Physics Tutor In New Jersey-AP Physics (C) 2022 ELECTRICITY & MAGNETISM Paper Solution

AP Physics C: Electricity and Magnetism Free-Response Questions Set 1 Answers FORMULA

r

E

One-I

E:

⑧

E.

Que-2

& = Do

Eds

=

9/to

E2

E*

=KA

> =</R

BeMF, e-dft

ADVANCED PLACEMENT PHYSICS C TABLE OF INFORMATION CONSTANTS AND CONVERSION FACTORS

Neutron mass, mn

27

kg

10

27

kg

9.11 10

31

1.67

Electron mass, me Avogadro’s number, N 0

1

8.31 J (mol K)

Boltzmann’s constant, k B

1.38

10

23

e

1.60

10

19

C

1 electron volt, 1 eV

1.60

10

19

J

Electron charge magnitude,

kg

10 23 mol

6.02

R

Universal gas constant,

10

1.67

Proton mass, m p

Speed of light, Universal gravitational constant, Acceleration due to gravity at Earth’s surface,

c

3.00

10 m s

G

6.67

10

11

1u

1.66

10

27

kg

931 MeV c 2

Planck’s constant,

h

6.63

10

34

Js

4.14

10

25

8.85 10

12

e0

Vacuum permittivity, Coulomb’s law constant, k Vacuum permeability, 1 atmosphere pressure,

PREFIXES Factor Prefix Symbol

mole, hertz, newton, pascal, joule,

1 4 pe0 m0

Magnetic constant, k

m kg s A K

kg 2

9.8 m s2

g

1 unified atomic mass unit,

meter, kilogram, second, ampere, kelvin,

N m2

J K

hc

UNIT SYMBOLS

8

m0 4 p

1 atm mol Hz N Pa J

1.99

Jm C

2

1.24 Nm

109 N m 2

9.0

7

(T m) A

1 10

7

(T m) A

1.0

105 N m 2

watt, coulomb, volt, ohm, henry,

W C V W H

15

eV s

3

10 eV nm

2

C2

10

4p

10

1.0

105 Pa

farad, tesla, degree Celsius, electron volt,

F T C eV

VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES q 0 30 37 45 53 60 90

109

giga

G

sin q

0

12

35

2 2

45

3 2

1

106

mega

M

cos q

1

3 2

45

2 2

35

12

0

103

kilo

k

tan q

0

3 3

34

1

43

3

10 2

centi

c

10 3

milli

m

6

micro

m

10 9

nano

n

12

pico

p

10

10

The following assumptions are used in this exam.

I. The frame of reference of any problem is inertial unless otherwise stated.

II. The direction of current is the direction in which positive charges would drift.

III. The electric potential is zero at an infinite distance from an isolated point charge.

IV. All batteries and meters are ideal unless otherwise stated.

V. Edge effects for the electric field of a parallel plate capacitor are negligible unless otherwise stated.

ADVANCED PLACEMENT PHYSICS C EQUATIONS MECHANICS

Ãx

x

ax t

Ãx 0

x0

1 2 at 2 x

Ãx 0 t

Ãx2

Ãx 0 2

2ax x

a

F m

Fnet m

F

dp dt

J

p

F dt

m FN

DE

W

K

1 2 mà 2

P

dE dt

P

F v

DUg

Ã2 r

r

F t net I

I 2

I

r dm

mr

Us

1 k Dx 2

x

xmax cos( wt

T

2p w

p

Iw

1 2 Iw 2

w

w0

at

q

q0

w0 t

2p

Tp

2p

UG

1 2 at 2

1 f

Ts

FG

2

FE q

E

DV

1 4pe0

qV

DV

Q C

Cp

m k

UC

r2

Gm1m2 r

N =

qi ri

i

P Q q R r t U

= = = = = = = V= v = r = F = k =

1 q1q2 4pe0 r

k e0 A d i

= = = = = = = = = = = =

Ci

1 C i i

area magnetic field capacitance distance electric field emf force current current density inductance length number of loops of wire per unit length number of charge carriers per unit volume power charge point charge resistance radius or distance time potential or stored energy electric potential velocity or speed resistivity flux dielectric constant

FM

qv

B d

dQ dt

1 QDV 2

1 C DV 2

2

B m0 I

dB

m0 I d r 4p r 2

R

r A

F

I d

E

rJ

Bs

m0 nI

I

Nevd A

FB

I

DV R

e

E d

e

L

UL

1 2 LI 2

g

Gm1m2

n

E dr

UE

C

F I J L

dV dx

Ex

V

e

Q e0

E dA

I f

A B C d E

1 q1q2 4pe0 r 2

FE

1 Cs

kDx

rw r

acceleration energy force frequency height rotational inertia impulse kinetic energy spring constant length angular momentum mass power momentum radius or distance period time potential energy velocity or speed work done on a system position coefficient of friction angle torque angular speed angular acceleration phase angle

Fs

2

mi xi mi

x cm

K

w2r

t

a

L

F dr

mg Dh

ac

Ã

Dp

mv

Ff

t

x0

a = E= F = f = h = I = J = K = k = = L = m= P = p = r = T = t = U= v = W= x = m = q = t = w = a = f =

ELECTRICITY AND MAGNETISM

Rs 1 Rp

P

Ri

i

i

I DV

1 Ri

B

B dA

dI dt

d FB dt

ADVANCED PLACEMENT PHYSICS C EQUATIONS GEOMETRY AND TRIGONOMETRY Rectangle A bh Triangle

A

1 bh 2

Circle

A

pr2

C

2p r

s rq Rectangular Solid V wh Cylinder 2

V

pr 

S

2p r 

CALCULUS

A = area C = circumference V = volume S = surface area b = base h = height  = length w = width r = radius s = arc length q = angle

df dx

d f du du dx

d n x dx

nx n

d ax e dx

aeax 1 x

d ln ax dx d sin ax dx

a cos ax

d cos ax dx

s

x n dx

r

q 2p r 2

eax dx

Sphere

dx

a sin ax 1

n

ln x

V

x

S

4p r 2

cos ax dx sin ax dx

Right Triangle

b2

c2

sin q

a c

cos q

b c

tan q

a b

a

c q

a 90°

b

xn 1, n

1

1 ax e a

4 3 pr 3

a2

1

a 1 sin ax a 1 cos ax a

VECTOR PRODUCTS   A B AB cos q   A B AB sin q

1

PHYSICS C: ELECTRICITY AND MAGNETISM SECTION II Time—45 minutes 3 Questions Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions, which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in this booklet in the spaces provided after each part.

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1. A nonconducting sphere of uniform volume charge density is surrounded by a thin concentric conducting spherical shell, as shown in the cutout view. The sphere has a charge of −Q and the shell has a charge of +3Q. The radii of the inner sphere and spherical shell are R and 4R, respectively, as shown in the cross-section view. (a) Determine the charge on the outer surface of the shell.

-

3

+

ENet: Router

+

ginner

-

O ->

Monter=Ewet-Einner 30-0

=>

=

2@

↳

@ out ev

inner

(b) Using Gauss’s law, derive an expression for the electric field a distance r from the center of the sphere for r < R. Express your answer in terms of Q , R, r, and physical constants, as appropriate.

Apply gauss

the over

gEds= E.

I

GMU"-e

aDf=Us #=

FRs

to

-

(9).

-

-R-

ppFrs FT

-O-

--Us b

UC R

NERGY PGNSITY

->

REMAIN

-

WILL

constant

(c) The magnitude of the electric field at r = R is 8N / C. Calculate the value of the electric field at r = 2R. E=

afr E

=>

ECYou

FRC=4

&

Ef

2N/c

(d) Derive an expression for the absolute value of the potential difference bet ween the outer surface of the sphere and the inner surface of the shell. Express your answer in terms of Q , R, and physical constants, as appropriate.

voafco-ORtitoOn No-Vi

to

=

-

+ fat-fr]

3 -

G R-

Vc

-

8(K) (1-1) ER(E)- TOR =

Q

-

405-

Vo

(e) i. On the following axes that include regions I, II, and III, sketch a graph of the electric field E as a function of the distance r from the center of the sphere.

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UK o

E102) continuous increase

Linearly

⑮> R

QLV <4R R and

at

approaches

towards

200

v=4R, maximum and

decreasing

ii. On the following axes that include regions I, II, and III, sketch a graph of the electric potential V as a function of the distance r from the center of the sphere.

O-4R

-

always

increasing

decreasing 8>4 R

continuous

graph

across all the three

region.

Begin your response to QUESTION 2 on this page.

2. The plates of a certain variable capacitor have an adjustable area. An experiment is performed to study the potential difference across the capacitor as it discharges through a resistor. A circuit is to be constructed with the following available equipment: a single ideal battery of potential difference *Ve ∆V0, a single voltmeter, a single resistor of resistance R, a single uncharged variable capacitor set to capacitance C, and one or more switches as needed.

(a) Using the symbols shown, draw a schematic diagram of a circuit that can charge the capacitor and may also be used to study the potential difference across the capacitor as it discharges through the resistor.

Y

3R

The capacitor is fully charged by the battery. At time t = 0, the capacitor starts discharging through the resistor. −t

Rs

(b) Show that the potential differenceAVc ∆V across the capacitor as a function of time t is * ∆V (t) = ∆V e RC as CC0 the capacitor discharges.

V-IR

&

=

=

0

IR

8-CR) => Doe"RC

Q:

&

-eYRC

18-00 (E) Rc AVCK)

=

=

O

0

(o) 8X

*Vect) vo

goto logfor to -

RC

c

proved

=

O.etc

c) The experiment is performed using a resistor of R = 150 kΩ. Data for the potential differenceC ∆VC across the below.

of

slope In

(Fri) -

On

·P P2

graph

_m=f=

(+0.24,-0.32), Pn

A= -

the

lof

-

I

=

13 3

072 + 0.32 -

-

0.72).

0.54-0.24

(1)

I

-

t

=

(0.54

1338

R

D

1

A.TT) 7150x18

=

x

=>

5. ON F

133

(d) The capacitor is adjusted so that the surface area of the plates is increased, and the experiment is repeated. Would the slope of the best-fit line in the second experiment be more steep, less steep, or unchanged compared to the slope of the best-fit line in part (c)? g Less steep ___ Unchanged Briefly justify your answer. ___ More steep ___ S= When

the

I

Hence 3

A A

is

Increased

will also

increase

slope will

be less

H22p

(e) The ideal battery is then replaced with a non-ideal battery with internal resistance r, and the experiment is repeated. vNo i. Would the slope of the graph in this final experiment change compared to the graph in part (c)? ___ Yes ___ Briefly justify your answer.

Capacitor desistor

is

R

still

hence

discharge through

the slope will

remain

same.

ii. Would the vertical intercept of the graph in this final experiment change compared to the graph in part (c)? ___ Yes ___ No~ Briefly justify your answer.

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chaye,

because

resistance

affect

of

the

across the

the

the

internal

battery

potential

changing

does not

difference

capacity.

3. A lightbulb of resistance R = 10.0 Ω is connected to a rectangular loop of wire of negligible resistance near a very long current-carrying wire. The rectangular loop has a length L = 4.0 cm and a width W = 2.0 cm and is positioned so one of the longer sides of the loop is a distance d = 1.0 cm above and parallel to the long wire, as shown. The current in the long wire is initially flowing to the right and is given by= I(t) = C −Dt Dt, where C = 10.0 A and D = 2.0 A/s. At time t = 5.0 s, the current in the long wire is instantaneously zero as the current changes direction. (a) What is the direction, if any, of the magnetic field produced by the induced current in the rectangular loop as the current in the long wire changes direction?

W Out of the page ___ No direction, because the field is zero Justify your answer. ___ Into the page ___ Direction

of

the

current reversed

magnetic also

flux

inside the

hence

loop

decreases then a per lens Law the induced current produced In such

way that It will produce magnetic field into the loop. (b) Calculate the magnetic flux through the loop due to only the long wire at time t = 3.0 s.

So

a

FCt=ty-b(t)

(a0 S =

=

$(2 d5.=

&

1072(3)

=

4 A

/ lerTar S195=A.2I(K If

Reth /loge (th)) 4xx102x9x.04 lofe (-3) =3-52x158+m &

-

-

(c) Calculate the current through the lightbulb at time t = 3.0 s. e.

danMor(spt)

=>

->

-ML

lose

-ModL F

=

=

9x18X2X.o 2 TX

=18 x

10

189

A

lose

(

(W) f

<c-pt).

loge (w)

lose (vi)

(d) A group of students attempts to experimentally verify whether the current through the lightbulb is consistent with the current calculation from part (c). The current in the rectangular loop is measured to be greater than the current calculated in part (c). Which of the following could explain this discrepancy?

Select one answer. ___ The students did not account for Earth’s magnetic field. ___ The rectangular loop is tilted and is not in the same plane as the wire. ___ The resistance of the lightbulb is greater than the recorded value. ___ The long side of the rectangular loop is shorter than the recorded value. ___ The current in the long wire changes at a faster rate than expected.

Briefly justify your answer.

If

the

shayes will be

fence, and

current in the

wive

faster rate, there greater change of magnetic

at

so

the induced

induced

emf

will

current

be

larger

(e) Later, the same rectangular loop with lightbulb is rotated such that a short side of the loop is 1.0 cm above and parallel to the long current-carrying wire, as shown. The current in the wire is again initially flowing from left to right and given by I(t) = C − Dt, where C = 10.0 A and D = 2.0 A/s. The current through the lightbulb in the loop’s new orientation at time t = 3.0s is #z I2. Which of the following correctly relates the current Miz I2 to to I1, the current through the lightbulb in part (c)? ___I <I ___I =I ___I >I F12 212121 > In => I New orientation Justify your answer.

E some part of recradle *

=

further from the away staight wire, which means are

the

magnetic field

through

recrayle will

be

less,

rate

of

fence

of

change

resulting

the

hence decreases

smaller current

AP® Physics C: Electricity and Magnetism 2022 Free-Response Questions

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