--\177\"

Here

in the referenceframe fixed to the mirrorthe latter obtains the smallermomentum. 5.268.sin (0/2) E/mcV gl, 0 = 0.5% 5.269.AO/Oo = --(i --e -vM/Rc\177) 0, i.e. the frequency of the photon decreases. 5.270.V = 2nhc(i -- i/\177)/eA\177 = i6 kV. = 3t kV. 5.271.V = nhc/edsina 5.272. 2ahlmc(\177-- 1)----2.8pm, where ,/----i/\177 t --(v/c)Z. 5.273.332 nm, 6.6.\1770 m/s. 5.274.A = 2nch (\177-- t) -- 1.9eV. 5.275.\177= 4.4V. 5.276.T\177 =h(\177o+\177)--A\177 =0.38eM 5.277.w 2nch]/e\177 = 0.020. 5.278.v\177 = 6.4.i0m/s. 5.279.0.5V; the polarity of the contactpotentialdifference to that of externalvoltage. is opposite 5.280. h/\177, the Comptonwavelength for the given particle. \177

\177

k,\177,,

\177

5.244.Tr,\177n=(n/Vn2--l--l)mc\177.; 0.14MeV

respectively.For muons.

ncos0

5.245.T

cos 0--l \177

n\177

bTl/(b\303\267

and

0.26GeV

(q\177--\177/\177) \177

mc\177=0.23MeV. --i) ----1.75kK.

5.247. TIAX) 5.248. 3.4 5.249.5.i09kg/s, about l0

\177

-----

\177m.

\177

years.

5.250.T=\177/3cRp/cM =2.i0 K, where R is gas constant, M is the molarmass of hydrogen. \177

348

t6\177h

\177)

5.261. dp/dt =

\177-

\177,\177

5.256. uv b

= (kT/n\177c

u\177

P\177/Sn\177c\177r\177=

induced by a charged particle moving with velocity V excitesthe atomsof the medium turningthem into sources of light waves. Let us considertwo arbitrary points A and B along the path of the particle.The light waves emitted from these points when the particlepassesthem reach the point P (Fig.44)simultaneously and amplify each other provided the time taken by the light wave to propagatefrom the point A to the point C is equal Fig. 44. to that taken by the particleto fly over the distance AB. Hence, we obtain cos0 = v/V, where v = c/n is the phase velocity of light. It is evident that the radiation is possibleonly .if V v, i.e.when the velocity of the particleexceedsthe phase velocity of light in

T\177=

\177m.

a\177ca

\177.

5.243. The field

the

\177w

5.255. (a)

n\177

\177

\177/\177c50

\177

= 1.44

\177

1.0nJ/K. = 3T/a= 7.85.10s-1 ; (b) kw = 2nca/5T

= (b) S = 16gTaV/3c

=4\177T\177V/c;

the universal

349)))

5.281. Let us write tim

energy and momentum conservationlaws

in the reference frame fixed to the electronfor the moment preceding = ray, where the collision with the photon: + raocs racs, ra----m or v = c. (v/c) s. From this it follows that The results have no physical meaning. 5.282. (a) Light is scatteredby the free electrons;(b) the increase of the number of electronsthat turn free (the free electrons have the bindingenergy much lower than the energy transferred to them by the photons);(c) the presence of a non-displaced componentis due to scatteringby the strongly bound electronsand the nuclei. 5.283. = 4nkc [sin (02/2) sin (0\177/2)]/(\177 1) = pm. 5.284.T ----? = 0.20MeV. = 5.285.(a) rad/s; 2nc/(\177 + 2nMmc)= 2.2.1020 \177

h\302\242o

oVi-

h\302\242o/c

v=0

\177

h\302\242o\177/(1

6.i5.AN/N =

(0.7\177

o\177

6.i6.A(r = n (ZeZ/T)2 cots (0o/2) = 0.73kb. 6.i7.(a)= 0.9MeV; (b) deride2 = A(r/4n sin (0/2)= 6.18.t (3mcZ/2e%) In = 15ns. 6.19.t 13ps. 6.2i.r, ]/nli/rao), E, = nlio), where n N\177

\177

m2cZrZ/l\177e

\177

0.6\177

kb/sp.

\177)

\302\242

\177,

-----

6.22.

1.2

\177)

rl,

\302\242o'

2ncMX (b) T-- i+\177,rnc/2ali--60keV.

5.286.

\"--

q- 0.3 where Z i and Z z are the atomic numbers copperand zinc, Mi and M2 are their molarmasses, is Avogadro'snumber. \177

\177e

\"\177r

--\177

sin (0/2)

1\177-2 (lie)linch)

He+

5.287.sin (0/2)= V mc (p-p')/2pp'. Hence0 = 120 = )ie) [i+ V 1+ 2mc2/Tsin (0/2)]T/2= 0.68MeV. 5.288.

106m/s

E b, eV

T, eV

(Pi' V,

(pl, V

k, nm

2.18 13.6 13.6 13.6 10.2 121.5

52.9 26.5

0.144MeV.

4.36

54.5

30.4

40.8

\302\260.

\177

6.23. = meaZS/Iian

\177-=

\177

2\1771)

-----

\177m.

\177

(h)

]/l cos

sin (0/2)

\177_n\177_2n

cos(O/2)

where n --V i + Uo/T.

\177tm,

\177)

\177Apt/A.\177g

\177-

\177

(Ze\177/T)

-\177.

\177

(2\177c/A\177)

R/A\177

\177R/A\177

\177,

\177

atom.

6.42.(e -- e')/e 3hR/8mcs = 0.55.i0-\177%, where m is the mass of the atom. s m/s, where m is the mass of 6.43.v----2V]iR/m = 3.1.10 \177

the

electron.

6.44.v = 3RA\177/8n cos0 = 0.7.10m/s. \177

(ZeS/mv\177)

,t50

\177

a/\177hR

\177

\177nZSe\302\242/T

\177

E\177

-\177.

Z.\177g

\177

\177c/RA\177

\177

6.8.(a) (0/2) 1/2. 6.9.3.3-10 6.10.d = (4]rST2/nlZeesina (0/2)-\177 1.5 where n is the concentrationof nuclei. 6.11.Zpt = V 78. s cot = = 2.0.10 6.12.(a) 1.6.10\177; nnd (b)N (00/2)Iox where n is the concentrationof nuclei. = 0.006, 6.i3.P = nnd where n is the concentration of nuclei. s tan s 6.14.AN/N = I -(0o/2)\1770.6. =

\177

6.35.Z= V(i76/15) Li++. =3, --l)/(2ZV --l) = 0.479m. 6.36. (Z = 6.37. 54.4eV =79 eV. 6.38.E =Eo\1774hR(He+). 6.39.v--V =2 (h\177--4hR)/m= 2.3.i0 m/s, where \177=2nc/L = 20.5eV. 6.40.T\177, 6.41.v 3hR/4mc = 3.25m/s, where m is the mass of the

2 V 2mT/[l+(2bT/ZeZ)Zl. = mpea/m\177bZT =:4 eV.

6.7.b=

\177

t5Z\177A\177

\177

\177m.

\177/5\177

\177

\303\267

T\177

: :

\177.\177

\177/zn

= 0.034pm. 6.4. p,\177 = (ZeS/T)cots (0/2)= 0.23pm; = [i (a) cosec(0/2)lZe2/T = 0.56pm.

6.5.p 6.6.

\177.

\177

m\177e\177/ch

\302\260.

6.2.b 0.73pro. 6.3.(a)

\177t\177lM,

\177t\177

i_\177ho)/mce

\177

\302\242o

\303\267

-\177-

s-'.

----2.07.10 = e/2mc, = 6.24. nehl2mc, = t25 kG. 6.25.B = 6.27.The Brackettseries, = 2.63 6.28.(a) 657, 487 and 434 nm; (b) a. i.5.10 6.29.For n >> i sin 0 naac/lR, whence 0 60% 6.30.He+. 6.31.N (n -- 1). 6.32.97.3,102.6 nm. and 121.6 6.33. 5. 6.34.R 176nc 2.07.10s

5.289. (2,nMmc)(]/1 2mc2/T,\177-i) = 3.7 pm. 5.290.tan (p ---, (p = 3t 5.29i p--2\177(l+\177)mc=3.4 cm. eB 5.292.AL (l(4h/mc) sin:(0/2)--- i.2pm. 6.1.r ----3eS/2E= 0.16nm, ----(2nc/e)V mra = 0.24

6.45.(a) nli

V\177/m;

(c)

E. =

(d) En 351)))

6.46.Eb =

where is the reducedmass of the system. If the motion of the nucleusis not taken into account, thesevalues (in the caseof a hydrogen atom) are greaterby m/M where m and M are the massesof an electronand a pro0.055%, \177xe4/2h

2, R

\177xe4/2]i

\177x

\177

ton. 6.47.ED -- = 3.7 meV, --)'D = 33 pm. 6.48.(a) 0.285pro, 2.53keV, 0.65 rim; (b) I06pro, 6.8eV, 0.243 6.49.123,2.86and 0.186 pm. 6.50.0.45keV. 6.51.For both particles 2zdi (i+ m,\177/md)/\177/2rant ----8.6pro. Er\177

)\177H

\177m.

)\177-----

6.52. 6.54.First, let us )\177----

6.53.), --2nh/\177'\177_128pro. 2)\177,)\177z/V\177,,

The conditiondT/d}\177=O provides \177pr-=nh/V'm--\177-=0.09 nm. \177= 2nh/1/2rnT (t A- T/2mcZ), T 4rnc\177A\177/\177--= 20.4keV(for an electron)and 37.5 MeV (for a proton).

6.55. 6.56.T.=(]/\177---l) mcz = 0.21MeV. 6.57.k = A-rnc\177a/nh= 3.3pro. = 2.0. 6.58.v = 4nhl/mbAx i0 m/s. <\177

\177

...

20=D/2l. 6.63.(a) n= =1.05; (b) V/V\177i/q(2+\177l) =50. 6.64. = where n = t, 2, 6.66.1.10 ', 1.10and 1.10 cm/s. 6.67.Av h/ml =-l.i0m/s; = 2.2.10m/s. 6.69.At i0-\177 s. 6.70. I eV. Here we assumed that p Ap and Ax = l. 6.7i.Av/v ,-,h/lV 2rot= t.i0-'. 6.72.F a. 6.73.Taking into account that p Ap MAx h/x, we get E = T U From the condition dE/dx = 0 + we find x o and then = hoe, where o) is the oscillath tan

Vi\303\267//\177/y n\177n2h\177/2rnl

E\177

\177,

-\177\302\260

\177

v\177

\177

,\177

\177

\177lral\177/h

\177,

Tm\177,\177

h\177/2ml

\177

-=--

\177

\303\267

h\177/2rnx

\177

,\177

6.77.P = t/3+ lf\177/2n = 0.61. A

6.78.\177=

Em\177

l/-k\177m

6.74.Taking into account that --

we get E 352

p\177/2m

e\177/r

\177

h\177/2rnr

p \177

\177-,

e\177/r.

the value

and Ar \177r, From the condition

\177-,h/Ar

\303\267

\177

\303\267

]

..... ....

cos(nnx/l),if n= 1,3, 5

Asin(nnx/l), if

n=2, 4,

HereA = 6.80.dN/dE=(l/nIi)Vm/2E; if E = t eV, then dN/dE = 0.8.10 levelsper eV. 6.8t.(a) In this casethe SchrSdingerequation takes the form V'\177-\177.

\177

\177 \177

o\177 +k\177\302\242=0 Og\177

k\177.=2mE/ii\177.

Let us take the origin of coordinates at one of the cornersof the well.On the sidesof the well the function (x, y) must turn into zero (accordingto the condition),and thereforeit is convenientto seekthis function insidethe well in the form (x, y) = a sin kix sink2y, sinceon the two sides(x _-- 0 y --_0) \177---- 0automatically.The possiblevalues of and are found from the condition of turninginto zero on the opposite sidesof the well: (ll,y) = 0, = (g//1)nD nl = 1,2, 3, . = -4- (n/l\177) (x, = O, n2 = 1,2, 3 \303\227

\177

ar\177d

k\177

....

k\177

..,

\177

\177b

\177:

k\177

\177

l\177)

n\177,

k\177

The substitutionof the wave function into the SchrSdinger equation leadsto the relation + = k whence k\177

\177,

k\177

+ and 49.4 units of (b) 9.87,24.7,39.5, = 19.5%. 6.82.P = 1/3--

\177,

or'sangular frequency. The rigorouscalculationsfurnish

\177

E.is

(n\177llx

n\177ll\177)

=\177h\177/2m.

h\177/rnl

kx\177/2.

\177

h\177/ml

\177,

--me'/2Ii=

\177,

\177

sin 0 = 0.23-4-0.04nm, where k = 3 and the

angle 0 is determinedby the formula

E,\177,\177

.\177

\177m.

\177

nhk/\177/2rot

53 pro,

A(i\177)

o\177\302\242

V-2\177 4.9 z (V-\177- i) d sin 0 = 0.t5keV. = 6.61.d nhk/\177-\177-\177 cos(0/2)= 0.21nm, where k = 4.

6.62.d

Ap\177

\177

n\177h\177/2rne

Ii\177/rne

\177

\177,\177/l/i

\177

6.75.The width of the image is A A' hl/p6, where A' is an additionalwidening associated with the uncertainty of the momentum (when the hydrogen atomspass through the slit),p is the momentum of the incidenthydrogen atoms.The function has the minimum when 6 0.0tmm. 6.76.Thesolutionof the SchrSdingerV-h\177--\177--should be sought e quation in the form T = (x)-](t). The substitution of this function into the initial equation with subsequentseparationof the variablesx and t results in two equations.Their solutions are (x).--e i\177, where k = V\177-\177/h, the energy of the particle,and (t) e where o) = E/h. Finally, T :aei(\177x-\177t), where a is a certain constant.

d)\177

-a/\1772,

6.59.Ax = 2\177hl/d 6.60.v o =

re]]

-\177\302\260t,

()\177)

(\177)

find

.-\177

find the distribution of molecules over de Brogliewavelengths.From the relationf (v) dv = --9 where 1(v) is Maxwell'sdistributionof velocities,we obtain = A\177-% a = 2n\177h\177/nkT. \177

dECr = 0 we = --13.6 eV.

6.83.(a) E =

\177.

\177/4\177

n\177h2/2rna

integersnot equal to zero:(b) AE = (n\177

\303\267

n\177

\303\267

n\177)

\177,

where

n\177,

n a are

(c) for the 6-th level = 14and E --+ the numberof statesis equal o six (it is equal to the number of permutationsof a triad 1,2, 3.) n\177h\177/rna\177;

\177

n\177

23--9451

\303\267

n\1772

7\177h\177/ma\177;

353)))

us integrate the SchrSdingerequation over a small there is a discontinuity interval of the coordinatex within which = 0: x at the point in U (x), for example

6.84.Let

(_5)=J (+5)_.\177..z 0\177

0\177

\177

Since the discontinuity

5-+0. What

is finite

\177'\177

-\177-(E--U)\177dx.

p\177

the integral tends to zero as

\177,

--\177

\303\227\177

= be-x\177+ ce From the condimust satisfy the standard and boundary conditions. of the wave finiteness the for = the and 0 requirement tion (0) =

asin(kx\303\267

a),

\303\227\177

\177(x)

P\177

Their common solutions \177(x)

of the particle and are the probabilities Then inside the outside and well. being

2rn

follows is obvious. 6.85.(a) Let us write the SchrSdingerequation for two regions 2 = 2mE/h O x < l, k\177 =0, k = 0, = 2m (Uo E)/h2. x l, \177

6.86.Supposethat

\177b\177e-

I 0

\177

2\177-3\177 a\177

sin\177

kx dx

where the ratio b/a can be found from the condition\177 (/) =

\177

(l).

= then = Now it remainsto take into accountthat = 2/(4 3n) = t4.9%. The penetrationof the particleinto the regionwhere its energy E U is a purely quantum phenomenon.It occursowing to the wave propertiesof the particleruling out the simultaneousprecise magnitudesof the coordinateand the momentum,and consequently the precisedivision of the total energy of the particleinto the potential and the kineticenergy. The latter could be done only within the limitsset by the uncertainty principle. 6.87.Utilizingthe substitution indicated,we get P\177

\303\267P

\177

P\177

\303\267

\177

\177\"

+k\177=0, where

\177

= 2mE/\177

\177.

shall seek the solution of this equation in the form = = a sin (kr + a). From the finiteness of the wave function Z at the point r = 0 it follows that a = 0.Thus, = (a/r)sin kr. From where the boundary condition (to)=0 we obtain kro = n = 1,2, . Hence,E, = = n = 1,2 6.88.(a) \177(r)= t sin (nnr/ro) ; (b) r = ro/2; 50%. We

\302\242

\177;

n2\1772h\177/2mr\177.

1/2-K\1770

6.89.(a) The solutionsof the

tion

Fig. 45. And a=0 and c=0. the

function it follows that continuity of (x) and itsderivative at whence tan kl \177

\177

finally, from the point x l we obtain \177

--k/\303\227,

)\177

(r):

.... n\177,

\177

SchrSdingerequationfor the func-

= A sin (kr + a), where k = Y 2mE/t\177, = Be\177+ Ce-:%where = V'2m (u0-E)/\177. r > r0, Since the function (r) is finite throughout the space,a = t? = 0.Thus, =- A sin kr q)2 = C r

< ro,

9\177

\303\227

9\177

\302\242

sin kl = =l::kl V t\1772/2ml\177Uo. sidesof the last equation Plottingthe left-hand and right-hand the straight line crosses which at the can find we points (Fig. 45),

to the eigenthe sinecurve.The rootsof the equation corresponding intersection those from points found values of energy E are the rootsof that equation arelocated for which tan (kl)\177 \177 O, axisare shown in the even quadrants(thesesegmentsof the abscissa rootsof the the that the from seen is plot heavy in the figure).It not always exist. equation, the bound statesof the particle,do of The dottedlineindicatesthe ultimateposition the straight line. =- (2n t) n\177h\177/8m. = n \177h2/8rn, (b) (/\177Uo)t 354 (kl)\177

i.e.

i.e. rn\177,\177

(l\177Uo),\177

m\177n

rv\177

0 and

\177

and From the continuity of the function or the point r = r0 we get tan kro = \177

its derivative

--k/\303\227\177

sin kro = -4-W \177/2mr\177Uokro. As it

tion

(b) r 2\177*

this equawas demonstratedin the solution of Problem6.85, determinesthe discontinuousspectrumof energy eigenvalues.

o\177U0

n\177h\177/8m. 355)))

= \177 6.90.a = rncol2h, E = the where with level i.e. principalquantum \177--mea/8h 6.91.E number n = 2. being at the interval 6.92.(a) The probabilityisofdPthe electron From the condih\177o/2,

\302\242o

\177,

nucleus (r) 4nr\177dr. = tion for the maximum of the function dP/dr we get (c) (U)= (b) (F)= = 6.93. (p/r)4gr dr =-e/rl,where p =-\177is the space wave function. chargedensity, is the normalized solutionsof the Schr6dingerequation 6.94.(a) Let us write the the barrierin the following \177orm: to the left and to the right of x < O, (x)= i\177 + e-i\177x, where = \177/\177,

r, r + dr

from the

\177

r\177r

r\177;

\177e\177/r\177.

2e\177/r\177;

\177

\177o

\177

a\177e

\177

k\177

b\177

where = \1772m (E--Uo)/h. + be x > 0, \177 an amplitudeat and the Let us assume that the incident wave has x \177 0 there is reflected wave an amplitude Since in the region R is the coefficient = reflection 0. The a wave,

(x)= aae

i\177x

k\177

e-\177h\177x,

b\177.

only

travelling

to the incident stream,or, ratio of the reflectedstreamof particles of the of amplitudesof correspondratio the squares in other words, at the ing

b\177

waves. Due to the continuity of 0 we have at + bl = a2 and

\177

a\177

\177

(b\177/a\177)

and itsderivative

(k\177

b\177

(k\177/k\177)

k\177)\177/(k\177

k\177)

a\177,

point whence

\177.

the solution the Schrbdingerequa(b) In the caseof E \177 Uo takes the form barrier the tion to the right of where z = \177(Uo E)/h. -x\177, (X) = a2 o\177

\177

eu\177

\1772

b\177e

\342\200\242

0.The probability

From the finiteness of (x) it follows that of finding the particleunder the barrier has the density a\177

\302\242

\177

(x)\177

e-\177. Hence,

x\177ii

P\177

(x)

\177

i/2\177.

-2\177 ' 6.95.(a) D\177exp[ W V2m(Uo--E)] \"

term and --0.04for a P term. 6.98.\177= VhR/(Eo--eTt) --0.88. 5.3eV. 6.99. = hR/(\177 R\177I\177/2ucAE-0.68 and (2P \1772S). 6.i00.0.82=9m2a\177cAE/\177 (3S \1772P) = meV. 2.0 AE 6.i01. = 1.05\"10 rad/s. 6.102. 3Dale, 6.103. 3Pa/\177, 3/2,5/2, 6.104. (a) 2, 3, 4, 5; (b) 0, 2,3, 4, 5, 6; (c) 1/2,

6.97.--0.4ifor

an S

-3=

\177

)\177

E\177

\177m

\177

\177a

A\177

3S\177/\177,

7/2, 9/2. 356

3P\177/\177,

3D\177/\177.

the

For the state 6.105. state sO:0, 6.106. (a) 2F7/2,

]i'V'\177/2,

\177P:

]iV\177-\177/2,

and h

V'\177/2;

for

]iV\177;

6.107.In the F state Ms--li\177/g;for the D state it only found that M\177IiV-\177. 6.t08.3, 4, 5. 6.109. (a) t, 3, 5, 7, 9; (b) 2, 4, 6; (c) 5, 7, 9.

3i 6.110. 6.111. 6.1i2. 1Fa, 6.113. The sameas in the foregoingproblem. Thesecondand the third term. 6.114. 6 = 10. 6.115. g = 4 6.116. 4, 7 and 6.117. aFt. As. 6.118. 6.119. (a) (b) h h3 6.120. (a) aFt/e, V (b) 6.121. (a) Two d electrons;(b) five p electrons;(c) five

can be

\302\260.

\177D2. \177P1,

aPo.l.\177,

\177D2,

3D\177,\177,a,

\177F2.\177.\177,

-\177

aP\177.

\177Sat2;

i-\177o/2;

V-i\177/2.

\177F\177/z,

trons.

6.122.

(a) (b) 6.123. 4Fall. 6.124. 6.i25. ---6.126. N/No =

d elec-

\177P0,

\177----\177

V-\177-

(\177Sb/2).

----3\"i0-1V, where

R (l l/he). where g and go i.14\"10-4, are the statisticalweights(degeneracyratios) of the levels 3P and n%-\177\302\260\177/\177T

\177l

(g/go)e-\177\302\260\177/\177r

\302\242o

3S respectively (g----6,go = 2). = ----

6.i27.

l/v in

i.3

6.t28.N = E\177P/2ncIi 7.10 = 65 6.129. ----(nlio)/P)(g/go) \177

\177ts.

\177l

\177

\177.

ns, where g and go are the degeneracyratiosof the resonantand the basiclevel. (a) )9ind/P\177p = K. (b) T = Supposethat is the intensity of the passing ray. The decreasein this value on passingthrough the layer of the substance of thicknessdx is equal to --dI = dx =) (I/c)lice dx, \177

e-\177\302\260\177/\177T

6.130. 6.131.

1.7.10 I \177

\177/\177R;

\303\227I

(N\177Bx\177

N\177B\177I

and are the concentrations of atoms on the lowerand Hence upper levels,Bla and B\177 are the Einsteincoefficients. where

N\177

(l -Next, the Boltzmann distribution should be taken into consideration, as well as the fact that hoe >> kT (in this caseN 1 is approximately equal to No, the total concentrationof the atoms). u ----(lice/c)

g\177N\177/g\177Ni).

N\177B\177

6.132.

A\177,Dop/A\177,\177,\177

,\177,

4\177,Vpr/\177,

,\177,

10a,

where

v\177

---:V2RT/M. 357)))

L = t54 pm. 6.133. for Co; (b) \1775keV. for A1, i80pm 6.134. (a) 843pm Three. 6.135. 6.136. = 15kV. 6.t37.Yes 6.t38.Z = 1+2l/(n--i)eVl/3hB(n--t/l/t/z) = 29. Z = t + #4Ato/3R = 22, titanium. 6.139. = 5.5keV. 6.i40,Eb = (Z --l) + 2ncMLt, where = 0.5 keY, /ia) (2nc/coAX 6.i41.Er. i) -(Z i) = i.45keY, T 6.142. 1) (Z o

\177

a/\177R

\177

2nc\177/\177

<E> =

\177B

IOBlOzl=

,\177ax\"

\177

(-= E.exp(-' exp(--(zEo) exp (--Eo/kT) + t/2), a = l/kT.The summation is carriedout E\177

\177

E\177/kT)

\177xp

\177]

\177

0-\177\"

as follows:

o\177

(-i--exp

\177

In exp

exp(--ccE\177)=acz

(3/Y-g\177

nlgyt\177lcr

(l\177

2/\177)

i\177oV.

\177

A\177.

.i(the ratio5.5kG.the corresponding -=

Land\177

6.i64.Ate = (\177i.3,:t:4.0,:i:6.6).10and, six 2ncliA\177,/gtta\177l\177?

'\302\260

s-\177

s\177x

factors);

(ra\177gi

rn\177ga)

\177

V-\177Mmd

,na\177eB/mc

-\177

R (\177io/kT) (e\177/\177

components.

\"\177

-} exp(lfo/kT)--t

= 0.56R,

\177'_ t)\177

where R is the universal gas constant.

is the 6.i77.d----\177/2h/\177Ao)=0.i3 nm, where molecule. = 42g and 387 nm. 6.178.= Xo/(l -\177x

reducedmass

of the

\177

= 4.96N/era. = t.37.101\177rad/s, 6.179.o) = nc = = increase 6.180. W ill 3.9times. 0.067. I,I\177 exp(--lio)/kT) 6.181. (a) SeeFig.46a in which the arrows indicatethe motion \302\242.oLol2nc)

(L\177

\303\227

\177,IL\177L\177

directionsof the nuclei in the moleculeat oscillationfrequenciesare oh, o)z, with

samemoment.The o)a being`the frequency of two independentoscillations in mutually perpendicular planes. Thus, there are four different oscillations. (b) See Fig.46b; there areseven different oscillations: three longitudinalones(to, and four transversal ones for each two oscillations frequency. 6.182. = (liar) d(o. 6.183. = (S/2nv o do). the

\302\242oa,

(o\177,

\302\242%)

(o\177),

(o)\177,

6.i65.(a) Six (1) and --four (2); (b) nine=(1)i.0-i0 (2). . 6.i66.Ate = n s , where m is the = 1.57.10 6.i67.to = 4 molecule. of the 6.i68.2 and 3. \177

(E) C Votb= N oOT

-\177

2T6/gY\177t\177lt

358

the definition

----

1.2-10tad/s, where g is the Land6

\177

(b) B

\177

-\177,

\177\302\260

F = 2I\177/cr = 3\"10 N. 6.154. = i5 kG/cm. + OB/Oz= 6.155. into 6 sublevels;(c) does does not It splits 6.t56.(a) split;(b) not split(g = 0). 6.t57.(a) 58 [xoV; (b) AE = 2gYi\177B = 145 normal (both 6.t58.(a) Normal;(b) anomalous;(c) normal;(d) termshave identicalLand6 factors). 6.t59.L =AE/2\177vB = 3; XF3. 6.t60. =L\177eB/2nmc = 35 pro. 6.161.= 4.0 kG. kG. 6.t62.B \177iAto/gt\177s= 3 2 6.t63. = (a)

E,=

-\177

Bm\177a

6.176. According to (E>

\177

I/3e-\177(\177-2B)/\177T

N\177,,b/N\177o\177

where he) (v over v taking the values from 0 to

\177x

\177_\177

\177B/gB/li

= 4.t0 N.

o)\177td\177/li,

the

2.26. i0 m/s. 6.I43.(a) g== 2, with the exceptionof the singletstate, where = 0/0;(b) g t. 6.144. (b) 0;(c)1;(d) 5/2;(e)0/0. (a) --2/3;

6.i45. 6.i46.Ms = 2 6.i47. 6.i48. 6.i49. 6.i50.M = 6.151. = 6.i52.o) factor. 6.t53.F,,ax=

= is the reducedmass where 6.174. a. molecule;(a) 36;(b) i.7.10\177;(c) 2.9.t0 = 3.i'10 where 6.17'5. -= h/2I,I is the moment of inertia of the molecule. E\177/E\177ot

\177

\177.

\177

per eV.

a/\177iR

\177/\177R

-*\302\260

\177

6.t69.M = V indUEC = 3.5h,where m is the massof the mole6.t70.I li/Ao) ----1.93.\1770 g.cm2, d = t12 pro. 13levels. 6.171. 6.172.N l/2-/\177/h = 33lines. where I is the moment of inertia 6.173. dN/dE\177, l/I/2liZE, the molecule.In the case of J= l0 dN/dE=l.O.lOlevels

cule.

dN\177

dN,\177

s-\177

mass

6.184. = dN\177

\177)

\177

(V/n\177v

\302\242o

do).

6.185. (a) O = (Mk) aVno;

= (n/e)

\177)

(b) O

-----

(Mk) v

4V\177-\177-\177o;

/ i8n2no/(V[1346.i86.O=(]i/k)\177

concentrationof the atoms.

2v\177

=470 K,

\177)

6.20|.AE = 6.202.(a) =

where no is the

2n\177]i\177lmV

dn\177

(ma/n\177]i

-22 eV. 1/a ----2oi0 (392n) 2 = a) v du;

(b)

<v)lv,\177,\177

6.203. ----8\177 dk. n 513 6.204.p = a/an (T) = ----= 4.5 eV. kT A 6.205. 2) (\177T/AT where 6.206. n:]/l+Uo/T:i.02,

\177

3/4.

-\177

dn\177

\177/\177-\177tt\17715rn)

(\177

5\"

i0 atm. \302\242

he/2rn.A is the work function.

2/\177 _\177

\177,\177

(3\1772n)

6.207.E,\177 --

2kT\177T\177

T\177--

\177

T\177

0.33eV.

0.05K-i, where p eAEol2kT, a pt OT _ 6.208. AE o is the forbidden band width. Alno and 0.06eV. 6.209.AE:--Z\177--\177--\177=i.2 acl\177

\177

kTZ)\177r-

Fig. 46.

\177,

concent6.187. v\177k@/]i\1776-'\177-\177-o=3.4 kin/s, where no is the km/s, ration of the atoms. The tabulated values are: v11=6.3 v_\177

= 3.1km/s. 6.t88.The oscillationenergy of a moleof a \"crystal\" is

=O.Ois. \177:t/ln 6.210. (p-n = hBVlelpYH= 5\"i0 cm -a, Uo 6.211. (P--P\177)P\177

P\177

p\177)

\177

= 0.05me/(V's). 6.212. -- = i/\177B = 0.20m\177/(V.s). 6.213. n+/n ---- = 4.0. 6.2t4.(a) P = i--exp(--\177t); (b)\177=

O/T

lu\177

u\177l

\177

where x:\177+/kT.Hencethe molarheat capacity is

\177.

O/T \177

\342\200\242

ex --1

elf-I

the heat capacity C R. T >> = 2l/na V+\177:: +s; (b) N l/a, dN/d+ 6.189. (a) is equal to the number of the atoms in dU the chain. 6.190.Uo = 9RO/8\177 ----48.6J/g, where is the molar mass of copper. 220K;(b) C (a) O I0

When

\177

\177,

\177

6.191.

\303\227

\177

(c)

\302\242o,\177

4.i

capac-

peratures. = 6.194.(E> 31s kO. 6.t95.SeeFig.47.

6.|96. 6.i97.

Fig. 47.

6.198. =

(b)

1--2-\177/\177

0.93. 6.199. K. 6.200. \177

\1773.\1770

360

\177

<T>

z65%.

\177i\302\242omax

(a)

\177I\177T,\177.

28 meV, ]ikmax

T,\177ax

(3\177en)\177/\177/2m;

= l/i =

\177

6.227. -- 1.6hours,

exp(lnA\177--lnA\177) --

9.8hours; A/q) = 9.5days.

N\177/N1

i0. --

-----

T\177

t 6.228. --(T/ln2) In (i 6.229.(a) Ne (t) = N,o \177

(e-\177,l\177--

(b) tin= \"\177

\342\200\242

\177

T\177

\177

-\177

\177

\303\227

the heat

i.i.10s-1,

6.221.T 4.5.i09years, A i.2.i0dis./s. 4.1.i0years. 6.222. About 2.0.i0years. 6.223. 6.224.3.2.10and 0.8.i0dis/(s.g) respectively. 6 1. V ----(A/A') exp(--tIn 2/T) 6.225. 6.226. 0.19%. = = \177:

\303\227

ity is proportionalto T at these tem-

\177,\177/t

\177

i01\177

becai\177se

\177

I\177

i0 J/(mol-K); rad/s. 6.t93.Yes,

i/\177,.

6.2i5.About 6.2i6.i.2-l0 6.2i7. = 16 s. 6.218.T 5.3 days. e 6.2t9.4.6.i0 part./min. 6.220. ln(i--\177) i.0 years.\177--(i/t) = =

\303\227

e-k\177);

(;',,\177/\177,,,) \"

6.230. (a)

(T\177/T1)

k\177--k\177

(t)

N\177

XN\177ot

6.23|. (t)=N,o ( i N\177

-\177

exp(--\177t); (b) t m \\

X,__X,

(--)\177t) 0.72.10 exp ---- 46\"I0u part./s. Here No e No (e is the initial number of Bi nuclei. 6.233. (b) eight alpha decays and six beta decays. (a)

6.232.--= /\177

No)\177,

-\177''

-\177'\177)

)\177,)

)\177,)\177/()\177

\177\302\260

Pb\177\302\260\177;

36i)))

i05 m/s; 0.020. v = V-2--A---\177/ra = 3.4. 6.234. t.6 MJ. 6.235. 0.82MeV. 6.236.

6.272.T --[-ma)/ma ----3.3MeV. 6.273.Betweeni.89and 2.06MeV. 6.274.Q --3.7MeV. 6.275.t.88and 5.75MeV respectively. 6.276.4.4MeV; 5.3.t0m/s. \177

respectively. 6.237.(a) 6.1era;(b) 2.l.10and 0.77.t05 and for K-capture, decay 66.238.Q -- (Mp--Ma)c z --decay. (Mp Ma 2ra) c for 6.239.0.56MeV and 47.5eV. 6.240.5 MJ. 0.32and 0.65MeV. 6.241. 6.242.T -- (Q + 2mc\177)/MNC 2 = 0.tt keV, where Q = -=- (MN -- Mc 2m) c m is the mass of an electron. 6.243.40 km/s. 6.244.0.45c, where c is the velocity of light. the mass of the 6.245.Ae/e E/2mc 3.6.t0 where rn is nucleus. 6.246.v e/rnc 0.22kin/s, where rn is the mass of the nucleus. 6.247.v gh/c = 65 ---ex 4.6 m. 6.248.hmin 6.249.T ---d- (M --m)\177/4mM cos O] 6.0MeV, where of an alpha particleand a lithium nucleus. m and M are the masses 0.89;(b) ----2ra/(rn M) = 6.250. 4mM/(m @ M) (a) ----2/3.Herern and M are the massesof a neutron and a deuteron. = 30 where ral and are the 6.251. arcsin deuteron. a and massesof a proton 2.iOn kg/cma, t- i0as nucl./cma. 6.252. 6.253. (c)(z; (d) CI (a) d; (b) MeV. 56.5 Bes, 6.255. and 8.7MeV; (c) t4.5MeV. tt.5 8.0 6.256. MeV; (b) (a) 6.257. -- ----0.22MeV. MeV, where e is t2ec----1t.9 E = 20eNe--2.4e\1776.258. nucleus. the in nucleon corresponding binding energy per a.m.u. 6.259.(a) 8.0225 a.m.u.; (b) t0.0135 -- -t-a 6.260. (Ea + Q 6.26\177. (a) 8.2.t0 kJ, 2.7.t0kg; (b) t.5 kg. 5.74.10\177kJ, 2.10kg. 6.262. 2.79MeV; 0.85MeV. 6.263. -- = t7.3MeV. 6.264.Q --2 X -6.265. (t + Tp --(t -Q cos0 ------1.2MeV, where rapCo, 6.266.(a) --1.65MeV; (b) 6.82MeV; (c) --2.79MeV;

\177

:: : :: : : :

'/\177.Q

\177,

\177

-\177,

\177

-\177

I\177m/s.

hc\177/g

\177

T\177/[i

\177

\177-

\177l

\1771

0,\177ax

(rn\177/rn\177)

\302\260,

a\177.

F\177;

E\177

\177he

E\177

8e\177

E\177)

\177\302\260

\177

7e\177l

: \177'l\177)

\177lp)

\177l,

3.tt MeV.=

E\177).

(E\177

\303\227

(d)

T\177

]//-\177'lp\177l\177TpT\177

-----

\177l\177,

rn\177,/rno.

0.92.t0m/s, VLl 0.53.t0\177m/s. t.9 MeV. 6.268. Q+(t- c)/\" 8.5MeV. 6.269. T\1776.267.

rn\177

-----

\177

v\177,

rn\177/rn

(ra\177

--\177/\177,

T,\177

6.277.T,= r\177+rn,

[\177+

,\177,

E\177,

tnt,

m2, m3,

are

m\177

-\177

\177

rn,m, T =2.2MeV, where the masses of neutron, a C nucleus, an * (m\177--m\177)

T\177h

]

t\177

alpha particle,and a Be nucleus.

6.278.By Eb/2mc 6.279.E = Q \177

0.06%,where m

6.5MeV.

\177/aT

\177

is the mass of a deuteron.

: :

mc 6.280. = 16.9,t7.5 and 17.7MeV, ma+mc T,:16.7, where Eb is the binding energy of a deuteronin the transitional nucleus. 6.\177I. = (M/Np d) In 2.5kb, where M is the molar mass of cadmium,N is the Avogadro number,p is the density of cadmium. 6.282. Io/I exp 1 + nd] = 20,where n is the concentration of heavy water molecules. 6.283. w = {l--exp[ -+ nd]} + = 0.35, where n is the concentrationof Fe nuclei. E\177

\177

E\177

-\177

\177

: :

\177)

[(2\177

(\177,

\177,/(\177,

\177a)

\177)

2; (b)a w = ATeCt In 2 = 2.i0-a. = 3-10 years; (b) Nmax J\177NoT/ln 2 = i.0-i0 la, where N0 is the number of Au nucleiin the foil. 6.\1776.N = (i --e = 6.i0 6.287.J = where X o (l --e is the decay constant, N0 is the number ofpart./(cm\177.s), Au nuclei in the foil. 6.288. N = No i.3.10where i is the number of generations. 6.289. N = vP/E 0.8.10s 6.284.(a) T =

6.285. (a) t

(w/k) In

\177/\177J

::: -x\177)

t\177

Jn\177/\177.

Aex\177/\177N

-x\177)

ki-\177

\177,

\177a

T: -\177.

6.290. (a) N/No 4.10\177; (b) \177/(k--i)=10s. 6.291. 0.05,0.4,and 9 GeV respectively. 6.292. (l) = V + 2)= i5m. = 6.293. lmc/VT(T@2mc =26 ns, where m is \1770

\177

(\177

\177o

the

rest

pion. 6.294.J/Jo= exp[ --lmc/% T (T + = 0.22,where m is the rest mass of a negativepion. = (ran -= 6.\1775\". Ev = 29.8 MeV. = --ran) --m\177]/2mz4.1=MeV, 6.\1776\".T 19.5MeV. = 52.5MeV. 6.297*. = (m, -6.298\". m=mpWT-\177m[+ T(T+2mp)=iii5 MeV, a A particle. --m[)/(mn@ T) = 22 MeV. 6.299*. Ev = mass of a

\177)

\177

2mc\177)]

m\177)\177/2m,

T\177

\177

[(m\177

Tma\177

m\177)\177/2m,

\177/\177

(m\177

----_

i.\177_mn/rnc

6.270.9.1MeV, 170.5 \302\260.

362

* In the quantity

m\302\242s

answers to Problems 6.295-6.299 marked is abbreviated as m.)))

[by

asterisk the

= 0.94GeV, --2 (mz Tz) 6.300%m = ]/'m'\177 neutron. (0/2) --i], Ev = m\177/2 sin (0/2).For 6.301\". = [cosec = Ev O ----60 the energy ---cos(0/2) i/V i \"b 2reC,whence O = 99 6.303*. 6.304*. (a) eth = 4me = 2.04MeV; (b) = 320MeV. = 5.6GeV; (b) Tth=m\177 6.305*. (a) = 0.28GeV. 6.306. (a) 0.90GeV; (b) 0.77GeV. E particle. 6.307.S =--2,Y Processes 2, and 3 are forbidden. 8.308. forbidden. 6.309.Processes2, 4, and 5 are terms of energy; in other proin forbidden I is Process 6.310. of baryon cessesthe following laws of conservationare broken: of charge lepton of (4), strangeness charge(2),of electriccharge(3), muon and charge(6). (5), and of electron -\177

\177-

m\177

(m\177

\177-

T\177)

ru\177

T\177

::--i, \177

\302\260

T\177

Tt\177

APPENDICES

m\177.

\302\260.

6m\177

t. Basic Trigonometricalformulas

\302\260

i i

sin s coss secs t tan s cosecs \177--cots \177 \177z-{-

\177z=

\302\242z--

\302\242z=

si\177

sin (a -4cos(a :h

tan

\177.SeC

\177

c\302\242

\177

-=-

\177

\177)

(a =h

tan

\177.COS\177

COS

sin a cos -4-cos sin cosa cos :Fsin a sin

----

\177)

cot (a :t=

a.cot a = i \177

= t tan tan \302\242z

\177)

=i=

tan

a-tan q= cot cot 1 cot :t= cot

=t=

\302\242z

\177z

sin \177:

sin etA-sin

\177=2

COS

sin ct-- sin

\177

cos

2\302\242z-----

--2 tan

tan

2\302\242z----

cot

sin

coss a--sin

--tan s \302\242z--

2\302\242z----

\"-\177--=

]//t--cos 2

cos._5- =

sinh

ct-=--

cosb

ct-\177

cos 2

\302\242z--[5

_\177_

cos a--[5 cosa\177-cos[5=2cos--\177 \177

2+-\177

cos

c\177--

tan

c\177

\302\261

cos = --2sin --\177sin

cot cz

\302\261

(a -4-[5)

sin

tan

cot [5=

COS

\302\261

sin sin

COS[5 (c\177

c\302\242

-4-[5) sin

2sin sin \177=cos \177)-- cos 2 cos cos[5----cos [5)-4-cos 2 sin cos = sin (a--[5)-4-sin (a \177z

(\177z--

\177z

(c\302\242--

\177

tq-cos\177

sin

= 2 cos--\177sin

\302\242z

t cots 2 cot a

\177

marked by an asterisk the quan* In the answers to Problems6.300-6.305 2 m. tity mc is abbreviatedas

sin 2 a

\177

(\177z-{\177-

[5) [5)

2 ecc-\177-e-ct

2 365)))

2.Sine Function O'

{

20'

O.OOOO 0.0058 2 3

4 5 6 7

8 9

ii 12

13 14 i5 16 17

19 20 21 22 23

24 25

26 27

28 29

30 31 32 33 34 35 36 37

38 39 40

42 43 44 366

0.0175 0.0233

0.0349 0.0523 0.0698 0.0872

\177po

0.0116 45 0.0291 46 47

0.0407 0.0465

0.0581 0.0640

3.Tangent Function

Values

48 0.0756 0.0814 49 50 0.0929 0.0987 5i 0.1045 0,11030.t161 0.1334 52 0.12190.1276 0.1507 53 0.13920.1449 0.1564 0.16220.1679 54 55 0.1736 0.i794 0.i85i 56 0.1965 0.2022 0.19080.2136 0.2i96 57 0.2079 58 0.2250 0.2306 0.2363 0.24i9 0.2476 0.2532 59 60 0.2588 0.2644 0.2700 61 0.2756 0.28120.2868 62 0.2924 0.2979 0.3035 0.3090 0.3i45 0.320i 63 64 0.3256 0.33ii 0.3365 65 0.3420 0.3475 0.3529 66 0.3584 0.3638 0.3692 67 0.3746 0.3800 0.3854 0.3907 0.39610.4014 68 0.4067 0.41200.4173 69 0.4226 0.4279 0.4331 70 71 0.4384 0.4436 0.4488 72 0.4540 0.4592 0.4643 73 0.4695 0.4746 0.4797 74 0.4848 0.4899 0.4950 75 0.5000 0.5050 0.5tO0 76 0.5150 0.52O0 0.5250 77 O.5299 0.5348 0.5398 78 0.5446 0.5495 0.5544 79 0.5592 0.5640 0.5688 0.5736 0.5783 0.5831 80 81 0.5878 0.5925 0.5972 82 0.61tl 0.60180.6065 83 0.6i57 0.6202 0.6248 0.6293 0.6338 0.6383 0.6428 0.6472 0.6517 85 0.6561 0.6604 0.6648 86 87 0.6777 0.66910.6734 0.6820 0.6862 0.6905 89 0.6947 0.6988 0.7030

0.707i

20'

0.7112

0.7193 0.7234 0.7314 0.7353 0.7431 0.7470

0.7547 0.7585 0.7660 0.7698 0.7771 0.7808 0.7880 0.7916 0.7986 0.8021 0.8090 0.8i24 0.81920.8225 0.8290 0.8323 0.8387 0.8418 0.8480 0.8511 0.8572 0.8601 0.8660 0.8689 0.8746 0.8774 0.8829 0.8857 0.89100.8936 0.8988 0.9013 0.9063 0.9088 0.91350.9159 0.9205 0.9228 0.9272 0.9293 0.9336 0.9356 0.9397 0.9417 0.9455 0.9474 0.9528 0.95110.9580 0.9563 0.96130.9628 0.9659 0.9674 0.9703 0.9717 0.9744 0.9757 0.9781 0.9793 0.98160.9827 0.9848 0.9858 0.9877 0.9885 0.9903 0.9911 0.9925 0.9932 0.9945 0.9951 0.9962 0.9967 0.9976 0.9980 0.9986 0.9989 0.9994 0.9996 0.9998 0.9999

\177o

3 4 5 6 7

8 9

t0 11 12 13 14 15 16 17

18 i9 20 21 22 23 24 25

26 27

28 29

31 32 33 34 35 36 37

38 39 40

41 42 43

20'

40'

Values

\177

\177*

O.0000 0.0058 0.01t6 45 0.0175 O.0233 0.0291 46 O.0349 0.0407 O.0466 47 48 O.0524 O.0582 0.064t O.0699 O.0758 0.08t6 49 50 O.0875 O.0934 0.0992 0.10510.ttt0 0.tt69 5i 52 O.1228 O.1287 O.t346 53 O.1405 O.i465 0.t524 54 O.t584 0.1644 O.t 703 55 O.1763 O.1823 O.t 883 0.1944 O.20O4 0.2065 56 O.2i26 O.2186 O.2247 57 58 O.2309 0.2370 O.2432 O.2493 O.2555 0.2617 59 60 0.2679 O.2742 0.2805 6i O.2867 O.2931 0.2994 0.3057 0.3i2i 0.3185 62 63 0.3249 0,33i4 0.3378 64 O.3443 0.3508 0.3574 0.3640 0.3706 0.3772 65 O.3839 0.39O6 O.3973 66 0.4040 0.4t08 0.4176 67 0.4245 0.43i4 0.4383 68 0.4452 O.4522 0.4592 69 70 0.4663 0.4734 O.4806 7i 0.4877 0.4950 O.5022 O.5095 0.5i69 O.5243 72 0.53i7 O.5392 0.5467 73 O.5543 O.5619 0.5696 74 5 774 O. 5 851 O. 5 930 O. 75 0.6009 0.6088 0.6168 76 O.6249 O.6330 0.6412 77 O.6494 0.6577 0.6661 78 79 O.6745 O.6830 O.6916 O.7002 O.7089 0.7177 80 8t O,7265 O.7355 O.7445 O.7536 O.7627 0.7720 82 O.7813 O.7907 0.8002 83 0.8O98 0.8195 0.8292 84 0.839t 0.8491 0.859t 85 O.8693 0.8796 O.8899 86 87 0.9004 0.91100.92t7 O,9325 O.9435 O.9545 88 89 O.9657 0.9770 0.9884

20'

&O'

1.00001.012 t .036 1.048 t.072 t .085 1.124 t.ttt t. t.t50 t64 t.t92 t.206 t.235 t.250 t.280 t .295 t.327 1.343 1.376 t.393 t.428 t .446 t.483 t.50t t.540 t.560 t.600 1.62t t.664 1.686 i.732 t.756 1.804 t.829 t.907 t .963 1.991 2.050 2.08t 2.i45 2.t77 2.246 2.282

t. 024 1.060 1.098 t. t.137 178 t. 220 t.265 i.3ii t. 360 t .4tt

2.475

2.560

1.84\1771

2.356 2.394 2.5t7 2.605 2.65t 2.747

2.904

3.078

2.798

2.960 3.140

3.27i 3.340 3.487 3.566

3.82t 4.0ti 4.tt3 4.331 4.449 3.732 4.705

4.843

5.i45 5.309 5.671 5.87i

t.464

i..580 520 t

t,643

t.709

t .780

1.855

1.935 2.020

2.ii2 2.2ii

2.318 2.434

2.699 2.850 3.0i8 3.204 3.4i2 3.647 3.9i4 4.2i9 4.574 4.989 5.485 6.084 6.827

6.3t4 6.56i 7.770 7.1t5 7.42fi 9.010 8.t44 8.556 9.5t4 t0.08 10.71 tt.43 12.25 13.20 t4.30 i5.60 17.17 24.54 i9.08 21.47 42.96 28.64 34.37 57.29 85,94 171.9 367)))

4. Common N

it t2 t3

i4 t5 t6 17

t8 t9

20 2t

22 24 25 26 27

28 29 30 3t

32 34 35 36 37

38 39

4t 42

43 44 45 46 47

48 49

50 5t 52 53 54

368

0000 0043 0O86 04i4 O453 O492 0792 0828 0864 ii39 1t73 t206 146i t492 t523 i76t t790 t8t8 204t 2068 2095 23o4 2330 2355 2553 2577 260t 2788 28t0 2833 30i0 3032 3054 3222 3243 3263 3424 3444 3464 36t7 3636 3655 3802 3820 3838 3979 3997 40t4 4t50 4i66 4t83 43t4 4330 4346 4472 4487 4502 4624 4639 4654 4771 4786 4800 49i4 4928 4942 505t 5065 5079 5i85 5t98 52tt 53i5 5328 5340 544t 5453 5465 5563 5575 5587 5682 5694 5705 5798 58O9 582t 59ti 5922 5933 6o2i 603t 6O42 6t28 6t38 6149 6232 6243 6253 6335 6345 6355 6435 6454 6532 6542 655t 6628 6637 6646 672t

6730

6739

Logarithms

5599

0t70 02t2 0253 0569 0607 0645 0934 0969 t004 t27t t303 t335 1584 1614 t644 t875 t903 t93t 2t48 2t75 220t 2405 2430 2455 2648 2672 2695 2878 2900 2923 3096 3tt8 3139 3304 3324 3345 3502 3522 354t 3692 371t 3729 3874 3892 3909 4048 4065 4082 4216 4232 4249 4378 4393 4409 4533 4548 4564 4683 4698 47t3 4829 4843 4857 4969 4983 4997 5t05 5tt9 5t32 5237 5250 5263 5366 5378 5391 5490 5502 55i4 5611 5623 5635

5832 5944 6053 6t60 6263 6365 6464 656t 6656

5843 5855 5866 5955 5966 5977 6064 6075 6085 6t70 6t80 6t9t 6274 6284 6294 6375 6385 6395 6474 6484 6493 657t 6580 6590 6665 6675 6684

0t28 0531 0899 t239 t553 t847

2122

238O

2625 2856 3075

3284 3483 3674

3856 4031 42O0

4362 4518 4669 4814 4955 5092 5224 5353 5478 5717

5729

6749 6758

6812 682t 6830 6839 6902 69tl 6920 6928 6937 6990 6998 7007 70t6 7024 7t10 7076 7O84 7093 7t0t 7t60 7t68 7t77 7185 7t93 7243 725t 7259 7324 7332 7340

(Continued)

5740

5752

6767

6776

6866 6946 6955 6857

7O33

7O42

7tt8 7t26 7202 72t0 7267 7275 7284 7292 7348 7356 7364 7372

0294 0334 0374 0682 0719 0755 t038 t072 tt06 t367 t399 t430 t673

t703

t732

t959 t987 20t4 2227 2253 2279

2488 27i8 2945 3160 3365 3560 3747

74t2 74t9 7490 7497 7566 7574 7642 7649 7716 7723

7427

7435 75t3 7589 7664

7789

7803

7860 7931

7796

7868 7938 8000 8007 8069 8075

2504 2529

2742 2765 2967 2989

3181320t

8t36 8142 8202 8209

3385

3579 3598 3766 3784

8267

8274

7505 7582 7657 7731

4099 41164t33 4265 438t 4298 4425 4440 4456 4579 4594 4609 4728 4742 4757 487t 4886 4900

50il 5024 5038 5t45 5t59 5t72 5276 5289 5302 5403 5416 5428 5527 5539 5551 5647 5658 5670 5763 5775 5786 5877 5888 5899 5988 5999 6010 6096 6107 6it7 620t 62t2 6222 6304 6314 6325 6405 64t5 6425 6503 65t3 6522 6599 6609 66t8 6693 6702 67t2 6785 6794 6803 6875 6885 6893 6964 6972 698t

9736 9782 9827

7059 7067 7t35 7143 7t52 72t8 7226 7235 73O0 73O8 73t6 7380 7388 7396 7O5O

24--9451

8082 8149 8215 8280 8344

7443 745t 7459 7466 7475 7520 7528 7536 7543 755t 7597 76O4 76t2 76t9 7627 7672 7679 7686 7694 770t 7745

7752

7760

7767

78t0 78t8 7825 7832 7839 7882 7889 7896 7903 79t0 7945 7952 7959 7966 7973 7980 80t4 802t 8028 8O35 804t 8048

9741 9745 9786 9791

9832 9872 9877 99t7 9921 996t 9965

7875

833t 8338 8395 840t 8407 8457 8463 8470 85t9 8525 8531 8579 8585 859t 8639 8645 8651 8698 8704 8710 8756 8762 8768 88t4 8820 8825 887t 8876 8882 8927 8632 8938 8982 8987 8993 9036 9042 9047 9090 9096 9t01 9t43 9149 9t54 9t96 920t 9206 9248 9253 9258 9299 9304 9309 9350 9355 9360 94OO 9405 9410 945O 9455 946O 9499 95O4 95O9 9547 9552 9557 9595 9600 9605 9643 9647 9652 9689 9694 9699

3927 3945 3962

7738

8089 8096 8t02 8t09 8t56 8162 8t69 8176 8222 8228 8235 824t 8287 8293 8299 8306 835t 8357 8363 8370 8414 8420 8426 8432 8476 8482 8537 8543 8549 8555 8597 8603 86O9 86t5 8657 8663 18669 8675 8716 8722 8727 8733

7774

7846 79t7 7987

8055

8t168t.22

8t82 8189 8248 8254 83t2 83t9 8376 8382 8439 8445 8500 8506 856t 8567 862t 8627 868t 8686 8739 8745

8785 879t 8797 8802 8842 8848 8854 8859 8899 89O4 89t0 89t5 8954 896O 8965 897t 9O09 90i5 9020 9025 9063 9069 9074 9079 91t7 9t22 9t28 9133 9i70 9t75 9t80 9186 9222 9227 9232 9238 9274 9279 9234 9289 9325 9330 9335 9340 9375 9380 9385 9390 9425 9430 9435 944O 9474 9479 9484 9489 9523 9528 9533 9538 957t 9516 958t 9586 96t9 9624 9628 9633 9666 967t 9675 9680

8774

8779

9703 9750 9795

9708 97t3 97t7 8722 9727 9754 9759 9763 9768 9773 9800 9805 98O9 98t4 98t8

883t 8837 8887 8893 8943 8949 8998 90O4 9053 9O58 9t06 9tt2 9t59 9t65 9212 92t7 9263 9269 9315 9320 9365 9370 94t5 9420 9465 9469 9513 95t8 9562 9566 9609 96i4 9657 966t

9836 9841 988t 9886 9926 9930 9969 9974

9845 9850 989O 9894 9934 9939 9978 9983

9854 9859 9863 9899 9903 9908 9943 9948 9952 9987 9991 9996

369)))

5. Exponential

0.00 0.05 0.10 0.15 0.20 0.25 t).30

0.35 0.40 0.45

0.50 0.55

0.60 O.65

0.70 0.75

t.0000

1.05t3 0.95t2 t.t052

t.16t8 t.22t4

1.2840 t.3499 1.4t9t t .49t8 t .5683 6487 7333 t

t. t.

.8221

t.9155 2.0t58 2.tt70

0.9048 O.8607 0.8t87 O.7788 O.7408 O.7047 O.6703 0.6376 0.6O65 O.577O 0.5488 O.522t 0.4966 0.4724 O.4493 0.4274 0.4066 0.$867 0.3679 O.3499 0.3329 O.3166

2.00 2.05 2.10 2.t5 2.20 2.25 2.30 2.35 2.4O

2.45 2.5O

2.55

2.60 2.65 2.70 2.75

t .65

2.80 2.85 2.90 2.95 3.00 2.7t83 3.05 2.8577 3.t0 3.0042 3.15 3.t 582 3.3201 0.3012 3.20 3.25 O.2865 3.4903 3.30 O.2725 3.6693 3.35 0.2592 3.8574 3.40 O.2466 4.0552 3.45 0.2346 4.263t 3.59 O.2231 4.48t7 0.2t23 3.55 4.7tt5 0.20t9 3.60 4.9530 0.192t 3.65 5.2070

t.75

0.t738

0.80 0.85 0.90 0.95 t.00 t.05

1.t0

t.15 .20 t.25 t.30 t.35 t.40

t .45

.50 t.55 t

1.60 1.70

.80 t .90 t t .85

i.95 370

t .0000

Functions

2.2255 2.3396 2.4596 2.5857

5.4739 5.7546 6.0496 6.3598 6.6859 7.0287

O.t827

3.70

0.t653

3.8O

O.t572 O.1496 O.1423

3.75

3.85

3.90 3.95

(Continued)

7.389t

0.t353

4,00

8.t662

0.1225 0.tt65 0.1t08

4.t0 4.t5 4.20 4.25 4.30

7.7679

8.5849 9.0250 9.4877 9.9742

10.486

0.1287 O.t054

0.t005

t4.t54

0.O9537 0.O9072 0.08629 0.08208 0.07808 0.07427 0.07065

15.643

0.06393

1t.023 ii.588 i2.t82 t2.807 t5.464

t4.880

t6.445 t7.288

t8.t74

19.t06

20.O86

2t.tt5

22.t98 23.336 24.533 25.790

27.1t3

28.503 29.964

3t.500

33.tt5 34.813

0.06721

0.06081 0.05784 O.O55O2 0.05234 0.04979 0.04736 0.04505 0.04285 0.04076 0.03877 0.03688 0.03508 0.03337

0.03175

44.701

0.03020 0.02872 0.02732 0.02599 0.02472 0.02352 0.02237

46.993 49.402

0.02024

36.598 38.475 40.447 42.52t

5t.935

0.02128 0.01925

4.05

4.35 4.4O

4.45

4.50 4.55

4.60 4.65 4.70

4.75

4.80 4.85 4.90 4.95

5.O0

5.O5

5.10 5.15 5.2o 5.25 5.30 5.35 5.40 5.45

5.50 5.55 5.6O

5.65 5.70 5.75

5.80 5.85 5.90 5.95

54.598 57.397 60.340 63.434 66.686 70.t05 73.7O0 77.478 8t .45t 85.627 90.0t7 94.632 99.484

104.58 t09.95

t15.58 121.5t

127.74 t34.29

14t.t7

0.01832 6.0 6.1 0.01742 6.2 0.01657 6.3 0.0t576 6.4 0.015OO 6.5 0.O1426 6.6 O.01357 0.01991 6.7 6.8 O.0t228 0.01168 6.9

0.0ttit

7.0

O.01005 O.00956 O.00910 O.OO865 O.O0823 O.00783 O.00745 O.(}0708 0.00674

7.2 7.3

0.01057

7.1 7.4

7.5 7.6 7.7

7.8 7.9

8.0 8.t 0.00610 8.2 8.3 172.43 O.OO580 8.4 O.00552 t8t .27 8.5 t90.57 0.0O525 8.6 O.00499 200.34 8.7 2i0.6t O.(}0475 8.8 22t .4t O. 8.9 O.00450 232.76 9.0 O.00409 244.69 9.t O.00389 257.24 9.2 0.O0370 270.43 9.3 0.00352 284.29 9.4 O.00335 298.87 $t4.19 0.00318 9.5 9.6 O.00303 330.30 9.7 O.(}0288 $47.23 9.8 O.00274 365.04 9.9 383.75 O.00261 t48.4t

t56.02 t64.02

0.00641

0(\177152

i0.0

403.43 445.86 492.75 544.57

60t .85

665.t4 735.t0

8t2.41 897.85 992.27

t096.6 i2t2.2 t539.4 t480.5

O.00248 O.00224 O.0O203

0.00184 0.00t66

0.001503 0.001560 O.00t231

0.001114 0.001008 O.000912 O.000825 0.000747 O.000676

t636.0 0.000611 t808.0 0.000553 t998.2 O.0005OO 2440.6 2697.3 298t.0 3294.5

3641.0 4023.9

4447.t 4914.8

5451.7 6002.9 6634.2 7532.0

8103.1

8955.3 9897.t 10938 12088 13360 14765

16318

18034 19930 22026

O.OOO453 O.0004t0 O.00057t O.000335 O.000304 0.000275 0.000249 0.000225 O.000205 O.000t84 O.000i67

0.000t5i

0.000136 O.000123 0.000tt2 0.000101 0.00009i 0.000083 O.000075 0.000068 O.00006t 0.000055 O.000O50 0.000045

37t)))

6.Greek Alphabet A,

\302\242z--

I. t--iota K, lambda A, ),--

alpha

beta F, y--gamma A, 6--delta E, e--epsilon

\1775--

\177--zeta

\177--eta

p--rho o--sigma T, r--tau

M, N, \177,

O, H,

\177t--

Derivative

Function

sin x

T, v-- upsilon q)--phi

\177nu

v--nu

COS X

X, W,

\177--xi

o--omicroa n--pi

g\177,

\177--

chi

\177--

psi

constants

Function

COSX

--sin x

tan x

cot x

arcsin x

\177

arctan x

sinh In u

lIla

cosh x

tanh

Approximations

coth x

Approximate formulas

i t

l+z

\177

arccot x

ex ax

Derivative

arccos x

sin S x

omega

\302\242o--

In x

Numerical

Derivative

COS

\177x

Constants and

Integrals

\302\242,

7. Numerical

Function

\303\227--kappa

0--theta

9.Derivativesand

cosh z x

sinh

coshSx l

sinh'x

(for (z<< 1)

dx= n-[-I

n=3.14i6

(n .\177--1)

n\177=9.8696

]\177'= 7725

e=2.7183 log e=0.4343 In i0=2.3026

Sill

x dx

\177

--COSx

eesx dx = sin x tan x dx = --In cosx

--arcsinx

cot x dx = In sin x Integration

8.SomeData on Vectors

parts:

udv \177

Some Definite Integrals

1, x

\177e

-xdx=

-\177-

d -\177-

372

(a a) =

c\302\242

-\177-

da -\177-

(ab) =

da , n

\177

, \177-

t, n=t

db -\177-

\177

x\"dz \177

\177ex--

[ ----

n-\1770

n=2

2.31,n=-t/2

n=t 2.405,n=-2 nt/tS, n=3

[nZl6' 24.9,n----4

a= t 0.225, a=2 t.t8, = a=3 2.56, ex--t 4.9t, a=5 6.43,a= t0 373)))

t0.Astronomical Body

12.Thermal Expansion Coe|ficients

Dat,

(at room temperatures) Mean density, t03 kg/m 3

Mass, kg

Meanmradius,

Period about

rotation axis, days of

Linear expansion coefficient;

Solids

i.97.

6.95. l0s 6.37.i0 t.74.i0

Sun Earth Moon

i .4i

25.4

3.30

27.3

5.52

5.96.i02\177 7.30.i02\177

\177

Planets of

s\302\260

l\177)

\177

Mean distance from the

solar system

Sun,

10\177

\177

.00

0.24t

i49.50

Mars Jupiter

227.79 777.8

t.000 t.88t tt.862

108.i4

29.458

Neptune

t64.79 Material

it. Density of Substances Solids

Diamond Aluminium Tungsten

Graphite Iron (steel) Gold Cadmium

Cobalt

Ice

Copper Molibdenum Sodium

Nickel Tin Platinum

Cork

Lead Silver Titanium Uranium

Porcelain Zinc

p' g/cm3

3.5 2.7 19.t

t.6

7.8

Liquids

Ben zene Water

Glycerin Castor oil Kerosene

t9.3 Mercury 8.65 Alcohol 8.9 Heavy water 0.9t6Ether 8.9 Gases(under standard t0.2 tions) 0.97 Nitrogen 8.9 Ammonia 7.4 Hydrogen 2t.5 Air 0.20 Oxygen tt.3 Methane 10.5 Carbon dioxide 4.5 Chlorine 19.0 2.3

[ P,

OT'

6= Vl

i.8 ii.0

Mercury Ethyl alcohol

13.Elastic Constants.Tensile Strengtb

84.0t3

2867.7 4494

5.0

i0.0

Kerosene

8.5

\177,

2.i

Glycerin

\177

10-4K-I

Water

i6.7

glass

0.6t5

1426.i

Saturn Uranus

Common

Note.

57.87

22.9 i8.9

Aluminium

Siderial period, years

Mercury Venus Earth

10-aK-1

Brass Copper Steel (iron)

Bulk expansion coefficient;

Liquids

c\177,

g/em 3

0.88 i.00 1.26 0.90

Aluminium

Copper Lead Steel(iron)

Glass

Young's modulus E, GPa

Shear modulus

0.79

130 16 200

0.29

8i 30

\177t

0.34 0.34 0.44

5.6

0.25

Tensile strength

GPa

Note. Compressibilit)\"

\177:

0.i0

0.30 0.0i5 0.60 0.05

Water

i.i O.72

condi- p, kg/m

ratio

G. GPa

O.80

i3.6

Poisson's

Compre\177sibil-

ity,

\177,

GPa-l

0.0t4 0.007

0.022

0.006

0.025 0.49

\177

i.25

14.Saturated Vapour

Pressure

0.77

0.09 i.29; i.43 0.72 i.98

3.2i

0 5

Pressure,kPa

0.6i 0.87

i.22

2.33

i.70

\302\260C

25 3O

35 40 50

Pressure,kPa

3.i5

Pressure,kPa

\302\260C

4.23

7.35

8O 9O

5.60

t2.3

i00

i9.9

3i.0 47.3

70.0

i0i.3

7.0

375)))

374

t5. Gas Constants (under

i7, Permittivitles

standard conditions)

Relative

Gas molecular

mass

Heat

con-

ductivity mW

(relative values) Van der

dia-

meter

\177I,

4 40

He Ar

28 32 44 i8 29

02 C02 H20 Air

.67

iM.5

i6.2 .67 1.4i i68.4 .40 .40

.30 .32 .40

24.3 24.4

i8.9 22.t 8.4 i6.7 i9.2 14.0 9.0

23.2 i5.8 24.i i7.2

Weals

constants

Molecular Viscosity ttPa-s

0.20 0.35

\177 ' mol i .30 0.032

0.35

Note. This table quotes the mean values of molecular diameters. per[orming more accurate calculations, it should be remembered the values oI d obtained from the coefficientsoI viscosity, heat conductivity, and diffusion, as well as the Van der Waals constant b,

perceptibly from one another.

i6.SomeParametersoi Liquids and Specific heat * Substance

capacity

g.K

Ice

4.18 2.42 0.14 2.42 0.90 0.46 2.09

Copper Silver Lead

0.i3

Water

Glycerin Mercury Alcohol Aluminum

Iron

\177o

Porcelain

2.7

Ebonite

Resistivity (at 20 oc) p, n\177.m

Conductor

Temperature \177,

-1

coefficient

4.5

Aluminium

Tungsten Iron

50 90 20

4.8 6.5 4.0 4.3 4.2

i90 i5

4.i

mN/m

284 853

490

* Under standard conditions. ** Under standard atmospheric pressure. 376

specific heat of melting q, J/g

6.0 6.0

Glass

Surface tension *

2250

O.39 O.23

2.0 2.0 3.5 2.3

Gold Copper Lead Silver

Solids

Specific heat of vaporigation ** q, Jig

Plexiglas Polyethylene

Alcohol

18.Resistivitiesof Conductors

When that differ

Paraffin

7.5

Mica

t.00058

Kerosene

Dielectric

Water Air

0.24 0.027 1.35 0.039 0.35 i.35 0.032 0.40 3.62 0.043 0.30 5.47 0.030 0.27 0.37

Dielectric

i9.Magnetic

Susceptibilitieso[ Pare- and Diamagnetics

66 22

32i

Paramagnetlc

stance

sub-

270

333 i75

88 25

Oxygen Ebonite

Aluminium Tungsten

Platinum

25--9451

lO-a

0.0i3 0.38 1.9

Nitrogen Air

Liquid

\177t-1,

oxygen

23 176

360 3400

Diamagnetic stance

Hydrogen

Benzeno Water

Copper

Glass

Rock-salt Quartz Bismuth

sub-

W-i,

iO-a

--0.063 --7.5 --9.0

--i0.3 --i2.6 --i2.6

--i5.1

--i76

377)))

22. Work

20. Re[ractiveIndices

i.00029

i.33

Air

Water

Metal

Substance

Function

i.50

Glass

to depend on the

3.74

2.29 4.62 t.89

Barium Bismuth

nature

Note. Sincethe refractive indicesofare knownthe valueso[ n listed here light, of the substance and the wavelength should be regarded as conditional.

Cesium

Cobalt Copper

4.25 4.47

23. K Band Iceland spar Wavelength

\177,

Colour

red

656 589

orange yellow green blue

527

486

1.653 t.550 i.485 i.655 1.55t i.486 i.658 1.553 i.489 1.664 t.556 i.49i i.668 1.559 i.495 1.676 t.564 1.498 i.683 t.568

indigo

violet

400

21.Rotation o| the Natural

\177,

Plane o| Polarization

(the thickness o[ the plate

in quartz

rotation

% deg

\177,

(p, deg

i.484

687

is t

1.54t i.542 i.544

1.547

t.550 t.554

t.558

mm)

t99.0 2t7.4 219.4

295.65 226.9t 220.7

274.7

t2t.t

49t

257.t

328.6

t43.3

Magnetic Liquid

Benzene Water

378

78.58

372.6 404.7

435.9

508.6

70.59 58.89

48.93

41.54

3t.98 29.72

23 26

30 42

589.5

2t.72

670.8

t6.54

17.32

656.3

6.69 3.4i 2.28

t040 t450 t770

(k----589nm). The Verdet Constant Liquid

V, ang. mln/A

2.59

0.0t6

Carbon disulphide Ethyl

alcohol

0.053 t.072

Metal

4.58

Gold Iron Lithium

Silver

4.84

Titanium Tungsten

4,27

Molybdenum Nickel

5.29

Platinum

A, eV

2.15

Potassium

4.36 2.39

4.28

2.27

Sodium

3.92

4.50 3.74

Zinc

Edge \177K'

226.8 t74.1 t60.4 t48.6 t38.0 128.4 6t.9

Element

\177K'

47 5O

48.60

42.39

74 78 79

Pt Au

82 92

Mass absorption coefficient Air

i0 20 30 40 5O 7O

V, ang. min/A

A, eV

Pb U

t7.85

i5.85 t5.35 t4.05

t0.75

\177t/p,

cm2/g

60 Rotation

Element

(p, deg

k, nm

Metal

24. Mass Absorption Cee|ficlents (X-ray radiation, narrow beam)

\177.,

344.t

Absorption

Quartz

Metals

Various

A, eV

Aluminium

2.42

Diamond

0.48

0.75

t.3

t.6 2.t

2.6

t00 t50 200

25O

8.7

Water

0.t6 0.t8 0.29 0.44 0.66

i.0 t.5

2.1

2.8 3.8 t2

28 51

AI umlnlum

0.i6

0.28 0.47

i.i

2.0 3.4

5.t 7.4

1t t5

46 102 t94

Copper

0.36 t.5 4.3 9.8

48 70

98 t31 49 t08 198

Lead

3.8 4.9

14 3t

54 90 t39)))

28.Units

25. Ionization Potentials of Ate.ms Ionization potential (p, V

Atom

5.39

9.32 8.30

Be B C

ii.27

Ionization potential \177p,

9 10 1t

i7.42 t0.44

eV,

Atoms

Light

Excessof mass

Isotope

of atom A, a.m.u.

0.00867 0.00783

Hi Ht Ha

0.0t4i0 0.0i605 0.0i603 0.00260 0.0i5i3 0.0i60t 0.0i693 0.0053i 0.0i2i9 0.01354 0.0i294

He He \177

\177

Li Li

\177

Be Be Be Bexo \177

\177

Excessof mass Isotope

6 7

Note. Here Ar is mass number.

atom A,

a.m.u.

C11

0.01i43

C1\177

0.00335 0.00574 0.00307

C13 N is N\177

015 017 Fx9

Ne2o Na2 s NaS\177

0.00930

MgS

the relative atomic mass

(in

0.000il 0.00307 --0.00509 --0.00087 --0.00160 --0.00756 --0.0i023 --0.00903 --0.01496 a.m.u.),A is

the

Im, lumen

Ix, lux

Radionuclides

G, gauss

g, gram

Isotope

Cobalt Co

38 84 86 88 92

Strontium Polonium

5.2years

6\302\260

Radon

9\302\260

'I\302\260

2\1772

Radium Ra226 U 236 Uranium

28 years i38 days

3.8days i620 years 4.5.109 years

Mx, maxwell

V, volt W, watt

N, newton

Wb, weber

DecimalPrefixes Factor

1012 i0\177

t0-1

Name

of prefix

teragigamegakilohectodecadeci-

Name of

Factor

Symbol

eentimillimicro-

10-2

T G

t0-s

t0-6

Symbol

prefix

nano-

pico-

t0-19

da d

femto-

atto-

SI and CGSUnits Name of unit

Length Half-life

T, tesla

reel,mole

F, farad

Time Kind of decay

St, stokes

m, metre rain, minute

dyne

electron-volt

ohm

P, poise Pa.pascal tad, radian S, siemens s, second sr, steradian

l, litre

Physical quantity

27.Half-LifeValues

\1772,

Hz, hertz J, joule K, kelvin

unit

C, c\177)ulomb cd, candela D, diopter

5.14

dyn,

26.Mass of

henry h, hour

B, bel b, barn

2i.56

Ne Na

\1770

mass

i3.62

Oe, oersted

ampere

a.m.u.,atomic

i4.54

24.58

Atom

i3.59

PhysicalQuantities certain quantities

Names and symbols of

Velocity

Acceleration Oscillation frequency Angular velocity Angular frequency Mass Density

Force

Conversion factor

i SI unit/l

SI cm

100

cm/s cm/s

t00 t00

\177

t i t 103

dyn

t09

rad/s

kg kg/m a

g/cm s

S-1 N

CGS unit

S-I

i0-3

381)))

380

29. The Basic Formulas oI Electrodynamicsin the SI and GaussianSystems

(Continued) Name of unit Physical

quantity

Pressure, stress Momentum Moment of Energy,

Power

dyn. cm

erg erg/s

kg.m

\177

V/m

C,m$ C/m A A/m

Current Current density

Resistance Besistivity Conductance

\177

flux

Magnetic field strength Magnetic moment

Wb A/m

A.m 2

CGSM unit CGSM unit cm

Inductance Luminous intensity Luminous flux Illumination

lm/m

cd/m

9 \177

I/(9.iO i/(9.in0

=3.tt.t0s 7

t0t.3kPa = 760 mm Hg = t00 kPa (precisely) bar Pa mm Hg=i33.3 1.atm \177i0t.3 J = 4.i8 cal

atm

\177

i ib \177

I0 t0-a i09 i

= i0 emz -x9 j = 1.6.t0 -la i.6.i0

Energy

density of

\177oule's

law

dis./S (curie)=3.70.t0 \177\302\260

Magnetic moment of a current rying loop

Magnetic

D = eE

D = eoeE

DndS= q \177

C = q/V

.\177DndS=4nq

eOeS d

4rid

CV\177

/2 ED

j=oE

-z\177

-\177

i Ci

the

Gausstheorem for the vector D Capacitanceof a capacitor Capacitance of a plane capacitor Energy of a system of charges of a caEnergy pacitor

D=Eq--4nP e=t+4n\303\227

electric field Ohm's law

=t0-s cm

{ erg t.66.t0 g I a.m.u. = 931.4MeV { t eV

D = eoE q- P

between E

Relation D and

4n.iO-\177 a

gxven

Electric dipole p

\303\227

iO4

\177

units

(\177

vector D Belation between e and

9.iO

here

E and

o',

3.10 n iOs

are

808

between

Relation

in field E between Relation P and E Relation between P, and E of the Definition

3.i05 9.iO1 3.10

\177

Brightness Note. The CGSelectric and magnetic Gaussian system. Some Extrasystem Units

382

3.iOn

Luminosity

year

i2\177.i0

A/m

Magnetization

1/300

i/(3.i0

\177.m

Magnetic induction

3.i09

CGSEunit CGSEunit CGSEunit CGSEunit CGSEunit

charge

t07

erg/K CGSEunit CGSEunit CGSEunit CGSEunit CGSEunit CGSEunit

t 4\177e

the

Gaussian system

Potential of the field of a point

field of a plane capacitor

C/m F

charge Strength

t07

g.cm\177

of the a point

Strength field of

i0s

\177)

Pa.s

Temperature Heat capacity, entropy Electric charge Potential Electric field strength Electric induction Electric dipole moment Electric polarization Capacit y

t07 t07 t07

erg/(s,cm g.cma/s

\177

Moment of inertia Dynamic viscosity

Magnetic

g.cm/s

N-m W W/m kg.mVs

Energy flux density Angular momentum

dyn/cma

kg.m/s

work

CGS unit Name

force

Conversion factor

i SI unlt/l

CGS

car-

dipole

Pm in the field

pro:IS

:--IS t

N:[pmB],W:--pmB)))

(Continued

Name

Blot

Savart's

and

law

\177t

Oaussian system

I[dl, r] 3

r] dB:ic I [dI, r3

Induction of the field produced

centre of a loop (c) in a solenoid in the

(b)

Definition

vector H

Maxwell's equations in inte\177

r \177

= B--

4\177J

Maxwell's equations tial

constant field

in differen-

form

Lorentzforce

F = q [vB]

law

Force of interaction of parallel currents Emf

Inductance Inductance solenoid of magnetic

Energy

= I [dl, B]

[vB]

--\177q

dF----/[dl,B]

field

producedby cur-

Hldl =

(in

\177-

bn)

-fi

V.D=4\177p V\303\227E=--\302\261fi C

V.B:O

-)c

n V \177

\177t,\177t

LI\177

Velocity

electromagnetic

,= c/V

v=t/Veo\177oe\177

wave in a medium

\177

t c

-----

BndS 0 \177

2Ill

i c

Relation between E and H in an

electromagnetic dt

wave

L= c(I)/I

L=q)/I the

\177ndS

2IlI\177

\177t\302\260

\177

DndS=4n S pdV

B:\177tH

\177o\177H

d(I)

induction

: f

\177

V\303\227n=i+fl

Relation between B and H

Ampere's

I 9dV

V.B=O

Relation between J and H Relation between and

Gaussian system

BndS 0 \177

Circulation of the vector H in a

\177t

E\177dl=-

2\177/\"

-\177

form

B= 4a nI

of tbe

DndS \177

current

Name

gral

direct

(a) by

(Concluded'

\177

4n\177

n*V

\177

Poynting

vector

S=[EII]

\177-

\177 [EH]

rent

density magnetic field

Energy

384

385)))

30. Fundamental Velocity of light Gravitational

Constants s m/s c=2.998.i0 -n 6.67.t0 { 6.67.t0-s cma/(g.s

vacuum

(Concluded)

m\177/(kg.s

constant

\177=

\177)

g = 9.807m/s i0 reel-1 NA = 6.023.

Free fall acceleration (standard value)

constant Molar volume of ideal gas at stp Loschmidt's number

Vo= 22.4 1/mol 2.69.t0\177 m-a no= { 2.69.t019 cm-a

8.3t4J/(K.mol) 0.082 1.atm/me|

B= 8.3t4.t07erg/mol

t.380.t0 J/K -16 erg/K t.380.t0 -\177a

constant

0\"965\"i0\177 xa

constant

Elementary

C/kg.equiv.

i0 CGSE/g.equiv. ( 2.90. -19 C t.602.t0 e= t0 4.803.

charge

CGSE

-1\302\260

0.9ti.i0kg 0.9tt.i0 g 0.5iiMn eV a\302\260

Electron rest mass

me=

Specificcharge of electron

-\1777

\177

i.76.i0C/kg

5.27-i017 CGSE/g i.672\"i0 kg -\1777

Proton rest mass

i.672.i0_\177a

rnp\177--

Stefan-Boltzmann constant Constant in Wien's displacementlaw

s C/kg 0.959.i0 [ 2.87.t0 mp -s CGSE/g o\1775.67.i0 a) cm.K b=0.29

Planck constant

Specificcharge of

._\177_e

proton

/\"

x\177

W/(m\177.K

.s i. 054. i.054.i0 i0-\177L -\1777

\177

erg.s

0.659.i0eV.s me i016 s-1 2.07. -1\177

\177

Rydberg

B _=

constant

...\177g-._=

t0 R' = R/2\177zc-- t .097.

\177

First Bohr radius

of energy Binding hydrogen atom Compton wavelength

electron of

Nuclear magneton

386

electron

em-x

t0 em rx = libCe = 0.529. eV E= me\1772Iiz= i3.56 :tc= h/rac---- 3.86t0-n em = i0-la cm re = e2/rnc2 2.82. = =2\177c 0.927.i0erg/G \177

Classicalelectron radius Bohr magneton

magnetic moment

Atomic mass unit Permittivity

of vacuum

-\177

-2\302\260

\177t\177v

eh = . \177 5.05 i0_2a erg/G

= 2.7928DN

9n------t.9t3 t.660.t0_\177 = 931.4 a.m.u. MeV 0.885. t0 -11 9-t09 t/4\177 .6H/m .257.t0 -----

\177

Universal gas constant

Faraday

Neutron

\177

Avogadro

Boltzmann

Proton magnetic moment

Permeability of vacuum

\177/4\177

= t0.7 H/m)))

\177

OTHERMIR PUBLISHERS'PHYSICS TITLES To the Reader Publishers welcome your comments on the book. and design of this content, translation We would also be pleased to receive any suggesMir

tions

you

care

Our address Mir

to make about

is:

Publishers, 2 Pervy

Rizhsky

GSP, Moscow, i29820, USSR

our

Optics A. MATVEEV,

D.Sc.(Phys.)

publications.

Pereulok,

This is the third volume in the seriesof textbooks written by the author for physics students at collegesand universities. The first two books (MolecularPhysicsand Electricity and Magnetism) were published in

1985and 1986. The material presentedin

this

book is developedon the basis of the

electromagnetic theory. Fourier transformations have beenused for monochromatic, nonmonochromatic and random radiation. The description of traditional aspectsis supplemented by a discussion of problems from the branches that have beendevelopedintensively over the last 15-20years.These include Fourier optics,Fourier analysis of random signals, matrix methods in geometric optics,holography, lasers,nonlinear phenomena, etc. Contents.Electromagnetic Waves. Nonmonochromatic and Random Radiation. Light Propagation in IsotropicMedia.GeometricOptics. Interference.Diffraction. Fourier Optics.Light Propagation in Anisotropic Media.Dispersion.Generation of Light. Nonlinear

Phenomena in

Optics.)))

Textbookof Elementary Physics. Vol. I.Mechanics,Heat, Molecular Physics Edited by

G. LANDSBERG,Mem. USSR

Acad. Sc.

English translation of the tenth revised and enlarged edition of oneof the most popular textbooks on physics in the Soviet Union. The distinguishing feature of this textbook is the detailed analysis of and phenomena occurring in the physical aspectsof the processes nature and in engineering. While preparing the present edition of the book,the authors have introduced the modern terminology and the latest system of units of measurement (SIunits). Intended as a textbook for lecturers and students of preparatory faculties, high schooland polytechnical students, as well as for those with a view to take entrance exams in interested in self-education physics at the university level. Contents.Mechanics:Kinematics. Dynamics. Statics.Work and Energy. Curvilinear Motion. Motion in Noninertial ReferenceSystems and Inertial Forces.Hydrostatics. Aerostatics. Hydro- and Aerodynamics. Heat and Molecular Physics:Thermal Expansion of Solidsand Liquids. Work, Heat and the Law of Energy Conservation.

This is the

Molecular Physics.Propertiesof Gases.Propertiesof Liquids. Properties of Solidsand Transition from the Solidto Liquid State of Aggregation. Elasticity and Strength. Propertiesof Vapours. Physics of Atmosphere. Heat Engines.

Textbookof Elementary Physics. Vol.

II.Electricity

Edited by

and Magnetism

G. LANDSBERG, Mem. USSR

Acad. Sc. This is the

English translation of the tenth revised and enlarged edition of one of the most popular textbooks on physics in the Soviet Union. The distinguishing feature of this book is that it contains a detailed and phenomena ocdescription of the physical aspectsof the processes curring in nature and in everyday life. While preparing the present edition of this book, the authors have introduced the modern terminology and the current system of units and measurements. Intended as a textbook for lecturers and students of preparatory faculties, high-school and polytechnical students, as well as for those interested in self-education with a view to take entrance examinations in physics at the university level. Contents.Electric Charges.Electric Field.Direct Electric Current. Thermal Effects of Current. Passageof Current Through Electrolytes. Chemical and Thermal Generators.Passageof Current Through Metals, Gasesand Semiconductors. Magnetic Phenomena.Magnetic Field. Magnetic Field of ElectricCurrents. Terrestrial Magnetic Field. ForcesActing on a Current-Carrying Conductorin a Magnetic Field. Electromagnetic Induction. Magnetic Propertiesof Bodies. Alternating Current. Generatorsand Electromagnets.)))

MENDELEEV'S PERIODIC I

]

Groupsof

[IV

III

]

Lithium

Elements

]

Aluminium

40.08

29 Cu Copper

H 1

]

Silver

196.967

Francium

[12231

112.40

Barium

79 Au Gold

Fr 10

a Praseodymium

t40.12

Hf 72 Ta

Lanthanum

137.34 138.91 SO Hg 81 T1

200.59 204.37 Ra 88 AC 89 kctinium

226.0254

Tantalum

178.49

180.948

207.19

[\177[Ku

Radium

140.907

Nd 60

104

Protactinium

232.038\177231.0359

Neptunium

238.03

237]

Bismuth

Manganese 54,9380

Co 27

Iron

Cobalt

55.847

58.9332

Krypton

Rh 45

Molybdenum

Ruthenium

Rhodium

95.94

[99J

101.07

Telluri um

[Pd 46 Palladium

102.905

106.4

Xe 54

Iodine

127.60

Xenon

126.9045 Re 75 Os 76 Rhenium

Tungsten

Osmium

186.2

Poloni um

190.2

131.30 77

[Pt

Iridium

Platinum

192.2

85 At

196.09

Rn 86 Rad on

Astatine

[209]

83.80

Technetium

58.71

Kr Ru 44

183.85

Nickel

79.904

Mo 42

Ni 28

Bromine

78.96

,52

39.948

35 Br

Selenium

208.980 105

LANTHANI

62 Eu 63 Gd 64 um Europium

**ACTINI

[244]

Gadoliniu_-n

151.96 157.25

150

Am Uranium

83 Bi

Argon

35.453

Mn 25

51.996

20.179

[2221

12101

\177

Pm 61 145]

Chromium

,', Kurchatoviurt [261]

1227]

NeodymiumlPromethiumtSar

144.24

121.75 73

Hafnium

82 Pb Lead

Thallium

Mercury

Antimony

118.69

rh 901Pa91 Fhorium

Arsenic

Tin

114.82 57

32.064

Ar 18

Chlorine

Sulphur

Neon

18.9984 17 C1

16 S

33 As

Niobium

* 8

50.942

Zirconium

Indium

Ba 56

Cesium

132.905

VII

47.90

Yttrium

Cadmium

Cs 55

Vanadium

Germanium

Gallium

Titanium

88.906 91.22 92.906 48 Cd 49 In 50 Sn 51 Sb

107.868

Ti 22

15.9994

30.9376

Fluorine

Oxygen

Phosphorus

28.086

1.00797

Helium 4.00260

Hydrogen

87.62

Strontium

85.47 7

65.38 69.72 72.59 74.9216 Y 39 38 Zr 40 Nb 41

Rubidium

44.956

Zinc

Rb 37

Scandium

30 Zn

63.546

Silicon

26.9815

Calcium

39.098

Carbon Nitrogen 10.811 12.01115 14.0067 13 AI 14 Si 15 P

Magnesiun 24.305

Potassium

VIII

VII

[

\177

Boron

Mg 12

Sodium

5 B

9.0122

22.9898

Beryllium

Nil,

III

6.94

Cerit

OF THE ELEMENTS

\177e

'\177/LE

243]

Lutetium

Dysprosium

158.925 62.50

DES Bk 97

Cf 98

Curium

Berkelium

Califo

247]

12471

[251]

95 Cm 96

Americium

DES Tb 65 Dy 66 Ho 67 [Er 68 Tm 69 70 Lu 71 IYb Terbium Holmium Erbium Thulium 164.930 167.26 I

]Es

rniumtEinsteinium

112.541

IYtterbium 168.934 1173.04

174.97

1Ol

]Fm 1OOlMd Fer mi /

j[257[

um/Mendeleviurn\177Nobelium)[Lawrenciur\177 [[2581)))

ELEMENTARY

PARTICLES

MASS

PARTICLE

SYMBOL*

LIFE

MEAN MeV

Electron Muon

\303\267

\177t-

Pions K

Kaons

{ Eta

K-

meson

Proton

Neutron

Lambda hyperon

Sigma hyperons

\177-

\177o

Xi hyperons

Omegahyperon

Eo \177-

* Symbols on the right-hand side denote

t\177

\177

.\177o

\177+

\177o

r;o \177+

antiparticles

0,511 105,66

139,6 135,0 493,8

0 1

206,77

-6 2,2 10

273,2

-8 2,55.10

\342\200\242

966,3

498,0 548,8

1074

938,26

1836,1

974,5

1/2

-1

1/2

-1

0 0 0

2.10 -16

264,2

SPIN

SPIN h

Photon

Neutrino

ISOTOPIC

CHARGES

2,4.10

1/2

\177

1/'2

+1/2

1/2

-1/2

+1 0

1/2

+1/2

1/2

-\177/2

1838,6

1.10

1115,4

2182,8

2,6\"10

1/2

1189,4

2328

0,8\"

1/2

23\1772

1,6.10

1/2

-1

1/2

-1 -1 -1 -1

1/2

-l

1/2

-1/2

-2

1/2

+1/2

3/2

-1

-2 -3

939,55

1197

1192

-3

-a\302\260

q\302\260

2333

1321

2585

1314

2572

1675

3278

-14

<10

-10

1,7.10 3'10-m \17710

-lo

MODE

m\177

0 0 +1

1 1

DECAY

\177

-19

0 0 0 0

1,23.10-8 -8 10-10_10

PRINCIPAL

no\1772y

K0-\1772n,

n \177p A\302\260--,p

e-+\177e

+n-

!z--n+ n\177

\302\260--*A0

+-lt\"

\177:-'-A0+

E0\"0\177A0+

f\177E

+ n,

A0+K

and corresponding antiparticles have the identical values of mass, mean life, spin, and isotopic spin T, while their electric Q, lepton L, and baryon B charges, projections of isotopic spin Tz, and strangeness S,have the values that are opposite in sign.)))

Note. Particles