PEMI – 1 ELECTRO MAGNETIC INDUCTION AND ...einsteinclasses.com/Electro M_I_Ac.pdfalways be some magnetic flux through the closed loop of a current-carrying circuit. But the effect

PEMI – 1

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road

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ELECTRO MAGNETIC INDUCTION AND ALTERNATING CURRENT

C1 Magnetic Flux

Like electric flux, magnetic flux, B, through a surface Sd

is defined as

S

B Sd.B

. If B

is uniform

then B = S.B

and it represents total lines of induction crossing through a given surface S.

C2 Magnetic Induction and Faraday’s Laws

If the magnetic flux through a circuit or closed loop changes, an emf and a current are induced in the circuit.This phenomenon is known as electromagnetic induction and the law which governs this phenomenon isknown as Faraday’s Law. This law states that the magnitude of induced emf in a circuit is equal to the time

rate of change of the magnetic flux. Mathematically, dt

d|e|

. As cosBAA.B

. Hence if there

is any change in magnetic field (B) or area (A) or orientation () then there is induced emf. If somesituation, more than one of these may contribute in induced emf, in this case magnitude of induced emf iswritten as

dt

dsinBA

dt

dA)cosB(

dt

dB)cosA()cosBA(

dt

d|e|

This induced emf creates an induced current in the circuit whose magnitude is given as

R

|e|

circuitofresistancenet

emfinducedI . Also the charge flown =

R

.

Practice Problems :

1. A circular coil (constant radius) of total length L having number of turns N is rotated about thediameter in a uniform magnetic field B with an angular velocity . Initially the magnetic field isperpendicular to the plane of the coil. The maximum value of the emf induced in it is

(a)N2

BL2

(b)

2

NBL2

(c)N4

BL2

(d)

4

NBL2

2. A thin circular ring of area A is held perpendicular to a uniform magnetic field of induction B.A small cut is made in the ring and a galvanometer is connected across the ends such that the totalresistance of the circuit is R. When the ring is suddenly squeezed to zero area, the charge flowingthrough the galvanometer is

(a) 2AB/R (b) AB/R (c)R4

AB(d)

R3

AB

[Answers : (1) c (2) b]

C3 Lenz’s Law

The direction of induced emf is governed by Lenz’s Law. This law states that an induced emf is always inthe direction that opposes the change of magnetic flux that induced it. Incorporating this law into Faraday’s

Law, the induced emf is given by dt

de

. The negative sign indicates that the induced emf opposes the

change of the flux.

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Note the Lenz’s Law is based on conservation of energy principle.

Practice Problems :

1. In the figure the flux through the loop perpendicular to the plane of the coil and directed into the

paper is varying according to the relation = 6t2 + 7t + 1 where is in milliweber and t is in

seconds.

Choose the correct statement :

(a) At time t = 2s, the current flowing through R is 10mA from left to right

(b) At time t = 2s, the current flowing through R is 10mA from right to left

(c) The current through R is always increasing linearly

(d) both (a) and (c) are correct

2. A rectangular coil (having resistance per unit length 10/3 /m) of 100 turns and size 0.1 m × 0.05 mis placed perpendicular to a magnetic field of 0.1 T. If the field drops to 0.05 T in 0.05 s then

(a) the magnitude of average induced current is 4mA

(b) the total charge flown in the coil is 5µC

(c) the total charge flown in the coil isindependent of time during which the field will change


3. A solenoid has 2000 turns wound over a length of 0.3 m. Its cross-sectional area is 1.2 × 10–10m2.Around its central section a coil of 300 turns is wound. If an initial current of 2A flowing in thesolenoid is reversed in 0.25 s, the emf induced in the coil will be

(a) 6.0 × 10–4 V (b) 6.0 × 10–2 V (c) 4.8 × 10–4 V (d) 4.8 × 10–2 V

[Answers : (1) d (2) c (3) d]

C4 Motional Electromotive Force

If a conductor with length L moves with speed v in a uniform magnetic field with magnitude B, and if thelength and velocity are both perpendicular to the field, the induced emf is e = vBL. More general, when a

conductor moves in a magnitude field B

, the induced emf in the direction is given by

a

b

ld).Bv(e

Practice Problems :

1. An electric potential difference will be induced between the ends of the conductor shown in thediagram when it moves in the direction

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(a) P (b) Q (c) L (d) M

2. A conducting square loop ABCD of side L and resistance R moves in its plane with a uniformvelocity v perpendicular to one of its sides. A magnetic induction B, constant in time and space,pointing perpendicular and into the plane of the loop exists everywhere, then

(a) The current induced in the loop is zero

(b) There is no induced emf in the rod BC and AD

(c) There is an induced emf BLv in each rod AB and CD

(d) All the above statements are correct

[Answers : (1) d (2) d]

C5 Induded Electric Field :

When an emf is induced by a changing magnetic flux through a stationary closed path, there is an induced

electric fleld E

of non-electrostatic origin such that

dt

dld.E B

Properties of Induced Electric Field

1. It is not a Coulomb field.

2. The lines of induced field form closed loop. Therefore, it is called a circuital field or vortex field.

3. This field is nonconservative and cannot be associated with a potential.

Practice Problems :

1. Consider a cylindrical space of radius R in which a time varying magnetic field is confined. Find thedependence of induced electric field on the distance r from the centre inside the space and outsidethe space ?

[Answers : (1) inside E is directly proportional to r and outside it is inversely proportional to r]

C6 Self inductance and Inductors

Any circuit that carries a varying current will have an emf induced in it by the variation in its own magneticfield. Such an emf is called a self-induced emf. Self-induced emf’s can occur in any circuit, since there willalways be some magnetic flux through the closed loop of a current-carrying circuit. But the effect is greatlyenhanced if the circuit contains a coil with N turns of wire. As a result of the current i, there is an averagemagnetic flux

B, through each turn of the coil. Here we defined the self inductance L of the circuit as

follows I

NL B

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The SI unit of inductance is the henry (H).

Self inductance of the solenoid

The inductance per unit length near the middle of a long solenoid of cross-sectional area A and n turns per

unit length is AnµL 2

0l

Self induced emf

The self-induced emf, using Faraday’s law, is given by dt

dILe

Practice Problems :

1. The current in a coil changes from 0 to 2A in 0.05 s. If the induced emf is 80 V, the self-inductance ofthe coil is

(a) 1 H (b) 0.5 H (c) 1.5 H (d) 2 H

2. A torodial solenoid with an air core has an average radius of 15 cm, area of cross-section 12 cm2 and1200 turns. Ignoring the field variation across the cross-section of the toroid, the self-inductance ofthe toroid is

(a) 4.6 mH (b) 6.9 mH (c) 2.3 mH (d) 9.2 mH

3. A coil is wound on a frame of rectangular cross-section. If all the linear dimensions of the frame areincreased by a factor 2 and the number of turns per unit length of the coil remains the same, self-inductance of the coil increases by a factor of

(a) 4 (b) 8 (c) 12 (d) 16

[Answers : (1) d (2) c (3) b]

C6 Energy Stored in an Inductor

If an inductor L carries a current i. the inductor’s magnetic field stores an energy given by 2Li

2

1U

C7 LR Circuits :

Applying Kirchoff’s voltage law across an inductor.

(a) If the direction of assumed current coincides with the direction of motion, the voltage across the inductor

falls and is given by dt

dIL .

(b) If the direction of assumed current is opposite to the diretion of motion the voltage across the inductor rises

and is given by dt

dIL .

Growth of Current in RL circuit :

Let us connect a coil of self-induction L with a resistance R across a cell of emf E as shown in figure. If theswitch S is thrown in contact at t = 0, current i in the circuit tends to grow. Hence an emf is induced acrossthe coil in such a direction as to oppose this current.

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By Kirchoff’s voltage law, we have

0dt

diLiRE

L

dt

iRE

di

i

0

t

0L

Rdt

iRE

Rdi

[log(E – iR)]0i =

L

Rt i(t) =

L

Rt

e1R

E

Here R

L is known as time constant of the circuit.

The current grown in the circuit exponentially as shown in figure.

Note the following points :

1. At t = 0, i = 0, we can say at t = 0, the inductor behaves like a breaking wire.

2. In steady state : At t , R

Ei , we can say at t , the inductor behaves like a connecting wire.

3. The rate at which the source or battery will supply energy = Ei, rate at which the energy is dissipated in

resistor = i2R and the rate at which the energy stored in the inductor =

dt

diLi . From conservation of

energy

dt

diLiRiEi 2 .

Decay of current in LR circuit : At t = 0, the current passing through the inductor is I0 and it is connected

across a resistor as shown in figure :

Using KVL, 0dt

dILiR dt

L

R

I

dI

i

0

t

0

dtL

R

I

dI L/t

0eII

Practice Problems :

1. In the following circuit initially there is no current through the inductor. Find the current passingthrough the battery at any time t. Also find the current through the battery at t = 0 and t = .

(a) (b)

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2. A solenoid has an inductance of 53 mH and a resistance of 0.37 . If it is connected to a battery, howlong will the current take to reach half its final equilibrium value ?

3. A solenoid having an inductance of 6.30 µH is connected in series with a 1.20 k resistor. (a) If a14.0 V battery is switched across the pair, how long will it take for the current through the resistor toreach 80.0% of its final value ? (b) What is the current through the resistor at time t = 1.0

L ?

4. At time t = 0, a 45.0 V potential difference is suddenly applied to a coil with L = 50.0 mH andR = 180 . At what rate is the current increasing at t = 1.20 ms ?

[Answers : (2) 0.10 s (3) (a) 8.45 ns; (b) 7.37 mA (4) 12.0 A/s]

C8 Energy Density of a Magnetic Field

If B is the magnitude of a magnetic field at any point (in an inductor or anywhere else), the density of stored

magnetic energy at that point is

0

2

Bµ2

Bu .

C9 Mutual Induction

When a changing current i1 in one circuit causes a changing magnetic flux in a second circuit, an emf e

2 is

induced in the second circuit; likewise, a changing current i2 in the second circuit induced an emf e

1 in the

first circuit. This is called mutual induction.

dt

diMeand

dt

diMe 2

11

2

The constant M, called the mutual inductance, depends on the geometry of the two coils and on the materialbetween them. If the circuits are coils of wire with N

1 and N

2 turns, respectively, the mutual inductance can

be expressed in terms of the average flux B2

through each turn of coil 2 that is caused by the current i1 in

coil 1 or in terms of the average flux B1

through each turn of coil 1 that is caused by the current i2 in coil 2

:

2

1B1

1

2B2

i

N

i

NM

The SI unit of mutual inductance is the henry, abbreviated H. Equivalent units are

1 H = 1 Wb/A = 1V.s/A = 1.s.

Mutual inductance of two solenoids one surrounding the other is given by µ0n

pn

sAl where n

p and n

s are

number of terms per unit length for primary and secondary coils and A is the cross-sectional area of primarycoil and l is the length of the primary coil.

C10 LC Circuit

An L-C circuit, which contains inductance L and capacitance C, undergoes electrical oscillations withangular frequency :

LC

1

Such a circuit is analogous to a mechanical harmonic oscillator, with inductance L analogous to mass m, thereciprocal of capacitance 1/C to force constant k, charge q to displacement x, and current i to velocity v.

Practice Problems :

1. A capacitor of capacitance 1 µ F is charged upto 10V and then connected across an ideal inductor of10 mH. Choose the correct statement :

(a) The angular frequency of LC oscillation is 104 rad/s

(b) At any moment total energy is 50µJ

(c) The current in the circuit changes with time sinusoidally

(d) All are correct

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2. A capacitor of 1 µ F initially charged to 10 V is connected across an ideal inductor of 0.1 mH. Themaximum current in the circuit is

(a) 0.5 A (b) 1 A (c) 1.5 A (d) 2 A

[Answers : (1) d (2) b)

C11 Back EMF in D.C. Motor : A motor is the reverse of generator – it converts electrical energy intomechanical energy. When currents is passed through a coil placed in a magnetic field, it rotates. As the coilrotates, the magnetic flux linked with changes, giving rise to an induced emf. This emf opposes the appliedemf () and is, therefore, called back emf (e). If R is the resistance of the coil, the current through it is given

by R

eI

.

Practice Problems :

1. In a dc motor, if E is the applied emf and e is the back emf, then the efficiency is

(a)E

eE (b)

E

e(c)

2

E

eE

(d)

2

E

e

[Answers : (1) b]

C12 Eddy Currents

When a metallic body is moved in a magnetic field in such a way that the flux through it changes or isplaced in a changing magnetic field, induced currents circulate throughout the volume of the body. Theseare called eddy currents.

C13 Alternating Current

An alternator or ac source produces an emf that varies sinusoidally with time.

Production of A.C.

Production of A.C. is based on Faraday’s law of electromagnetic induction. Suppose a coil of N turns, andarea A is rotated in a uniform magnetic field B with angular velocity . As the coil rotates, the flux throughit changes and therefore an emf is induced in it, given by =

0 sin t where

0 = NBA.

A sinusoidal voltage or current can be represented by a phasor, a vector that rotates counterclockwise withconstant angular velocity equal to the angular frequency of the sinusoidal quantity. Its projection on thehorizontal axis at any instant represent the instantaneous value of the quantity.

C14 Average and root mean square value of a.c.

For a sinusoidal current the average and rms (root-mean-square) currents are related to the currentamplitude I

0 by

00av I637.0I2

I

, 2

II 0

rms .

In the same way, the rms value of the snusoidal voltage is related to the voltage amplitude V0 by

2

VV 0

rms

The voltage v in an ac circuit is represented by v = v0sint and current in a.c. circuit is represented by

i = i0sin(t + ) where is the phase angle between the current and voltage.

C15 A.C. Circuit

Pure resistive a.c. circuit

The voltage across a resistor R is in phase with the current, and the voltage and current amplitude arerelated by V

R = IR

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Pure inductive circuit

The voltage across an inductor L leads the current by 900, the voltage and current amplitude are related by

VL = IX

L,

where XL = L is the inductive reactance of the inductor.

Pure capacitive circuit

The voltage across a capacitor C lags the current by 900; the voltage and current amplitudes are related by

VC = IX

C,

where XC = 1/C is the capacitive reactance of the capacitor.

LCR series circuit

In an ac circuit the voltage and current amplitudes are related by

V = IZ,

where Z is the impedance of the circuit. In an L-C-R series circuit,

222CL

2 )]C/1(L(R)XX(RZ ,

and the phase angle of the voltage relative to the current is

R

C/1Ltan

Practice Problems :

1. A 40 electric heater is connected to 200 V, 50 Hz main supply. The peak value of the electric currentflowing in the circuit is approximately

(a) 2.5 A (b) 5.0 A (c) 7 A (d) 10 A

2. An alternating voltage V = 2002 sin 100 t, where V in volt and t seconds, is connected to a seriescombination of 1 µF capacitor and 10 k resistor through an ac ammeter. The reading of theammeter will be

(a) 2 mA (b) 102 mA (c) 2 mA (d) 20 mA

3. Choose the correct statement :

(a) the current leads the voltage in phase if an ac source is connected across a capacitor

(b) the current lags behind the voltage in phase if an ac source is connected across an inductor

(c) the current and voltage are in same phase if an ac source is connected across a resistor.

(d) all are correct

[Answers : (1) c (2) b (3) d]

C16 Power in A.C. circuit

The average power input Pav

to an ac circuit is

cosIVcosVI2

1P rmsrmsav

where is the phase angle of voltage with respect to current. The quantity cos is called the power factor.

Practice Problems :

1. If a current I = I0 sin (t – /2) flows in a circuit across which an alternating potential E = E

0 sin t

has been applied, then the power consumed in the circuit depends on

(a) E0

(b) I0

(c) both (d) none

2. In circuit 1, an alternating current of 2 A flows for 10 minutes. In another similar circuit 2, a directcurrent of 2 A flows for the same time. If the heat produced in circuit 1 is X then the heat producedin circuit 2 is

(a) 0.5 X (b) 1.5 X (c) X (d) 2X

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3. A sinusodal alternating current flows through a resistor R. If the peak current is Ip, then the power

dissipated is

(a) Ip

2R (b) RI2

1 2p (c) RI

4 2p

(d) RI

1 2p

4. The impendence of a circuit consists of 3 resistance and 4 reactance. The power factor of the

circuit is

(a) 0.4 (b) 0.6 (c) 0.8 (d) 1.0

[Answers : (1) d (2) c (3) b (4) b]

C17 Resonance in LCR Circuit

In an L-C-R series circuit the current becomes maximum (for a given voltage amplitude) and the impedancebecomes minimum at an angular frequency

0 = 1/(LC)1/2 called the resonance angular frequency. This

phenomenon is called resonance. At resonance the voltage and current are in phase, and the impedance Z isequal to the resistance R.

Practice Problems :

1. In an LCR series circuit, the capacitance is changed from C to 4C. For the same resonant frequency,the inductance should be changed from L to

(a) 2L (b) L/2 (c) L/4 (d) 4L

[Answers : (1) c]

C18 Quality Factor

The Quality factor of an LCR series circuit is defined as R

LQ 0

where 0 is the resonance angular

frequency. It is an indicator of the sharpness of the current peak – higher the value of Q, sharper is the peak.

C19 Transformer

A transformer converts a low aleternating voltage to a high voltage and vice-versa. It is based on theprinciple of mutual induction. It consists of two coils wound on a soft iron core. The primary coil isconnected to an a.c. source.The secondary coil is connected to the load which may be a resistor or any otherelectrical device.

If the primary resistance is zero, then Ep is equal to the applied voltage. Further, if there is no flux leakage,

i.e., the same flux is linked with each turn of both the primary and secondary coils, then it can be shown that

p

s

p

s

N

N

E

E .

If Ns > N

p, then E

s > E

p and the transformer is called a step-up transformer.

If Ns < N

p, then E

s < E

p and the transformer is called a step-down transformer.

For an ideal transformer, Input power = Output power EpI

p = E

sI

s

p

s

p

s

s

p

N

N

E

E

I

I .

In actual transformers, there is some power loss. The main sources of power loss are :

1. I2R loss due to Joule heat in copper windings.

2. Heating produced due to Eddy currents in the iron core. This is reduced by using laminated core.

3. Hysteresis loss due to repeated magnetisation of the iron core.

4. Loss due to flux leakage.

When all the losses are minimized, the efficiency of the transformer becomes very high (90-99%).

Practice Problems :

1. In a step-down tranformer the input voltage is 22 kV and the output voltage is 550 V. The ratio of thenumber of turns in the secondary to that in the primary is

(a) 1 : 20 (b) 20 : 1 (c) 1 : 40 (d) 40 : 1

2. In a noiseless transformer an alternating current of 2 A is flowing in the primary coil. The number ofturns in the primary and secondary coils are 100 and 20 respectively. The value of the current in thesecondary coil is

(a) 0.08 A (b) 0.4 A (c) 5 A (d) 10 A

[Answers : (1) c (2) d]

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SINGLE CORRECT CHOICE TYPE

1. In the following figure there is a magnetic f i e l dof 0.2 Tesla along the positive x-axis.

The magnetic flux through the face BCDG is

(a) 0 (b) 1.8mWb

(c) 1.0mWb (d) 0.8mWb

2. A copper disc of radius R is rotated about its centrewith n revolutions per second in a uniformmagnetic field B. Choose the incorrect statement :

(a) If the field is in the plane of the disc thenthe induced emf between the centre andthe edge of the disc is zero

(b) If the field is in the plane of the disc thenthe induced emf between the centre andthe edge of the disc is non-zero

(c) If the field is perpendicular to the discthen the induced emf between the centreand the edge of the disc is BnR2


3. A player with 3 metre long iron rod runs towardseast with a speed of 30 km/hr. Horizontalcomponent of earth’s magnetic field is4 × 10–5 Wb/m2. If he runs with the rod inhorizontal and vertical positions, then thepotential difference induced between the two endsof the rod in the two cases will be

(a) zero in vertical position, 1 × 10–3 V inhorizontal position

(b) 1 × 10–3 V in vertical position, zero inhorizontal position

(c) zero in both positions

(d) 1 × 10–3 V in both positions

4. A square metal loop of side 10 cm and resistance1 ohm is moved with a constant velocity partlyinside a uniform magnetic field of 2 Wb/m2, directedinto the paper, as shown in the figure. The loop isconnected to a network of five resistors each of value3. If a steady current of 1 mA flows in the loop,then the speed of the loop is

(a) 0.5 cm/s (b) 1 cm/s

(c) 2 cm/s (d) 4 cm/s

5. Two inductors, each of inductance L, are connectedin series then effective self inductance (L

eff) is given

by

(a) Leff

= 2L (b) L Leff

3L

(c) 0 Leff

4L (d) 2L Leff

4L

6. A rectangular loop with a sliding connector of lengthl is located in a uniform magnetic fieldperpendicular to the loop plane as shown in figure.The magnetic induction is equal to B. Theconnector has an electric resistance R, the sides ABand CD have resistance R

1 and R

2 respectively.

Neglecting the self inductance of the loop, find thecurrent flowing in the connector during its motionwith a constant velocity.

(a)

21

21

RR

RRR

VB

l

(b)

21

21

RR

RRR

VB2

l

(c)

21

21

RR

RRR

VB3

l

(d)

21

21

RR

RRR

VB4

l

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7. An emf of 15 V is applied in a circuit containing 5H inductance and 10 resistance. The ratio of thecurrents at time t = and t = 1 s is

(a)1e

e2/1

2/1

(b)

1e

e2

2

(c) 1 – e–1 (d) e–1

8. A rectangular loop of sides 8 cm and 2 cm is lyingin a uniform magnetic field of magnitude 0.5 T withits plane normal to the field. The field is nowgradually reduced at the rate of 0.02 T/s. If theresistance of the loop is 1.6 , then the powerdissipated by the loop as heat is

(a) 6.4 × 10–10W (b) 3.2 × 10–10W

(c) 6.4 × 10–5W (d) 3.2 × 10–5W

9. A torodial solenoid with an air core has an averageradius of 15 cm, area of cross-section 12 cm2 and1200 turns. Ignoring the field variation across thecross-section of the toroid, the self-inductance ofthe toroid is

(a) 4.6 mH (b) 6.9 mH

(c) 2.3 mH (d) 9.2 mH

10. A coil is wound on a frame of rectangularcross-section. If all the linear dimensions of theframe are increased by a factor 2 and the numberof turns per unit length of the coil remains the same,self-inductance of the coil increases by a factor of

(a) 4 (b) 8

(c) 12 (d) 16

11. In the given circuit R is a resistor, L is an inductorand B

1 and B

2 are two bulbs. If the switch S is turned

off

(a) both B1 and B

2 die out promptly

(b) both B1 and B

2 die out with some delay

(c) B1 dies out promptly but B

2 with some

delay

(d) B2 dies out promptly but B

1 with some

delay

12. Two resistors of 10 and 20 and an idealinductor of 10 H are connected to a 2 V battery asshown. The key K is inserted at time t = 0. Theinitial (at t = 0) and final (at t = ) currents throughthe battery are

(a) A10

1;A

15

1(b) A

15

1;A

10

1

(c) A10

1;A

25

2(d) A

25

2;A

15

1

13. A tranformer is used to light a 140 W, 24 V bulbfrom a 240 V A.C. mains. The current in the maincable is 0.7 A. The efficiency of the transformer is

(a) 63.8% (b) 83.3%

(c) 16.7% (d) 36.2%

14.

Two conducting rings of radii r and 2r move inopposite directions with velocities 2v and vrespectively on a conducting surface S. There is auniform magnetic field of magnitude Bperpendicular to the plane of the rings. Thepotential difference between the highest points ofthe two rings is

(a) zero (b) 2rvB

(c) 4rvB (d) 8rvB

15. A magnet is moved with a high speed towards acoil at rest. Due to this, the induced emf, theinduced current and the induced charge in the coilare E, I and Q respectively. If the speed of themagnet is doubled, the incorrect statement is

(a) The induced current become 2I

(b) The induced emf becomes 2E

(c) The induced charge remains same

(d) The induced charge is 2Q

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16. The number of turns of primary and secondarycoils of a transformer are 5 and 10 respectively andthe mutual inductance of the transformes is 25 H.If the number of turns in the primary andsecondary are made 10 and 5 respectively, then themutual inductance of the transformer will be

(a) 6.25 H (b) 12.5 H

(c) 25 H (d) 50 H

17. In an LCR circuit, choose the correct statement

(a) current and voltage are always in phase

if LC

1

(b) current lags behind the voltage if

LC/1

(c) current leads the voltage if

LC/1

(d) All are correct

18. A small square loop of wire of side l is placed insidea large square loop of wire of side L (L >> l). Theloops are coplanar and their centres coincide. Themutual inductance of the system is

(a)L

µ22

20 l

(b)

L

µ2

20 l

(c)L

µ2

20 l

(d)

L

µ24

20 l

19. Two circular loops of radii a and b (b >> a) areplaced coaxially a distance r ( r >> b) apart. Themutual inductance between the loops is

(a) 2µ0a2b2/(2r3) (b) µ

0a2b2/(2r3)

(c) 3µ0a2b2/(2r3) (d) 4µ

0a2b2/(2r3)

20. A magnetic flux through a stationary loop with aresistance R varies during the time interval as

= at ( – t). The inductance of the loop is to be

neglected. The amount of heat generated in the loopduring that time

(a) a23/R (b) 1/2 a23/R

(c) 1/3 a23/R (d) 1/4 a23/R

21. A current I = 3.36(1 + 2t) × 10–2 A increase at steadyrate in a long straight wire. A small circular loop ofradius 10–3 m has its plane parallel to the wire andis placed at a distance of 1 m from the wire. Theresistance of the loop is 8.4 × 10–4 . Themagnitude of the induced current in the loop is

(a) 4 × 10–12 A (b) 8 × 10–12 A

(c) 12 × 10–12 A (d) 16 × 10–12 A

22. An ac source of angular frequency is fed across aresistor R and a capacitor C in series. The currentregistered is I. If now the frequency of source ischanged to /3 (but maintaining the same voltage),the current in the circuit is found to be halved.The ratio of reactance to resistance at the originalfrequency

(a)5

1(b)

5

2

(c)5

3(d)

5

4

23. A 200 km long telegraph wire has capacity of0.014 µF/km. If it carries an alternating current offrequency 5 kHz, the value of an inductancerequired to be connected in series so that theimpedance is minimum.

(a) 0.36 mH (b) 0.18 mH

(c) 0.9 mH (d) 0.3 mH

24. A 750 hertz, 20 V source is connected to aresistance of 100 ohm, an inductance of 0.1803henry and a capacitance of 10 microfarad all inseries. The time in which the resistance (thermalcapacity 2 J/0C) will get heated by 100C.

(a) 2.8 min. (b) 3.8 min.

(c) 4.8 min. (d) 5.8 min.

25. An LCR series circuit with 100 resistance isconnected to an ac source of 200 V and angularfrequency 300 rad/s. When only the capacitance isremoved, the current lags behind the voltage by 600.When only the inductance is removed, the currentleads the voltage by 600. The power dissipated inthe LCR circuit is

(a) 300 W (b) 400 W

(c) 500 W (d) 600 W

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EXCERCISE BASED ON NEW PATTERN

COMPREHENSION TYPE

Comprehension-1

A 10 ohm coil of mean area 500 cm2 and having1000 turns is held perpendicular to a uniform fieldof 0.4 gauss. The coil is turned through 1800 in(1/10)s.

1. The change in flux is

(a) 4 mWb (b) 6 mWb

(c) 8 mWb (d) 10 mWb

2. The average induced emf is

(a) 10 mV (b) 20 mV

(c) 30 mV (d) 40 mV

3. Average induced current is

(a) 2 mA (b) 4 mA

(c) 6 mA (d) 8 mA

4. The total induced charge is

(a) 100 µC (b) 200 µC

(c) 300 µC (d) 400 µC

Comprehension-2

An induction furnace uses electromagneticinduction to set up eddy currents in a conductor,thereby heating the conductor. Commercial unitsoperate at frequencies ranging from 60 Hz to 1 MHzand deliver powers from a few watts to severalmagawatts. Induction heating can be used forwelding in a vacuum chamber, to avoid oxidationor contamination of the metal. At high frequencies,induced currents appear only near the suface of theconductor — this is the “skin effect”. By creatingan induced current for a short time at anappropriate high frequency, a sample can be heateddown to a controlled depth. For example, thesurface of a farm tiller can be tempered to make ithard and brittle for effective cutting while keepingthe metal’s interior soft and ductile to resistbreakage.

To explore induction heating. Consider a flatconducting disk of radius R, thickness b andresistivity . A magnetic field B

max cos t is applied

perpendicular to the disk. Assume that thefrequency is so low that the skin effect is notimportant. Assume the eddy currents flow in circlesconcentric with the disk.

5. The maximum power delivered to the disk is

(a)

bBR 2.max

22

(b)

2

bBR 2.max

22

(c)

3

bBR 2.max

22

(d)

4

bBR 2.max

22

6. The average power delivered to the disk is

(a)

bBR 2.max

22

(b)

2

bBR 2.max

22

(c)

3

bBR 2.max

22

(d)

4

bBR 2.max

22

7. By what factor does the power change when theamplitude of the field doubles ?

(a) remains same (b) two times

(c) becomes half (d) four times

Comprehension-3

An air-core toroidal solenoid with cross-section areaA and mean radius r is closely wound with N turnsof wire. The wire is carrying a current ‘i’. The fieldof an idealized toroidal solenoid is confinedcompletely to the space enclosed by the windings.Assume that the magnetic field inside the solenoidis uniform across cross-section that is, neglect thevariation of magnetic field with distance from thetoroidal axis.

8. The magnetic field inside the solenoid is

(a)r2

NIµ0

(b)

r4

NIµ0

(c)r

NIµ0

(d)

r

NIµ2 0

9. Suppose N = 200 turns, A = 5.0 cm2 and r = 0.10 mthen its self inductance is

(a) 10 µH (b) 20 µH

(c) 30 µH (d) 40 µH

10. If the current in the toroidal solenoid increasesuniformly from zero to 6.0 amp in 3.0 µs. The valueof self induced emf is

(a) 80 V (b) 60 V

(c) 40 V (d) 20 V

11. The magnetic energy density depends on radius ‘r’as rn. The value of ‘n’ is

(a) –1 (b) –2

(c) 1 (d) 2

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Comprehension-4

Superconducting power transmission

The use of superconductors has been proposed forpower transmission lines. A single coaxial cablecould carry 1.00 × 103 MW (the output of a largepower plant) at 200 kV, dc, over a distance of 1000km without loss. An inner wire of radius 2.00 cm,made from the superconductor Nb

3Sn, carries the

current I in one direction. A surroundingsuperconducting cylinder, of radius 5.00 cm, wouldcarry the return current I.

12. In such a system, the magnetic field at the surfaceof the inner conductor is

(a) 50.0 mT (b) 40.0 mT

(c) 30.0 mT (d) 20.0 mT

13. In such a system, the magnetic field at the innersurface of the outer conductor is

(a) 50.0 mT (b) 40.0 mT

(c) 30.0 mT (d) 20.0 mT

14. The energy that would be stored in the spacebetween the conductors in a 1000 km superconduct-ing line is

(a) 1.29 MJ (b) 2.29 MJ

(c) 3.29 MJ (d) 4.29 MJ

15. The pressure exerted on the outer conductor is

(a) 118 Pa (b) 218 Pa

(c) 318 Pa (d) 418 Pa

Comprehension-5

In a circuit shown in the figure, switch S is closedat time t = 0. Thereafter, the constant current source,by varying its emf, maintains a constant current iout of its upper terminal.

16. The time constant of the circuit is

(a) L/R (b) zero

(c) L/2R (d) none

17. The current through the inductor as a function oftime is given by

(a) i(1 – e–2Rt/L) (b) 2i(1 – e–Rt/L)

(c) i(1 – e–Rt/2L) (d) i(1 – e–Rt/L)

18. The current through the resistor equals the currentthrough the inductor at time

(a) 2nR2

Ll (b) 2n

R

Ll

(c) 2nR

L2l (d) none

Comprehension-6

Choke Coil

A choke coil is an electrical instrument used forcontrolling current in an a.c. circuit. Choke coilconsists of an inductor with very small resistance.Choke coils are used with fluorescent mercury-tubefittings in houses. Let the inductance of theinductor is L and resistance is R. Let the voltageapplied is V = V

0 sin t.

19. The rms current through the choke coil is

(a))LR(2

V

222

0

(b)222

0

LR

V

(c)L

V0

(d)R

V0

20. The power consumed by the ideal choke coil is

(a) zero (b) very low

(c) very high (d) none

21. The power consumed by the choke coil is

(a) zero (b) very low

(c) very high (d) none

22. Iron cored chokes are used for reducing

(a) low frequency a.c.

(b) high frequency a.c.

(c) all types of frequencies

(d) none

23. Air cored chokes are used for reducing

(a) low frequency a.c.

(b) high frequency a.c.

(c) all types of frequencies

(d) none

24. In place of choke coil we can use to reduce the a.c.current

(a) resistor only (b) capacitor only

(c) both can be used

(d) neither can be used

Comprehension-7

D.C. motor

A D.C. motor converts direct current energy froma battery into mechanical energy. It is based on :torque will act on a current carrying coil placed inthe magnetic field.

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We know that, when a coil will be rotated in amagnetic field then there will be the induced emf.This is known as back emf.

Let the resistance of the coil is R, voltage of D.C.source is V and back emf is E.

25. The maximum current is

(a) V/R (b) E/R

(c)R

EV (d)

R

EV2

26. The current at any time is

(a) V/R (b) E/R

(c)R

EV (d)

R

EV2

27. The efficiency of the motor is

(a)V

E(b)

E

V

(c)E

V2(d)

E2

V

28. A motor having an amature of resistance 2.0 ohmoperates on 220 V mains. At its full speed, itdevelopes a back e.m.f. of 210 V.

(a) The current when the motor is switchedon is 110 amp

(b) The current when the motor is at fullspeed is 5 amp

(c) The efficiency of the motor is 95.5 %

(d) All the above

MATRIX-MATCH TYPE

Matching-1

A box P and a coil Q are connected in series with anac source of variable frequency. The emf of sourceis constant at 10 V. Box P contains a capacitance of1 µF in series with a resistance of 32. Coil Q has aself-inductance 4.9 mH and a resistance of 68 inseries. The frequency is adjusted so that themaximum current flows in P and Q.

Column - A Column - B

(A) The impedance of P at (P) 77

this frequency in ohm

(B) The impedance of Q at (Q) 97.6

this frequency in ohm

(C) The voltage across P in (R) 7.7

volt

(D) The voltage across Q in (S) 9.76

volt

MULTIPLE CORRECT CHOICE TYPE

1.

A small magnet M is allowed to fall through a fixedhorizontal conducting ring R. Let g be theacceleration due to gravity. The acceleration of Mwill be

(a) < g when it is above R and movingtowards R

(b) > g when it is above R and movingtowards R

(c) < g when it is below R and moving awayfrom R

(d) > g when it is below R and moving awayfrom R

2.

A square loop ABCD of edge a moves to the rightwith a velocity v, parallel to AB. There is a uniformmagnetic field of magnitude B, directed into thepaper, in the region between PQ and RS only. I, IIand III are three positions of the loop.

(a) the emf induced in the loop hasmagnitude Bav in all three positions.

(b) The induced emf is zero in position II.

(c) The induced emf is anticlockwise inposition I.

(d) The induced emf is clockwise in positionIII.

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3.

A conducting disc of radius r spins about its axiswith an angular velocity . There is a uniformmagnetic field of magnitude B perpendicular to theplane of the disc. C is the centre of the ring.

(a) No emf is induced in the disc.

(b) The potential difference between C and

the rim is 2

1Br2.

(c) C is at a higher potential than the rim.

(d) Current flows between C and the rim.

4.

A flat coil, C, of n turns, area A and resistance R isplaced in a uniform magnetic field of magnitude B.The plane of the coil is initially perpendicular to B.If the coil is rotated by an angle about the axisXY, charge of amount Q flows through it.

(a) If = 900, R

BAnQ

(b) If = 1800, R

BAn2Q

(c) If = 1800, Q = 0

(d) If = 3600, Q = 0

5. The SI unit of inductance, the henry, can bewritten as

(a) weber/ampere

(b) volt second/ampere

(c) joule/ampere2

(d) ohm second

6. Two different coils have self-inductance L1 = 8 mH,

L2 = 2 mH. The current in one coil is increased at a

constant rate. The current in the second coil is alsoincreased at the same constant rate. At a certaininstant of time, the power given to the two coils isthe same. At that time the current, the inducedvoltage and the energy stored in the first coil areI

1, V

1 and W

1 respectively. Corresponding values

for the second coil at the same instant are I2, V

2 and

W2 respectively. Then :

(a)4

1

I

I

2

1 (b) 4I

I

2

1

(c) 4W

W

1

2 (d)4

1

V

V

1

2

7. An inductance L, resistance R, battery B and switchS are connected in series. Voltmeters V

L and V

R are

connected across L and R respectively. When ‘S’ isclosed

(a) The initial reading in VL will be greater

than in VR

(b) The initial reading in VL will be lesser

than VR

(c) The initial readings in VL and V

R will be

the same

(d) The reading in VL will decrease as time

increases while that in VR will increase to

a maximum value

8. If L, Q, R represent inductance, charge andresistance respectively then the units of

(a) QR/L will be that of current

(b) Q2R3/L2 will be that of power

(c) QL/R will be that of current

(d) Q3R2/L will be that of power

9. An AC voltage of angular frequency is applied toa circuit which consists of an inductor of inductanceL and a capacitor of capacitance C in parallel. Thenacross the inductance

(a) current is maximum when 2 = 1/LC

(b) voltage is maximum when 2 = 1/LC

(c) current is minimum when 2 = 1/LC

(d) voltage is minimum when 2 = 1/LC

10. A capacitor is charged to a potential of V0. It is

connected with an inductor through a switch S. Theswitch is closed at time t = 0. Which of thefollowing statements are correct

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(a) the maximum current in the circuit is

L

CV0

(b) potential across capacitor becomes zero

for the first time at time LCt

(c) energy stored in the inductor at time

20CV

4

1isLC

2t

(d) maximum energy stored in the inductor

is 20CV

2

1

Assertion-Reason Type

Each question contains STATEMENT-1 (Assertion)and STATEMENT-2 (Reason). Each question has4 choices (A), (B), (C) and (D) out of which ONLYONE is correct.

(A) Statement-1 is True, Statement-2 is True;Statement-2 is a correct explanationfor Statement-1

(B) Statement-1 is True, Statement-2 is True;Statement-2 is NOT a correctexplanation for Statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True

1. STATEMENT-1 : A cylindrical magnet is placednear a circular coil. If the magnet is rotated aboutits own axis, no current is induced in the coil.

STATEMENT-2 : There is no change in magneticflux through the loop

2. STATEMENT-1 : Inserting an iron core in a coilincreases its self-inductance.

STATEMENT-2 : The self-inductance of the coildepends on the relative permeability.

3. STATEMENT-1 : Electric field lines produced bytime varying magnetic field is closed curves.

STATEMENT-2 : Electric field produced by timevarying magnetic field is non-conservative.

4. STATEMENT-1 : A transformer works on a.c.only and not on d.c.

STATEMENT-2 : In case of d.c., flux will beconstant and so no emf will be induced in thesecondary.

5. STATEMENT-1 : If an aluminium plate is movedrapidly through the region between the poles ofan electromagnet, it experiences a strongretarding force. However, if slots are cut into it,the force is greatly diminished.

STATEMENT-2 : It is due to the eddy currentflowing through the aluminium plate in largeramount, when the slots are not made.

6. STATEMENT-1 : A metal coil is kept stationaryin a non-uniform magnetic field. An emf is inducedin the coil.

STATEMENT-2 : emf will be induced only whenthe flux will change with time.

7. STATEMENT-1 : The figure shows an inductor Land a resistor R connected in parallel to a batterythrough a switch. The resistance of R is the sameas that of the coil that makes L. Two identicalbulbs B

1 and B

2 are put in series with L and R

respectively. When S is closed B1 lights up earlier

than B2.

STATEMENT-2 : The bulbs will be equally brightafter some time when the steady state is reached.

8. STATEMENT-1 : A light aluminium disc issuspended on a long string in front of the pole ofan electromagnet. When an alternating currentis passed through the winding of theelectromagnet, the disc is repelled

STATEMENT-2 : Due to change of magnetic fluxinduced (eddy) currents are set up in the disc inthe opposite direction. Hence the disc is repelled.

9. STATEMENT-1 : A capacitor of suitablecapacitance can be used in an a.c. circuit in placeof the choke coils.

STATEMENT-2 : Average power consume percycle in an ideal capacitor is zero.

10. STATEMENT-1 : A bulb connected in series witha solenoid is lit by a.c. source. If a soft iron core isintroduced in the solenoid then the bulb will glowdimmer.

STATEMENT-2 : On introduction of soft ironcode in the solenoid, its inductance increases.

11. STATEMENT-1 : Using the ordinary ammeter,we can measure the value of alternating current.

STATEMENT-2 : The average value of a.c. incomplete cycle is zero.

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12. STATEMENT-1 : Using hot wire instrument wecan measure the a.c. and d.c. both.

STATEMENT-2 : Both a.c. and d.c. produce heatwhich is proportional to square of the current.

(Answers) EXCERCISE BASED ON NEW PATTERN

COMPREHENSION TYPE

1. a 2. d 3. b 4. d 5. b 6. d

7. d 8. a 9. d 10. a 11. b 12. a

13. d 14. b 15. c 16. a 17. d 18. b

19. a 20. a 21. b 22. b 23. b 24. b

25. a 26. c 27. a 28. d

MATRIX-MATCH TYPE

1. [A-P; B-Q; C-R; D-S]

MULTIPLE CORRECT CHOICE TYPE

1. a, c 2. b, c, d 3. b, c 4. a, b, d 5. a, b, c, d

6. a, c, d 7. a, d 8. a, b 9. b, c 10. a, d

ASSERTION-REASON TYPE

1. A 2. A 3. A 4. A 5. A 6. D

7. D 8. A 9. A 10. A 11. D 12. A

INITIAL STEP EXERCISE

(SUBJECTIVE)

1. A square loop of wire (side a) lies on a table, adistance s from a very long straight wire, whichcarries a current I, as shown in figure.

(a) Find the flux of B through the loop.

(b) If someone now pulls the loop directlyaway from the wire, at speed v, what emfis generated ? In what direction(clockwise or anticlockwise) doesthe current flow ?

2. A square loop of wire with resistance R is moved atconstant speed v across a region whose sides aretwice the length of those of the square loop.

(a) Sketch a graph of the external forceF needed to move the loop atconstant speed, as function of thecoordinate x, from x = –2L tox = +2L. Take positive force tobe the right.

(b) Sketch a graph of the induced current inthe loop as a function f(x) ? Takeanticlockwise current to be positive.

3. The current in an ideal solenoid of radius R variesas a function of time. Find the magnitude of inducedelectric field at points (a) inside, and (b) outside the

solenoid. Express the result in terms of dt

dB. Also

draw the variation of electric field with ‘r’. Here‘r’ is the distance of a point from the axis of thesolenoid.

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4. Figure shows a square loop of side L perpendicularto the uniform field of a solenoid

Show that at any point on a side the component of

the induced electric field along the side is dt

dBL

4

1.

5. A long coaxial cable consists of two thin-walledconcentric conducting cylinders with radii a and b.The inner cylinder carries a steady current i, theother cylinder providing the return path for thatcurrent. The current sets up a magnetic fieldbetween the two cylinders. Calculate the energystored in the magnetic field for a length l of the cable.

6. In an L C circuit Qmax

= 100 µC; L = 40 mH;C = 100 µF. Find : (i) the equation for instant chargeon the capacitor; (ii) the equation for instantcurrent in the circuit; (iii) Plot the following graphs

(a) q versus t, (b) i versus t,

(c) UE versus t, (d) U

B versus t

7. A pair of parallel horizontal conducting rails ofnegligible resistance shorted at one end on a table.The distance between the rails is l. A conductingmassless rod of resistance R can slide on the railsfrictionlessly. The rod is tied to a massless stringwhich passes over a pulley fixed to the edge of thetable. A mass m, tied to the other end of the string,hangs vertically. A constant magnetic field B existsperpendicular to the table. If the system is releasedfrom rest, calculate

(a) the terminal velocity achieved by the rod,and

(b) the acceleration of the mass at the instantwhen the velocity of the rod is half theterminal velocity.

8. A coil A-C-D of radius R and number of turns ncarries a current i amp, and is placed in the planeof paper. A small conducting loop P of radius r isplaced at a distance y

0 from the centre and above

the coil A C D. Calculate the induced emf producedin the ring when the ring is allowed to fall freely.Express induced emf in terms of speed of the ring.

9. An inductor of inductance 2.0 mH is connectedacross a charged capacitor of capacitance 5.0 µFand the resulting L – C circuit is set oscillating atits natural frequency. Let q denote theinstantaneous charge on the capacitor, and I thecurrent in the circuit. It is found that the maximumvalue of q is 200µC. (a) When q = 100 µC what isthe value of |dI/dt| ? (b) When q = 200 µC, what isthe value of I ? (c) Find the maximum value of I(d) When I is equal to one half its maximum valuewhat is the value of |q| ?

10. Calculate the steady – state velocity with which aconducting wire of length l, mass m, and resistanceR will slide down without friction on two parallelconducting rails of negligible resistance. Thebottom ends of the rails and connected as shown infigure.

The inclination of the rails to the horizontal is ,and a uniform magnetic field of induction B isassumed to exist in the vertical direction.

11. A square frame with side a and a long straight wirecarrying a current I are located in the same planeas shown in the figure. The frame translates to theright with a constant velocity v.

Find the emf induced in the frame as function ofdistance x.

12. A long straight wire carrying a current I and aU-shaped conductor with sliding connector arelocated in the same plane as shown in the figure.The connector of length l, and resistance R slides tothe right with a constant velocity v.

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Find the current induced in the loop as a functionof separation x between the connector and thestraight wire.

13. A 3.56 H inductor is placed in series with a 12.8 resistor, and an emf of 3.24 V is then suddenlyapplied across the RL combination.

(a) At 0.278 s after the emf is applied what isthe rate at which energy is beingdelivered by the battery ?

(b) At 0.278 s, at what rate is energyappearing as thermal energy in theresistor ?

(c) At 0.278 s, at what rate is energy beingstored in the magnetic field ?

14. A long, straight wire has a constant current I. Ametal rod of length l moves at velocity v relative tothe wire, as shown in figure.

What is the potential difference between the endsof the rod

15. A metal disc of radius a = 25 cm rotates with aconstant angular velocity = 130 rad/s about itsaxis. Find the potential difference between thecentre and the rim of the disc if

(a) the external magnetic field is absent;

(b) the external uniform magnetic field ofinduction B = 5.0 mT is directedperpendicular to the disc.

16. Find the inductance of a solenoid of length l whosewinding is made of copper wire of mass m. Thewinding resistance is equal to R. The solenoiddiameter is considerably less than its length. Theresistivity and density of copper is and

0

respectively.

17. Find the inductance of a unit length of a cableconsisting of two thin-walled coaxial metallic cyl-inders if the radius of the outside cylinder is = 3.6times that of the inside one. The permeability of amedium between the cylinders is assumed to beequal to unity.

18. A superconducting round ring of radius a andinductance L was located in a uniform magneticfield of induction B. The ring plane was parallel tothe vector B, and the current in the ring was equalto zero. Then the ring was turned through 900 sothat its plane became perpendicular to the field.Find :

(a) the current induced in the ring after theturn;

(b) the work performed during the turn.

19. Two straight conducting rails form a right anglewhere their ends are joined. A conducting bar incontact with the rails starts at the vertex at timet = 0 and moves with a constant velocity of5.20 m/s along them, as shown in fig. A magneticfield with B = 0.350 T is directed out of the page.

Calculate (a) the flux through the triangle formedby the rails and bar at t = 3.00 s and (b) the emfaround the triangle at that time. (c) If we write the

emf as = atn, where a and n are constants, what

is the value of n ?

20. A conducting ring of radius a is rotated in auniform magnetic field B about P in the plane ofthe paper as shown in the figure.

(a) Find the induced emf between P and Qand indicate the polarity of the points Pand Q

(b) If a resistance R is connected between Pand Q determine the current through theresistor.

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21. A plane loop as shown in figure is shaped as twosquares with sides a = 20 cm and b = 10 cm and isintroduced into a uniform magnetic field at rightangles to the loop’s plane. The magnetic inductionvaries with time as B = B

0 sin t, where B

0 = 10 mT

and = 100 s–1.

Find the amplitude of the current induced in theloop if its resistance per unit length is equal to = 50 m m–1.

22. An inductor 20 × 10–3 henry, a capacitor 100 µFand a resistor 50 are connected in series across asource of emf V = 10 sin 314t. Find the energydissipated in the circuit in 20 minutes. If resistanceis removed from the circuit and the value of induc-tance is doubled, then find the variation of currentwith time in the new circuit.

23. A current of 4 A flows in a coil when connected to a12 V dc source. If the same coil is connected to a 12V, 50 rad/s ac source a current of 2.4 A flows in thecircuit. Determine the inductance of the coil. Alsofind the power developed in the circuit if a 2500 µFcapacitor is connected in series with the coil.

24. A choke coil is needed to operate an arc lamp at160 V (rms) and 50 Hz. The arc lamp has aneffective resistance of 5 when running at10 A(rms). Calculate the inductance of the chokecoil. If the same arc lamp is to be operated on 160 V(dc), what additional resistance is required ?Compare the power losses in both cases.

25. An ac source is connected to two circuits as shownin fig. (A) and (B).

Obtain current through the resistance R at reso-nance in both the circuits.

26. For the circuit shown in fig. current in inductanceis 0.8 A while in capacitance is 0.6 A.

What is the current drawn from the source ?

27. For the circuit shown in figure

Find the expressions for the impendence of thecircuit and phase of current.

28. A box contains L, C and R. When 250 V dc isapplied to the terminals of the box, a current of 1.0flows in the circuit. When an ac source of 250 Vrms at 2250 rad/s is connected, a current of 1.25 Arms flows. It is observed that the current rises withfrequency and becomes maximum at 4500 rad/s.Find the values of L, C and R. Draw the circuitdiagram.

29. An LCR circuit has L = 10 mH, R = 3 andC = 1 µF connected in series to a source of 15 cos tV. Calculate the current amplitude and theaverage power dissipates per cycle at a frequencythat is 10% lower than the resonance frequency.

30. A series LCR circuit containing a resistance of 120 has angular resonance frequency 4 × 105 rad s–1.At resonance the voltages across resistance andinductance are 60 V and 40 V respectively. Findthe values of L and C. At what frequency thecurrent in the circuit lags the voltage by 450 ?

PEMI – 22


New Delhi – 110 018, Ph. : 9312629035, 8527112111

FINAL STEP EXERCISE

(SUBJECTIVE)

1. A square loop of wire, with sides of length a, lies inthe first quadrant of the xy plane, with one cornerat the origin. In this region there is a non-uniform

time dependent magnetic field B(y, t) = y3 t2 k̂(where is a constant). Find the emf induced in theloop.

2. A wire frame of area 3.92 × 10–4m2 and resistance20 ohm is suspended freely from a 0.392 m longthread. There is a uniform horizontal magnetic fieldof 0.784 tesla and the plane of the wire frame isperpendicular to the magnetic field. The frame ismade to oscillate under the force of gravity bydisplacing it through 2 × 10–2 m from its initialposition along the direction of the magnetic field.The plane of the frame is always along thedirection of thread and does not rotate about it.What is the induced emf in the wire frame as afunction of time ? Also find the maximum currentin the wire frame.

3. Two long parallel horizontal rails, a distance d apartand each having a resistance per unit length, arejoined at one end by a resistance R. A perfectlyconducting rod MN of mass m is free to slide alongthe rails without friction. There is a uniformmagnetic field of induction B normal to the planeof the paper and directed into the paper. When avariable force F is applied, a constant current I flowsthrough it

R.

(a) Find the velocity of the rod and the applied force Fas a function of the distance x from R.

(b) What fraction of the work done per secby F is converted into heat ?

4. A long straight wire carries a current I0. At distances

a and b from it there are two other wires, parallelto the former one, which are interconnected by aresistance R (figure). A connector slides withoutfriction along the wires with a constant velocity v.Assuming the resistances of the wires, theconnector, the sliding contacts, and theself-inductance of the frame to negligible,

(a) Find the magnitude and the direction ofthe current induced in the connector

(b) Find the force required to maintain theconnector’s velocity constant.

5. In the figure shown il = 10e–2tA, i

2 = 4A and

vC = 3e–2tV. Determine

(a) iL and v

L

(b) vac

, vab

and vcd

(c) the energy stored in L and C, all asfunctions of time.

6. A wire shaped as a semi-circle of radius a rotatesabout its diamatric axis with an angular velocity in a uniform magnetic field of induction B as shownin the figure. The rotation axis is perpendicular tothe field direction. The total resistance of thecircuit is equal to R. Neglecting the magnetic fieldof the induced current, find the mean amount ofthermal power being generated in the loop duringa rotation period.

7. A very long conductor and an isosceles triangularconductor lie in a plane and separated from eachother as shown in the figure, a = 10 cm; b = 20 cm;h = 10 cm

PEMI – 23


New Delhi – 110 018, Ph. : 9312629035, 8527112111

(a) Find the coefficient of mutual induction.

(b) If current in the straight wire isincreasing at a rate of 2 A/s, find thedirection and magnitude of current in thetriangular wire. Diameter of the wirecross-section d = 1 mm. Resistivity of thewire, = 1.8 × 10–8 m.

8. A very small loop of radius a is placed at the centreof a very large loop of radius b as shown in thefigure. The large loop carries a constant current I

0

and is kept fixed in space. The small loop is rotatedabout its diametric axis with angular velocity . Ifthe resistance of the small loop is R and the selfinductance is negligible.

(a) Calculate the current in the small loopas a function of time.

(b) Find the torque required to rotate thesmall loop.

9. A wire loop enclosing a semi-circle of radius a islocated on the boundary of a uniform magnetic fieldof induction B. At the moment t = 0 the loop is setinto rotation with a constant angular acceleration about an axis O coinciding with a line of vector Bon the boundary. Find the emf induced in the loopas a function of time t. Draw the approximate plotof this function. The arrow in the figure shows theemf direction taken to be positive.

10. A metal rod of mass m can rotate about ahorizontal axis O, sliding along a circularconductor of radius a. The arrangement is locatedin a uniform magnetic field of induction B directedperpendicular to the ring plane. The axis and thering are connected to an emf source to form acircuit of resistance R. Neglecting the friction,circuit inductance, and ring resistance, find the lawaccording to which the source emf must vary tomake the rod rotate with a constant angularvelocity .

11. A - shaped conductor is located in a uniform

magnetic field perpendicular to the plane of theconductor and varying with time at the rateB = 0.10 T/s. A conducting connector starts movingwith an acceleration w = 10 cm/s2 along the parallelbars of the conductor. The length of the connectoris equal to l = 20 cm. Find the emf induced in theloop t = 2.0 s after the beginning of the motion, if atthe moment t = 0 the loop area and the magneticinduction are equal to zero. The inductance of theloop is to be neglected.

12. A closed circuit consists of a source of constant emf

and a choke coil of inductance L connected in

series. The active resistance of the whole circuit isequal to R. At the moment t = 0 the choke coilinductance was decreased abruptly times. Findthe current in the circuit as a function of time t.

13. A copper bar of the mass m and length l slides alongthe rails . The two upper ends of the rails areconnected by the capacitor of capacitance C. Thedistance between the rails is l. The entire system isplaced in a homogeneous magnetic field withinduction B directed vertically upward as shownin the figure. The self induction of the loop isassumed negligible.

PEMI – 24


New Delhi – 110 018, Ph. : 9312629035, 8527112111

Find the acceleration of the bar if coefficient of fric-tion between the buses and the bar is µ.

14. A metal rod OA of mass m and length l is keptrotating with a constant angular speed in avertical plane about a horizontal axis at the end O.The free end A is arranged to slide without frictionalong a fixed conducting circular ring in the sameplane as that of rotation. A uniform and constantmagnetic induction B is applied perpendicular andinto the plane of rotation as shown in figure.

An inductor L and an external resistance R areconnected through a switch S between the point Oand a point C on the ring to form an electricalcircuit. Neglect the resistance of the ring and therod. Initially, the switch is open.

(a) What is the induced emf across theterminals of the switch ?

(b) The switch S is closed at time t = 0

(i) Obtain an expression for thecurrent as a function of time.

(ii) In the steady state obtain thetime dependence of the torquerequired to maintain theconstant angular speed, giventhat the rod OA was along thepositive x - axis at t = 0.

15. A magnetic filed k̂a

yBB 0

is into the paper

in the +z direction. B0 and ‘a’ are positive constant.

A square loop EFGH of side ‘a’, mass m andresistance R in x – y plane, starts falling under theinfluence of gravity. Note the directions of x and yaxes in the figure. Find

(a) the induced current in the loop andindicate its direction, and

(b) the total Lorentz force acting on the loopand indicate its direction, and

(c) an expression for the speed of the loopv(t) and its terminal value.

16. Two parallel vertical metallic rails AB and CD areseparated by 1 m. They are connected at the twoends by resistances R

1 and R

2 as shown in figure. A

horizontal metallic bar of mass 0.2 kg slideswithout friction, vertically down the rails under theaction of gravity. There is a uniform horizontalmagnetic field of 0.6 T perpendicular to the planeof the rails. It is observed that when the terminalvelocity is attained, the power dissipated in R

1 and

R2 are 0.76 W and 1.2 W respectively. Find the

terminal velocity of the bar and the value of R1 and

R2.

17. A capacitor with capacitance of 10µF is periodicallycharged from a battery which produces a potentialdifference of 120V and is discharged through asolenoid 10cm long and with 200 turns. The meanmagnetic field induction inside the solenoid is3 × 10–4 T. How many times is the capacitor switchedover a second.

18. A circuit containing a two-position switch is shownin figure.

(a) The switch S is in position 1. Find thepotential difference V

A – V

B and the rate

of production of joule heat in R1.

(b) If now the switch is put in position 2 att = 0 find

(i) steady current is R4 and

(ii) The time when the current in R4

is half the steady value. Alsocalculate the energy stored in theinductor L at that time.

PEMI – 25


New Delhi – 110 018, Ph. : 9312629035, 8527112111

19. A metal bar AB can slide on two parallel thickmetallic rails separated by a distance l. A resistanceR and an inductance L are connected to the rails asshown in figure. A long straight wire carrying aconstant current I

0 is placed in the plane of the rails

and perpendicular to them as shown. The bar ABis held at rest at a distance x

0 from the long wire. At

t = 0, it is made to slide on the rails away from thewire. Answer the following questions.

(a) Find the relation smount i, di/dt and

d /dt, where i is the current in the

circuit and is the flux of the magnetic

field due to the long wire through thecircuit.

(b) it is observed that at time t = T, themetalbar AB is at a distance of 2x

0 from

the long wire and the resistance Rcarries a current i

1. Obtain an expression

for the net charge that has flown throughresistance R from t = 0 to t = T.

(c) The bar is suddenly stopped at time T.The current through resistance R is foundto be i

1/4 at time 2T. Find the value of

L/R in terms of the other givenquantities.

20(a). A thin non-conducting ring of mass m carrying acharge q can freely rotate about its axis. At theinitial moment the ring was at rest and nomagnetic field was present. Then a practicallyuniform magnetic field was switched on, which wasperpendicular to the plane of the ring and increased

with time according to a certain law B (t). Find the

angular velocity of the ring as a function of the

induction B (t).

(b) A thin wire ring of radius a and resistance r islocated inside a long solenoid so that their axescoincide. The length of the solenoid is equal to l, itscross-sectional radius, to b. At a certain momentthe solenoid was connected to a source of aconstant voltage V. The total resistance of thecircuit is equal to R. Assuming the inductance ofthe ring to be negligible, find the maximum valueof the radial force acting per unit length of the ring.

(c) The arrangement shown is placed in verticaluniform magnetic field. Two metal rod of length land mass m

1 and m

2 are pulled apart from rest by

a constant force F. Find the current in the resistoras a function of time ?

(d) A L-C circuit (inductance 0.01 H, capacity 1 µF) isconnected to a variable frequency ac source. Drawa rough sketch of the current variation as thefrequency is changed from 1 kHz to 2 kHz.

ANSWERS (SINGLE CORRECTCHOICE TYPE)

1. c

2. b

3. b

4. c

5. c

6. a

7. b

8. a

9. c

10. b

11. c

12. a

13. b

14. d

15. d

16. c

17. d

18. a

19. b

20. c

21. d

22. c

23. a

24. d

25. b

PEMI – 26


New Delhi – 110 018, Ph. : 9312629035, 8527112111

ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE)

1. (a)s

asln

2

Iaµ0

(b)

)as(s2

vIaµ 20

3. (a)dt

dB

2

r(b)

dt

dB

r2

R 2

5.a

bln

4

iµ 20

l

6. (i) 100cos(500t), (ii) –50sin(500t)

7. (a) 2)B(

mgR

l(b) g

2

1

8. 2/522

220

)yR(

yvrniRµ

2

3

9. (a) 104As–1 (b) 0 (c) 2A (d) 173 µC

10. mgR sin /(B2l2 cos2 )] 11.)ax(x

vIa2

4

µ 20

12.R

vIµ

2

1where,

xI 0

ind

l

13. (a) 518 mW (b) 328 mW (c) 191 mW

14.d

dln

2

Ivµ0

l15. (a) 3.0 nV; (b) 20 mV

16.0

0 mR

4

µL

l , where and 0 are the resistivity and the density of copper..

17. 0.26 µH/m.

18. (a) I = a2B/L; (b) A = ½2a4B2/L 19. (c) 1

20. (a) Ba2 (b) Ba2/R 21. 0.05 A

22. 0.52 cos 314 t 23. 17.28 W

24. r = 11 25. V/R, 0 26. 0.2 A

27.

2/12

2 L

1C

R

1

28. 1 µF, 0.049 H

29. 5.16 × 10–4

cycle

J30. 0.2 mH, µF

32

1, 8 × 105 Hz.

PEMI – 27


New Delhi – 110 018, Ph. : 9312629035, 8527112111

ANSWERS SUBJECTIVE (FINAL STEP EXERCISE)

1.5ta

2

1 2. e = 2 × 10–6 sin (10t) volt, 0.1 µA

3. (a) IBd

)x2R( , )x2R(

)Bd(

mI2BId

2

2

(b) 1

33dB

x2RmI21

4. (a)a

blnv

R

I

2

µ 00

(b)

R

v

a

bln

2

Iµ2

0

5. (a) iL = 4 – 2e–2t A; v

L = 16e–2t V

(b) vac

= 20e–2t –12; vab

= 17e–2t; vcd

= 12 + 16e–2t

(c) UL = 8(2 – e–2t)2 ; U

C = 9e–4t

6.

R

Ba

8

1P

22

avg

7. 1.22 × 10–8 H; 2.2 µA

8. tsinRb2

Iaµ;tsin

bR2

IaµI 2

2

02

002

0

9. ½(–1)n Bat, where n = 1, 2, .... is the number of the half-revolution that the loop performs

at the given moment t. 10. ½ (a3B3 + 2mg sin t)/aB

11. 12 mV 12. ]e)1(1[R

I L/Rt

13.mcosBC

cosµmgsinmga

222

l

14. (a) ½ Bl2 (b) R4

Btcos

2

Mg 42 ll

15. (a)R

avB0(anticlockwise) (b) )upward(

R

vaB 22

0 (c) mglogaB

mRe22

0

16. 1 ms–1, 3.0R,19

9R 21 17. 100 18. (a) 5V, 24.5 W

(b) (i) 0.6A (ii) 1.386 ms, 4.5 × 10–4 J

19. (a)dt

d (b)

l

0 Li2

Iµ

R

1 l(c)

2ln2

T

20. (a) )t(Bm2

q (b) 2

220

brR4

Vaµ

l(c)

tµR

lB

e1Bl

FI

22

21

21

mm

mmµ

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PEMI – 1 ELECTRO MAGNETIC INDUCTION AND ...einsteinclasses.com/Electro M_I_Ac.pdfalways be some magnetic flux through the closed loop of a current-carrying circuit. But the effect