Electromagnetic damping

ELECTROMAGNETIC DAMPING Introduction In this unit we are going to introduce you to an electrical instrument called galvanometer. You will learn in detail about it and you will be able to explore many interesting features about its construction and working.

Galvanometers are instruments which are intended primarily to indicate the existence of a current – steady or transient, in a circuit. These may be capable of measuring the current, only under certain circumstances, when calibrated.

The construction of galvanometers is based on the interaction between coils carrying current and magnets. These may be in general divided into two types as given below according to as the moving part is a magnet or a coil.

Moving-magnet galvanometer: In this type of galvanometer, the current is passed through a fixed coil which produces a magnetic field under which a magnet moves.

Moving-coil galvanometer: In this case the current is passed through a movable coil placed between the poles of a powerful magnet. The coil swings under the action of a deflecting couple acting upon it. We will discuss this particular type in detail here.

Moving coil galvanometers can be further classified as: Weston galvanometer or pivoted-coil galvanometer In this galvanometer, the coil swings between two pivots and the deflection of the coil is read by a pointer attached to it which moves on a circular scale. This type of a galvanometer is less sensitive but more convenient. These are extensively used in various bridge type circuits working with direct current, for obtaining null-point. You might have seen or worked with such a galvanometer, in particular, while performing an experiment with Carey Foster’s Bridge.

D’ Arsonval galvanometer or suspended-coil galvanometer Here the coil is suspended freely and the deflection of the coil is read using a mirror attached to it. You will learn about this type of galvanometer in detail. It shows very distinct and interesting behavior as a dead-beat galvanometer, a critically damped galvanometer and a ballistic galvanometer, under different circuit conditions. You will find the details about these types of behavior under the theory of this experiment.

Let us now look into the details of construction of this type of galvanometer. Figure 1 shows the sketch of D’ Arsonval galvanometer or suspended-coil galvanometer It consists of a narrow rectangular coil C of many turns of fine insulated copper wire suspended so as to move freely in a narrow annular space between the pole- pieces of a

powerful horse-shoe magnet NS. The pole-pieces of the magnet are hollowed out to make them cylindrically concave in shape. This makes the magnetic field radial, i.e. the field is always parallel to the plane of the coil. Within the coil is fixed a cylinder of soft iron which serves to concentrate the lines of force in the gap, thus making the field in the gap strong and practically uniform. The coil is wound on a light frame-work which is of metal (gener ally To binding alumi R To binding screw num) screw or of noncondu M cting materi al (bamb oo, ivory N S or C ebonit e); accord K ing to as a deadM-Mirror NSNS-Electromagnet beat C-Coil R-Rigid support or a ballist K-Spring ic action is requir ed.

Figure 1: Construction of suspended-coil galvanometer The suspension is a fine rectangular strip of phosphor-bronze, the upper end of which is attached to a rigid support R (also known as movable torsion head). It also serves as one of the current leads to the coil. The rectangular strip has very small torsional rigidity and hence the twist for a given couple is much greater and the stress on the material much less than with a wire of circular cross-section. It also offers a comparatively larger area for the dissipation of heat produced by the current. Connection to the other end of the coil is made by means of a very light metal spring K attached to the bottom of the coil. A small concave mirror M is attached to the top of the coil. This mirror generally has a radius of curvature equal to1 m. The whole arrangement is enclosed in a non-metallic case which is provided with a glass window in the front and leveling screws at the base.

A spirit level is also generally provided at the bottom rim to ensure the vertical orientation of the coil. The torsion head is connected to a terminal T1 of the galvanometer and the spring K is connected to the other terminal T2, through which the galvanometer can be connected in the circuit. A lock/unlock knob for the suspension is provided at the back of the galvanometer. A lamp and scale arrangement (Figure2) is used in front of the galvanometer mirror to record the deflection of the coil of the galvanometer.

M

S

L

L-Lamp S-Scale M-Mirror

Figure 2: Lamp and Scale Arrangement The suspended-coil type moving coil galvanometer which you are supposed to use in the lab, is known as the ‘ballistic galvanometer’ (B.G.). It is a specially designed moving coil galvanometer for measuring the total quantity of electricity or charge that passes through it due to a transient current or voltage of short duration, such as during charging and discharging of capacitor and in electromagnetic induction. When such a transient current passes through the galvanometer, its coil experiences an angular impulse and begins to perform angular oscillations about its axis. The maximum deflection (or throw) obtained is proportional to the total charge which passes through the galvanometer. As you will see in the theory, it can also work as a critically damped or a dead-beat galvanometer depending upon the circuit parameters.

For the purpose of making a moving coil galvanometer ballistic, i.e., to measure charge which passes for a short duration, its coil is required to get deflected under the action of an impulse imparted by a momentary current. Hence, the coil has to be light.

Torsion-head

Mirror Coil

Spirit level

Leveling screws

Figure 3: Moving coil ballistic galvanometer Further, the motion of the coil should not be damped. The coil is therefore wound on a light non-metallic frame made of bamboo or ebonite, to remove the damping effect of eddy currents in the metallic frame.

In order to measure the total charge, the whole transient current should have passed through it before the coil moves appreciably from its position of rest. Hence, the time period of oscillation of the coil should be fairly large. This is achieved by making Ι (moment of inertia of the coil) large and c (torsional constant, i.e., torsional couple per unit twist of the suspension fiber) small (as you will see from Equation. (4) in the theory).

For this reason, the suspension is made of thin rectangular strip of phosphor-bronze whose torsional constant is small.

As you have gathered a brief idea about the main instrument, ballistic galvanometer, we will now discuss in detail, the objective, working circuits and the theory underlying this experiment.

K1 R' BG

E

K'

Figure 4: Circuit diagram - A

Figure 5: Circuit Diagram- B

Apparatus (Description of the circuits) • • • • • • • • •

B.G. - Ballistic Galvanometer K1 and K - Tapping keys K - Plug Key R ,R1 and R2 - Resistance boxes with resistances upto10ΚΩ r - Resistance box with resistances upto 100Ω E –2V Lead accumulator Lamp and scale arrangement Stop watch Connecting wires

Theory With reference to Figure 1, let N be the number of turns of the coil, A, the area of the coil and B, the magnitude of uniform magnetic induction as produced by the cylindrical pole pieces of the horse-shoe magnet.

The passage of charge through the galvanometer is equivalent to the passage of a varying current for a certain short interval of time. Let i be the current in the coil at any instant.

The coil experiences a couple due to the horizontal and oppositely directed equal forces acting on the vertical sides. The moment of this couple or the torque acting on the coil is given by τ = NiBA .

(1) The torque acting for an infinitesimal time dt, gives the coil an angular impulse which is (torque × time) equal to NiBA dt . Therefore, if t be the total time for which the current flows through the coil, the total angular impulse given to the coil is t

= ∫ NBAi dt = NBAq 0

t

since

∫ i dt = q ,

the charge that has passed through the galvanometer. This impulse

0

produces angular momentum in the coil which rotates until the restoring couple of the suspension brings it to rest. The coil then swings back to its rest position as the suspension unwinds, and due to its inertia overshoots and twists the suspension in the opposite direction. A series of back and forth angular oscillations thus

results. The amplitude of oscillation keeps decreasing with time due to damping forces present.

Let us assume that the coil begins to oscillate after the impulse is over. Let ω be the angular velocity at start and Ι its moment of inertia about the axis of suspension. Then the angular momentum produced in the coil due to the angular impulse is Ιω , i.e. Ι ω = NBAq (2) 1 Now, the coil possesses a kinetic energy Ιω 2 at start. If damping is absent, then this 2 energy is entirely used for doing work in twisting the suspension. If c be the restoring couple per unit twist in the suspension, then the couple for a twist θ is cθ. The work done for an additional twist dθ is cθdθ. If the maximum twist is θ0, then the work required is θ0 1 = ∫ cθ .dθ = cθ 02 2 0 Equating it to the kinetic energy of the coil at start, we get 1 2 1 2 Ιω = cθ 0 2 2 2 Ιω = cθ 02 or (3) Now, if T be the time period of oscillation of the coil, we have Ι T = 2π (4) c T 2c Ι= or 4π 2 Multiplying Equation. (3) and Equation. (4), we get T 2 c 2θ 02 2 2 Ι ω = 4π 2 Tcθ 0 Ιω = or (5) 2π Comparing Equation (2) and Equation (5), we get Tcθ 0 NBAq = 2π T c or (6) q= . .θ 0 2π NBA q = k .θ 0 , (7) or T c . is a constant for the galvanometer, known as the ‘ballistic where k = 2π NBA constant’.

It is important to rewrite Equation (7) as q T c =k= .

Qs = 2π NBA θ0

where Qs is known as the ‘charge sensitivity’ of the ballistic galvanometer. Thus, charge sensitivity is the same as the ballistic constant.

Hence, Equation (7) clearly shows that the first throw of a ballistic galvanometer is a measure of the charge that has passed through it.

Equation of Motion of the moving system in Ballistic Galvanometer

The various couples acting on the moving system are as follows: (a) The deflecting couple, NiBA due to the transient current i flowing for a short time τ. d 2θ (b) The restoring couple, Ι 2 due to the inertia of the coil, where Ι is the moment dt of inertia of the coil. (c) The retarding couple due to damping has two components (i) due to air resistance and (ii) due to electromagnetic damping. The component due to air resistance is dθ proportional to angular velocity and can be taken as equal to b , where b is a dt constant. The component due to electromagnetic damping can be calculated as follows: The e.m.f. induced in the coil, dθ dθ =G , e = NBA dt dt where G = NBA . If R is the resistance of the galvanometer circuit, then the current due to this induced e.m.f. is e G dθ I' = = .

R R dt Couple on the coil due to this induced current is G dθ G 2 dθ ( NBA) I ' = NBA. . = .

R dt R dt The total retarding couple is then ⎛ G 2 ⎞ dθ dθ ⎜⎜ b + ⎟⎟. =δ R ⎠ dt dt ⎝ The torsional couple, cθ, due to the torsional twist of the suspension fiber. Here, c (d) is the torsional couple per unit twist of the suspension fiber.

The equation of motion may be obtained using the condition that under equilibrium, the deflecting couple is equal to the sum of the restoring couple, retarding couple and torsional couple. Hence, d 2θ dθ Ι 2 +δ + cθ = NBAi (8) dt dt We must consider the equation for two intervals: 0 < t < τ and τ < t < ∞. During the first of these intervals, the coil acquires momentum due to impulse given to it by transient current. It, however, is hardly deflected from its initial position because of the stipulation

that the transient current flows during a very small time τ in comparison with the free period T of the instrument, i.e., τ / T << 1.

Considering the motion of the coil for t >> τ; now i = 0, so that Equation (8) takes the form d 2θ dθ Ι 2 +δ + cθ = 0 dt dt d 2θ dθ or + 2 p. + q 2θ = 0 , (9) 2 dt dt δ 1 ⎡ ( NBA) 2 ⎤ where (10) 2 p = = ⎢b + ⎥ Ι Ι⎣ R ⎦ c (11) Ι Equation (9) represents a damped harmonic motion. We can solve this equation by assuming a solution of the form q2 =

and

θ = Aeα t where the parameter α is, in general, a complex number. Substituting this assumed solution back into the differential Equation (9), we get (12) α 2 + 2 pα + q 2 = 0 This quadratic equation has two roots

α = − p ± p2 − q2

(13)

The complete solution for θ is, therefore

θ = A1 e

⎛⎜ − p + p 2 − q 2 ⎞⎟ t ⎝ ⎠

+ A2 e

⎛⎜ − p − p 2 − q 2 ⎞⎟ t ⎝ ⎠

(14) The values of A1 and A2 can be calculated from the initial conditions, i.e., θ = 0 and dθ = 0 at t = 0. The exact physical nature of this solution depends on the relative values dt of p and q. In particular, the qualitative behavior of the system depends on whether the quadratic Equation (12) for α has one real solution, two real solutions or two complex conjugate solutions.

Conditions when a Moving Coil Galvanometer is Dead-beat or Ballistic Case I: Over- damping or Dead-beat*

When p 2 > q 2 or p > q , i.e., 2 ( 1 ⎡ NBA ) ⎤ ⎥> ⎢b + 2Ι ⎣ R ⎦

c Ι

R<

or

(NBA)2

4cΙ − b R < RCD , or where, the critical damping resistance, 2 ( NBA) RCD = 4cΙ − b

(15)

In this case, p 2 − q 2 is less than p and, therefore, powers of exponential in both the terms in Equation (14) will be real, say -α1 and -α2 and Equation (14) will now take the form,

θ = A1e −α t + A2 e −α t 1

2

(16) Both the terms are exponentially decreasing functions of time. Hence, the deflection θ will go on decreasing with time and the motion will be dead-beat or non-oscillatory. (refer to Figure 6) Case II: Critical damping

A critically damped system converges to zero faster than any other, without oscillating.

This condition is achieved when p = q; so that α has only one real value (α = -p) and the solution becomes

θ = Ae − pt The damping of a galvanometer can be varied readily by altering the total resistance R of the closed circuit in which it is used (Refer to Equation (10) where all other parameters except R are constant for a given galvanometer).

Critical damping is obtained by choosing R so that, when the galvanometer is deflected, the spot of light or pointer comes to rest in the minimum time without over-swing. The motion of the coil is just non-oscillatory and the frequency of oscillation is just zero. This is achieved when 2 ( NBA) R = RCD = 4cΙ − b Since all the terms which appear on the right hand side of the above equation are design constants of the suspension and the coil and

R = R1 + RG ,

(17) where R1 is the external circuit resistance and RG, the galvanometer resistance. Critical damping for a particular galvanometer can be obtained by varying R1 (see Figures 5 and 6).

Case III: Under- damping or ballistic

In this case, p < q.

Thus

p 2 − q 2 is complex and we can write it as jω, where

j = − 1 and ω = q 2 − p 2 Then Equation (14) can be written as ( − p + jω )t ( − p − jω )t

θ = A1e

[

+ A2 e

= e − pt A1e jωt + A2 e − jωt

(18)

]

θ being a real quantity, A1 and A2 must be complex conjugates so that the solution for θ can be written as θ = Ae − pt cos(ωt + β ) (19) dθ The constants A and β are determined from the initial conditions, i.e., θ = 0 and = ω0 dt at t = 0. Applying these conditions to Equation (19), we get A cos β = 0, (20) − pA cos β − ωA sin β = ω 0

− ωA sin β = ω 0

or

A Critical damping Ae-pt

Over damping Under damping 0

time(t)

Figure 6: Different cases of damping

From first condition of Equation (20), we find that cos β = 0 since A ≠ 0 ω π 3π 3π or β = , ,L . Also, because sin β is a negative quantity, β = and A = 0 . Hence, ω 2 2 2 Equation (19) becomes

⎛ ⎝

θ = Ae − pt cos⎜ ωt +

3π ⎞ ⎟ 2 ⎠

= Ae − pt sin ωt =

ω0 − pt e sin ωt ω

(21)

Equation (21) shows that the behavior of the system is now damped harmonic and the system will oscillate at the frequency ω with the exponentially decreasing amplitude given by Ae

− pt

. The time period T of these oscillations is given by 2π 2π T= = ω (q 2 − p 2 )12

(22)

We have found in this special case that for the galvanometer to be ballistic, p should be as small as possible. For this, according to Equation (10): The moment of inertia of the coil should be large. (a) The air damping should be small. (b) The electromagnetic damping should be small. Thus, the coil should be wound on (c) a non-conducting frame.

The resistance R of the circuit, including that of the coil should be large. For this, (d) the coil should have large number of turns.

* (At this stage, it should be clear to you that for the galvanometer to be dead-beat, just the opposite of the above mentioned conditions need to be satisfied).

In the ideal case, p should tend to zero. Under this condition, the second or the middle term in Equation (9) will be absent and motion will be undamped and simple harmonic with constant amplitude A and angular frequency ω = q (as given by Equation (11)). In actual practice, this case can be treated as equivalent to the working condition when the coil oscillates freely with R = ∞. This will be done in the lab, under the study of oscillatory behavior of the B.G., using circuit diagram – A. Correction for Damping in Ballistic Galvanometer

In deducing the relation between charge q and ballistic throw θ 0 (Equation (6)), it has been assumed that the damping is entirely absent and whole of the kinetic energy of the coil is used for twisting the suspension through an angle θ 0 . Actually, it is not so. The motion of the coil is damped due to viscosity of air and also due to opposing current induced in the coil and frame of the coil, which rotates in the field of permanent magnet of the galvanometer. Therefore, the coil oscillates with decreasing amplitude. Hence, the observed throw is smaller than its true value θ 0 , which would have been observed if the damping were entirely absent. A correction is therefore required.

Let θ1 ,θ 2 ,θ 3 , K be the successive throws observed at the end of first, second, third, … swings of the coil. θ1 ,θ 3 ,K are on one side of the rest position of the coil and θ 2 , θ 4 ,K on the other.

For Equation (21), the amplitude of motion is Ae amplitude θ 0 = A .

T π 2 For t = = , we have amplitude θ1 = Ae 4 2 q2 − p2 For t =

− pt

and for t = 0, we have

− pπ

3T 3π 2 θ = Ae = , we have amplitude 2 4 2 q2 − p2

5T 5π 2 = , we have amplitude θ 3 = Ae For t = 2 2 4 2 q −p

q2 − p2

−3 pπ q2 − p2

−5 pπ q2 − p2

and so on.

+ pπ

+ pπ θ1 θ 2 θ 3 q −p = = =L= e = e ω = eλ = d θ2 θ3 θ4 pπ T λ = log e d = = p. ω 2 2

Therefore,

or

2

(23)

(24)

Here, it is seen that the ratio of successive amplitudes θ1 ,θ 2 ,θ 3 , K , etc. is constant. This

log e d = λ

constant ratio d is called decrement and decrement.

is called the logarithmic

We have seen that the observed throw would have been θ 0 = A if there were no damping. The first throw θ1 is observed a quarter period (i.e., T/4) later and during all this time, the damping effect has taken place, so that − pπ

θ1 = Ae = Ae

2 q2 − p2 −λ

2

Hence, 2

θ0 λ 2 λ 1 ⎛λ ⎞ = e =1 + + ⎜ ⎟ +L θ1 2 1× 2 ⎝ 2 ⎠ λ = 1+

(approximately) 2 neglecting the higher powers of λ, which are very small in the case of ballistic galvanometer.

Therefore,

⎛ ⎝

θ 0 = θ 1 ⎜1 +

λ⎞ ⎟ 2⎠

(25)

Equation (6) can now be written as q=

cT ⎛ λ⎞ .θ1 ⎜1 + ⎟ 2π NBA ⎝ 2⎠

(26)

Experimental determination of the logarithmic decrement

The ballistic galvanometer coil is made to oscillate by passing a discharge through it and the successive amplitudes on the right and left are observed. Then

θ θ θ1 θ 2 = = L = 9 = 10 = d = e λ . θ2 θ3 θ 10 θ11

Multiplying all these factors, we have

θ1 = e10 λ θ11

θ θ 1 2.3026 log e 1 = log10 1 (27) 10 10 θ11 θ11 Alternatively, if throws on the same side are observed, i.e., if θ1 ,θ 3 ,θ 5 ,K are observed, then or

or

λ=

θ1 θ1 θ 2 = × = e 2λ = d 2 θ3 θ 2 θ3 θ θ 1 2.3026 λ = log e 1 = log10 1 2 2 θ3 θ3

As d2 =

θ1 θ 3 θ 5 = = = L, θ3 θ5 θ7

d2 =

θ1 + θ 3 + θ 5 + L θ3 + θ5 + θ7 + L

we can write

Hence

⎡θ + θ + θ + L ⎤ 1 λ = log e ⎢ 1 3 5 ⎥ 2 ⎣θ 3 + θ 5 + θ 7 + L ⎦

(28)

Thus, all the observed values of θn, or as many observed values of θn can be used to obtain the corresponding λ.

Q-factor

The efficiency of an oscillator is measured in terms of a quantity called Quality factor or Q-factor. It is a dimensionless parameter that compares the time constant for decay of the amplitude of an oscillating physical system to its oscillation period. Equivalently, it compares the frequency at which a system oscillates to the rate at which it dissipates its energy.

It is defined as 2π times the ratio of energy stored to the energy dissipated per cycle, i.e.

Energy stored Q = 2π × (29) Energy dissipated per cycle

Energy stored Power loss × T Energy stored =ω× , Power loss where T is the time period of oscillation and ω is the angular frequency. = 2π ×

(30)

A higher Q indicates a lower rate of energy dissipation relative to the oscillation frequency, so the oscillations die out more slowly.

Calculation of Q

The energy of a damped oscillator is spent in overcoming the frictional forces and hence, decreases continuously with time. Consequently, the amplitude of oscillation decreases.

The kinetic energy of the oscillator, at any time t, is given by 1 ⎛ dθ ⎞ K .E. = Ι⎜ ⎟ 2 ⎝ dt ⎠

2

(31)

where

θ = Ae − pt sin ωt

(21)

Hence, the time average value of kinetic energy per cycle is given by T 1 < K .E. >= ∫ K .E. T 0 1 2 2 − 2 pt Ιω A e (32) 4 where we have assumed low damping so that the time-dependent amplitude A(t ) = Ae − pt is regarded as constant over one cycle; hence taken out from integral.

Similarly, the average value of potential energy is given by 1 < U >= Ιω 2 A 2 e − 2 pt (33) 4 (since, the average values of kinetic energy and potential energy for a harmonic oscillator are the same). =

Average total energy is, therefore, < E >=

(

1 2 Ιω Ae − pt 2

)

2

(34)

The average power dissipation is d <E> dt = Ιω 2 A 2 pe −2 pt = 2p < E >

< P >= −

Therefore,

(35)

<E> <P> <E> =ω× 2p < E >

Q =ω×

=

ω 2p =

π pT

(36)

This expression can be expressed in terms of the logarithmic decrement λ using Equation (24) as

Q=

π 2λ

(37)

Also we may designate 2 p = Δω , i.e., resonance band width. So that Q=

ω 2p

=

ω Δω

(38)

Learning Outcomes While performing this experiment, you will be able to 1. Observe the oscillatory nature of motion of the coil of the galvanometer under minimum damping.

2. Time the period of oscillations (T) of the galvanometer coil.

3. Convert the oscillatory motion into dead-beat one.

4. Obtain the galvanometer resistance (RG).

5. Obtain the logarithmic decrement (λ) and Quality Factor (Q) for various stages of damping.

6. Investigate the dependence of logarithmic decrement (λ) and Quality Factor (Q) on the galvanometer circuit resistance (R).

7. Find out the Critical Damping Resistance (C.D.R.) of the galvanometer.

Pre-lab Assessment Answer the following questions

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

How does a ballistic galvanometer differ from a dead-beat one?

What are the different types of moving coil galvanometer?

What do you understand by logarithmic decrement?

Why and how do you determine logarithmic decrement?

What is electromagnetic damping and how can it be reduced in a B.G.?

How can the time-period of the suspended coil be increased?

How are the unwanted oscillations of the suspension of B.G. stopped?

How can you increase the sensitivity of the B.G.?

What is meant by â&#x20AC;&#x2DC;ballistic constantâ&#x20AC;&#x2122;?

How can the couple per unit twist of the suspension fiber be kept low?

What do you understand by E.M.F. of a cell?

Why secondary cells are generally preferred over primary cells?

Choose the correct answer

(1)

(2)

(3)

(4)

(5)

The coil of a suspended coil galvanometer has very high resistance. When a momentary current is passed through the coil, it (a) oscillates with decreasing amplitude (b) oscillates with the same amplitude (c) gets deflected and comes to rest slowly (d) shows steady deflection.

What does a ballistic galvanometer do? (a) It measures current. (b) It measures voltage. (c) It detects current. (d) It measures magnetic field.

In a moving coil dead-beat galvanometer, the coil is wound on an aluminum frame because (a) aluminum is a heavy metal (b) no eddy currents are developed in aluminum (c) on stopping the current in the galvanometer, the coil comes to rest rapidly (d) it increases the sensitivity of the galvanometer.

To make the field radial between the pole pieces of a magnet (a) the cylindrical (concave) pole pieces are used (b) the straight-cut pole pieces can be used (c) steel core should be used (d) laminated core should be used.

The pointer of a dead beat galvanometer gives a steady deflection because (a) its magnet is very strong (b) its pointer is very light (c) eddy currents are produced in its coil (d) the coil is wound on an ebonite frame

Answer in yes/no

(1) (2) (3) (4) (5) (6)

A moving coil galvanometer gets affected by a magnet placed close by.

A dead-beat galvanometer can be used to measure charge/transient current.

A soft iron core is used for the coil of a B.G. A lead accumulator can be recharged.

The coil of a dead-beat galvanometer can be wound over a non-conducting frame.

A large value of Q (quality factor) signifies under-damped oscillations.

Procedure 1. Release the lock/unlock knob for the suspension / mirror provided at the back of the B.G, so that the suspension fiber and hence, the mirror attached to it are free to move.

2. Level the B.G. with the help of the three leveling screws provided at its base and the spirit level provided on its bottom rim. (This is done to ensure that the coil hangs freely and its suspension is vertical).

3. Arrange the lamp and scale arrangement in front of the B.G. at a distance of 1m from the mirror of B.G. so that light from the lamp falls on the mirror of B.G. The crosswire in the reflected spot of light is adjusted at the zero (middle) of the scale.

Part – A (For study of oscillatory motion of the coil)

4. Connect the circuit according to the circuit diagram-A (Figure 4). The two terminals of the B.G. should be connected to a tapping key (K1), through which the other circuit connections should be made. (This key is used to arrest the undue motion of the coil).

5. Take out a high resistance of the order of 10 KΩ from the resistance box R'. (This should be done to safeguard that a large current does not flow through the galvanometer coil that may damage it).

6. Press the tapping key K' momentarily and release. (This means that the circuit is now open and the circuit resistance R' is infinite) 7. The coil will start oscillating about the mean position.

8. Note down n consecutive deflections, θn, on the scale, i.e. on both the sides of zero, until the deflection reduces to zero and record the observations in Table 1.

9. Again press the tapping key K' momentarily and release. As the coil starts oscillating, note the time taken by the spot of light to complete 20 oscillations, using a stopwatch. (In order to time oscillations accurately, make sure that you start your stop watch as soon as the spot crosses the zero position and not when it is at either of the extreme positions.)

10. Repeat the above step 4-5 times and record these observations in Table 2, for determination of time-period of the B.G. Part – B (To Study the effect of circuit resistance on damping)

11. Now reassemble the circuit according to Figure 5 (circuit diagram – B).

12. Keep R1 = 1 ΚΩ, R2 = 10 KΩ and adjust r to get full-scale deflection (a deflection of about 25cm on the scale) on plugging in the key K. 13. Open the key K now, the spot would come back slowly and if it does not overshoot the zero point on the scale, you are ready to start taking observations. If not, i.e. if the spot overshoots the zero mark slightly, then reduce R1 in steps of 100 Ω and again adjust r for full-scale deflection and check that spot does not cross-over the zero point. This corresponds closely to critical damping.

14. Now R2 and r are to be kept constant throughout the experiment and observations will be recorded by varying R1 only.

15. Plug in the key K. The spot of light will move to show a full-scale deflection as already adjusted. Note this deflection as θ1 (cm). As soon as it becomes stable, take out plug key K. Confirm that θ2 is zero. Clearly, θ3 is also zero. Record these observations in Table 3.

16. Now, R1 is to be increased in steps of 1 ΚΩ. Plug in the key K. The spot of light will now show a smaller deflection (as the resistance in the circuit has been increased).

Note the first deflection θ1. Remove the plug from key K. Note the consecutive deflections θ3, θ5 ….on the same side, till the last deflection, closest to zero.

17. Repeat the step 16 by increasing R1 in steps of 1ΚΩ each time, up to about 15 ΚΩ or 20 ΚΩ.

You have observed the cases of oscillatory motion, damped oscillation and over damped behavior of B.G. Now, to find the exact value of the critical damping resistance of the B.G., you are required to proceed as follows and record your observations in Table 4.

Part – C (For determination of critical damping resistance)

18. Start taking observations from that value of R1 in step 17, where θ3 was very small.

19. You should reduce R1 now in smaller steps and note θ1 and θ2 also (θ2 is the next deflection on the other side of the zero after the first deflection θ1).

20. Here, you are also required to note the time taken by the spot of light to come back to zero from θ1 (i.e. start the stop watch as soon as you plug out K and stop it as soon as the spot of light reaches the zero mark).

21. Repeat steps 19 and 20 by reducing R1in small steps.

22. You will eventually reach a step when θ2 will just be equal to zero.

23. You should stop taking further observations when θ2 remains zero; but the time taken by the spot to come back to zero starts increasing.

24. That particular observation when θ2 is zero and the time taken by the spot of light to reach back to zero is minimum corresponds to the condition of critical damping.

Observations Least Count of the scale used in the lamp and scale arrangement = ……. cm E.M.F. of the accumulator = …… V

Table 1: Oscillatory behavior of B.G.

S. No. 1 2 3 4 5 … …

Number of throw n 1 2 3 4 5 … …

Throw θn (cm)

20 ~ 30

Table 2: Time period of B.G.

S.No.

Time for 20 oscillations (sec)

Time period (T) (sec)

1 2 3 4 5 Mean T = ………..sec.

Therefore ω = 2π/T = ………..radian

Table 3: Determination of RG and effect of circuit resistance on λ and Q

Resistance R2 = ……. KΩ Resistance r = …….. Ω

S. No. 1 2 3 . . .

R1 (KΩ) ~1 2 3 . . .

θ1 (cm) ~ 25

θ3 (cm) 0.0

θ5 (cm) _

θ7 (cm)

θ9 (cm)

θ11 (cm)

θ13 (cm)

…

...

_ _ Decreasing

_ _

↓

~15

0.0

(The dashed lines indicates a pattern of the last deflection being observed closest to zero and hence the effect of the circuit resistance on damping of the oscillations of the coil.)

Table 4: Critical damping resistance (C.D.R.)

Resistance R2 = ……. KΩ Resistance r = …….. Ω S. No.

R1 (kΩ)

1 2 3 4 5 6 7 8 9 10

~ 3 to 5

Decreasing

θ 1 (cm)

θ2 (cm)

Time t (sec)

~1 Decreasing

Decreasing

0.0 0.0 0.0

Minimum

Increasing Increasing

(The value of R1 in the highlighted row corresponding to θ2 = 0 and the minimum time, is used in determining the CDR for the given B.G.)

Calculations Part A Graph 1

•

• •

Plot a graph between deflection θ n and the no. of deflection n as shown in Figure 7.

Take the serial number of observation (deflection), n, along x-axis and corresponding deflection θ n (on both sides of zero) along y-axis. Take the zero of y-axis in the middle of the graph.

Calculate λ using Equation (28).

Obtain the mean value of all observations of Table 2 to find the time-period T, of the B.G.

Part B

• • • •

Calculate 1

θ

for all observations of Table 3 and record them in Table 5 given 1

below.

Calculate λ for all observations of Table 3, using Equation (28) and record them in Table 5 given below.

Calculate p for all observations of Table 3, using Equation (24) and record them in Table 5 given below.

Calculate Q for all observations of Table 3, using Equation (37) and record them also in Table 5 given below.

Table 5: For plotting various graphs

S. No. 1.

2.

3.

4.

5.

6.

R1 (kΩ)

1

-1 θ1 (cm)

λ

p=

2λ (sec)-1 T

Q=

π 2λ

7.

Graph 2

Plot a graph between inverse of first throw θ1 and resistance R1, as shown in Figure 8.

The intercept on x-axis (R1 axis) gives the resistance of the galvanometer, RG. Graph 3

Plot a graph between λ and R1, as shown in Figure 9 (This represents the variation of damping with the circuit resistance).

Graph 4

Plot a graph between Q and R1, as shown in Figure 10.

Part C

Calculate the critical damping resistance of the B.G. using Equation (17). Here, R1 is that particular observation in Table 4 which corresponds to θ 2 = 0 and the minimum time, and RG is obtained from Graph 2.

Nature of graphs

θn

0

θn

n

Figure 7: Graph 1

1/θ1

0

RG

R1 (Ω)

Figure 8: Graph 2

λ

0 R1 (KΩ)

Figure 9: Graph 3

Q

0

R1 (Ω)

Figure10: Graph 4

Result • • • • • • • •

The effect of electromagnetic induction on the motion of the coil of the ballistic galvanometer was studied.

It was observed that the circuit resistance plays leading role in controlling the behavior of the galvanometer.

The resistance of the B.G., RG, was found to be = ….. Ω The Critical Damping Resistance, C.D.R., of the B.G. was found to be = … Ω The logarithmic decrement, λ, was found to vary inversely with the circuit resistance.

The Q-factor was found to vary linearly with the circuit resistance.

The galvanometer behaves as ballistic for circuit resistance larger than the C.D.R. The galvanometer behaves as dead-beat for circuit resistance smaller than the C.D.R.

Probable Sources of Errors 1. The accumulator may not be freshly charged. This will not send a steady current through the circuit.

2. The B.G. may not be properly leveled. In such case, the coil will not oscillate in a vertical plane and will lead to incorrect observations.

3. If the spot of light is not properly adjusted on the zero of the scale, all the observations for θ n as well as for the C.D.R. are liable to be incorrect.

4. The plugs of the resistance boxes may not be clean, or may be loose due to which the actual value of the resistance introduced in the circuit may be different from what is observed.

Glossary Ballistic Constant: A constant for the galvanometer (k) which gives the amount of charge that has passed through the galvanometer coil on multiplying it by the corrected first throw ( q = k .θ 0 ).

Ballistic galvanometer (B.G.): A galvanometer in which an impulse is applied to the coil of the galvanometer due to the passage of a transient current or charge and the subsequent motion of the coil is used to determine the magnitude of the impulse, and, hence the quantity of charge to be measured.

Charge Sensitivity: The charge in micro-coulombs which when sent through the coil will produce a deflection of 1 mm of the spot on the scale placed at a distance of 1metre from the galvanometer mirror.

Critical damping: Damping in a linear system on the threshold between oscillatory and exponential behavior.

Critical damping resistance (C.D.R.): The value of the total resistance in the circuit, at which critically damped condition is achieved. When resistance in the galvanometer circuit is larger than the C.D.R., the galvanometer becomes oscillatory and when it is smaller than the C.D.R., the galvanometer becomes dead-beat.

Damping: The dissipation of energy in motion of any type, especially oscillatory motion and the consequent reduction or decay of the motion.

Damped harmonic oscillation: The linear motion of an object, subject both to an elastic restoring force proportional to its displacement and to a frictional force in the direction opposite to its motion and proportional to its speed.

Dead-beat galvanometer: A galvanometer which comes to rest without vibration or oscillation.

Decrement: The ratio of the amplitudes of an under-damped harmonic motion during two successive oscillations.

Eddy current: A vortex-like flow of current running contrary to the main current.

Electromagnetic: Pertaining to phenomena in which electricity and magnetism are related.

Electromagnetic damping: Damping pertaining to phenomenon in which electricity and magnetism are related.

Electromagnetic induction: The phenomenon whereby an induced E.M.F. is produced whenever there is a change in the magnetic flux linked with the circuit.

E.M.F.: Abbreviated form of electro-motive force. It is the potential difference between the two plates of a cell when in open circuit. It is also so called because it is the driving force which sends the current through the cell.

Galvanometer: An instrument used to detect current or charge passing through it.

Impulse: The integral of a force over an interval of time. A pulse, which lasts for so short a time that its duration can be thought of as infinitesimal.

Lamp and scale arrangement: An arrangement comprising of a stand with a lamp and a scale, by which a small angular twist of the coil and hence of the small round mirror attached to it can be measured as a much larger linear deflection of the image of the spot of light on the horizontal scale.

Lead accumulator: It is a secondary cell or storage cell. Here, current from some other supply is first passed through it for some time and is stored as chemical energy. The same current is then obtained from it in the reverse direction through an external circuit. (i) It has very low internal resistance, (ii) its efficiency is very high and (iii) it gives a constant current.

Logarithmic decrement: The natural logarithm of the ratio of the amplitude of one oscillation to that of the next, when no external forces are applied to maintain the oscillation.

Moment of inertia: The sum of the products formed by multiplying the mass of each element of an object by the square of its distance from a specified line. Also known as rotational inertia.

Moving coil galvanometer: A galvanometer in which a current carrying coil is placed in a magnetic field and is subject to a force, the direction of which is normal both to the direction of current and the direction of magnetic field. This leads to the motion of the coil.

Moving magnet galvanometer: A galvanometer in which a current is passed through a fixed coil which produces a magnetic field, under which a small magnet / magnetic needle moves.

Oscillation: Any effect that varies periodically back and forth between two values.

Phosphor-bronze: An alloy with Copper-92.5%, Tin-7% and Phosphorous-0.5%. (i) It has great tensile strength, (ii) its torsional couple per unit twist is small and (iii) it is rustresisting.

Quality factor (Q-factor): A measure of the ability of a system with periodic behavior to store energy equal to 2Ď&#x20AC; times the average energy stored in the system divided by the energy dissipated per cycle.

Radial magnetic field: Achieved by making the magnetic poles curved into a concave shape. This is done in order to make the magnetic field always parallel to the plane of the coil, whatever be the position of the coil.

Steady current: A current which is constant with respect to time.

Time period: Time taken by a moving system to complete one vibration or oscillation.

Torque: For a single force, the cross-product of a vector from some reference point to the point of application of the force with the force itself. Also known as moment of force or

rotation moment. For several forces, the vector sum of the torques associated with each of the forces.

Torsion: A twisting deformation of a solid body about an axis in which lines that were initially parallel to the axis become helices.

Torsional rigidity: The ratio of the torque applied about the centroidal axis of a bar at one end of the bar to the resulting torsional angle, when the other end is held fixed.

Transient current/phenomena: A current pulse, or other temporary phenomenon occurring in a system for a short duration.

Post-lab Assessment Answer the following questions

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

How do you correct the observed throw for damping?

What do you understand by the sensitiveness of a galvanometer?

How do you obtain the galvanometer resistance?

Why do you multiply the expression for logarithmic decrement by 2.3026?

How do you stop the unwanted oscillations of the B.G.?

Can you use a ballistic galvanometer to measure current?

Why is the suspension used in the form of a rectangular strip?

Which material is used for making the B.G. coil and why?

What is the order of resistance of your B.G.?

With reference to the above question, what will be the order of the resistance if this galvanometer was of dead-beat type?

(11) Is the value of the Ballistic Constant k, a constant for a given galvanometer or does it depend upon the working conditions?

(12) What advantages does the lamp and scale arrangement for measuring deflections possess over the use of mechanical pointer?

Choose the correct answer

(1)

The motion of a torsional pendulum is simple harmonic. If Ι is the moment of inertia of the pendulum and c is the restoring couple per unit angular twist, then its time period will be c a) T = 2π Ι

c)

Ι c T = 2π Ιc

d)

T = 2π

b)

T = 2π

1 . Ιc

(2)

(3)

(4)

(5)

The equation of motion of a moving coil galvanometer is given as d 2θ dθ + 0.2 + 36θ = 0 . The period of oscillation will approximately be 2 dt dt π sec a) 6 π sec b) 2 π sec c) 3 π sec. d) 4 In the above equation, Q-factor will be a) 3 b) 15 c) 20 d) 30 In the above question, λ (the logarithmic decrement) will be a) 0.05 b) 0.5 c) 0.01 d) 0.1 The sensitivity of a moving coil galvanometer is large if a) the magnetic field is small b) the area of the coil is small c) the torsional constant of the suspension strip is small d) number of turns of the coil is small.

Answers to Pre-lab Assessment 1.

2.

Characteristics of Ballistic galvanometer: (i) The moment of inertia of the coil should be large. (ii) The air damping should be small. (iii)The electromagnetic damping should be small. Thus, the coil should be wound on a non-conducting frame. (iv)The resistance R of the circuit, including that of the coil should be large. For this, the coil should have large number of turns.

Characteristics of dead beat galvanometer: (i) The moment of inertia of the coil should be small. (ii) The electromagnetic damping should be large. Thus, the coil should be wound on a conducting frame. (iv) The resistance R of the circuit, including that of the coil should be small. For this, the coil should have small number of turns.

There are two types of moving coil galvanometer:

(i) Pivoted-coil type, where the coil swings between two pivots and the deflection is read by a pointer which moves on a circular scale. (ii) Suspended-coil type, where the coil is suspended freely by a thin phosphor bronze strip and the deflection is measured by a lamp and scale arrangement.

3. On account of the resistance of the medium (air) and electromagnetic damping, the magnitude of the amplitude of vibration of the coil decreases exponentially with time. The ratio of the successive amplitudes on either side of the position of rest is called ‘decrement’ and the logarithm of this to the base e is called logarithmic decrement.

4. It is determined to correct the observed throw for damping. The method adopted to determine it is to observe the ratio of the first and eleventh swing and then θ 2.303 λ= log10 1 .

10 θ11 5. Electromagnetic damping is the dissipation of energy in motion of an oscillatory system which causes decay of the motion due to the production of electromagnetically induced e.m.f. in the moving system. It can be reduced by winding the coil of the B.G. on a non-conducting frame made of ivory, bamboo or ebonite.

6. Time period of the suspended coil depends directly on the moment of inertia of the coil and inversely on the torsional constant of the suspension. Hence, to increase the time period, the moment of inertia of the coil should be increased and suspension should be made of the material having low torsional constant 7. The oscillations are stopped by short-circuiting the coil by closing the damping key. Eddy currents are produced in the coil in a direction which opposes the motion of the coil.

8.

The sensitivity of the B.G. can be increased by (i) increasing the number of turns in the coil, (ii) increasing the effective area of the coil, (iii) increasing the magnetic field, (iv) reducing the time period of the moving system, (v) reducing the couple per unit twist of the suspension.

9. It is a constant, which when multiplied by the throw of the galvanometer, gives the amount of charge which has passed through it.

10. By using the suspension of phosphor-bronze in the form of a thin rectangular strip.

11. E.M.F. of a cell is the abbreviated form of electro-motive force. It is the potential difference between the two plates of a cell when in open circuit. It is so called because it is the driving force which sends the current through the cell.

12. Secondary cell is preferred because (i) it gives strong current as its internal resistance is very low, (ii) its efficiency is very high i.e. it gives back most of the electrical energy used in charging it and (iii) it gives a constant current due to the large area of its plates and their close proximity.

Answers (multiple choice questions)

1.

1.

2.

3.

4.

(a) (c) (c) (a) (c)

Answers (Yes/No)

1.

2.

3.

4.

5.

6.

No No No Yes No Yes

Answers to Post-lab Assessment 1.

The observed throw is multiplied by ( 1 +

λ

).

2 2. A galvanometer is said to be sensitive when a small current / charge passing through it produces a large deflection in it.

3. A graph is plotted between 1/θ1 and R1. A straight line is obtained. On extrapolating it backwards, it gives an intercept on the negative side of resistance axis that gives the galvanometer resistance.

4. The logarithmic decrement is defined with respect to the base e. in order to convert its expression to the base 10, it is multiplied by 2.3026.

5. The unwanted oscillations of the B.G. can be arrested by short circuiting the coil, which is achieved by closing the tapping key K1. Eddy currents are produced in the coil in a direction so as to oppose the motion of the coil.

6. Yes, a B.G. can be used to measure current, but we will have to wait for a long time for the coil to give a steady deflection.

7. It is used because (i) the rectangular strip has a very small torsional rigidity and hence the twist for a given couple is much greater, and the stress on the material much less than with a wire of circular cross section and (ii) a strip possesses an additional advantage of offering comparatively large surface area for the dissipation of the heat produced by the current.

8. Thin enamelled copper wire is used, as it has a low specific resistance.

9. It is of the order of 150 Ω.

[Note: The resistance is generally written on a plate attached to the case of the B.G. by the manufacturers].

10. The resistance of the dead-beat galvanometer may be of the order of 25 Ω.

11. If k is expressed in coulombs per radian, it is independent of the working conditions; but when it is expressed as coulomb per mm deflection at a given distance of the scale from the mirror, it will depend on this distance.

12. By using a lamp and scale arrangement, a small angular twist of the coil and hence of the mirror attached to it, can be measured as a much larger linear deflection of the image of the spot of light on a horizontal scale.

Answers (multiple choice questions) 1.

2.

3.

4.

5.

b c d a c