PEMI – 1
ELECTRO MAGNETIC INDUCTION AND ALTERNATING CURRENT
C1
Magnetic Flux
Like electric flux, magnetic flux, B, through a surface

 

dS is defined as B   B.dS . If B is uniform
S

then B = B.S 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 is
known 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, | e |
 
d
. As   B.A  BA cos  . Hence if there
dt
is any change in magnetic field (B) or area (A) or orientation () then there is induced emf. If some
situation, more than one of these may contribute in induced emf, in this case magnitude of induced emf is
written as
| e |
d
dB
dA
d
(BA cos )  (A cos )
 (B cos )
 BA sin 
dt
dt
dt
dt
This induced emf creates an induced current in the circuit whose magnitude is given as
I
induced emf
|e|


net resistance of circuit R . Also the charge flown = R .
Practice Problems :
1.
2.
A circular coil (constant radius) of total length L having number of turns N is rotated about the
diameter in a uniform magnetic field B with an angular velocity . Initially the magnetic field is
perpendicular to the plane of the coil. The maximum value of the emf induced in it is
(a)
BL2 
2N
(b)
NBL2 
2
(c)
BL2 
4N
(d)
NBL2 
4
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 total
resistance of the circuit is R. When the ring is suddenly squeezed to zero area, the charge flowing
through the galvanometer is
(a)
2AB/R
(b)
AB/R
(c)
AB
4R
(d)
AB
3R
[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 in
the direction that opposes the change of magnetic flux that induced it. Incorporating this law into Faraday’s
Law, the induced emf is given by e  
d
. The negative sign indicates that the induced emf opposes the
dt
change of the flux.
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PEMI – 2
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 :
2.
3.
(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
A rectangular coil (having resistance per unit length 10/3 /m) of 100 turns and size 0.1 m × 0.05 m
is 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
(d)
both (a) and (c) are correct
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 the
solenoid 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 the
length 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 e 
a



 ( v  B ).d l
b
Practice Problems :
1.
An electric potential difference will be induced between the ends of the conductor shown in the
diagram when it moves in the direction
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PEMI – 3
(a)
2.
P
(b)
Q
(c)
L
(d)
M
A conducting square loop ABCD of side L and resistance R moves in its plane with a uniform
velocity 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
 
 E.d l  
d B
dt
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 the
dependence of induced electric field on the distance r from the centre inside the space and outside
the 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 magnetic
field. Such an emf is called a self-induced emf. Self-induced emf’s can occur in any circuit, since there will
always be some magnetic flux through the closed loop of a current-carrying circuit. But the effect is greatly
enhanced if the circuit contains a coil with N turns of wire. As a result of the current i, there is an average
magnetic flux B, through each turn of the coil. Here we defined the self inductance L of the circuit as
follows L 
N B
I
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PEMI – 4
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
L
 µ0 n 2 A
l
Self induced emf
The self-induced emf, using Faraday’s law, is given by e  L
dI
dt
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 of
the coil is
(a)
2.
(b)
0.5 H
(c)
1.5 H
(d)
2H
A torodial solenoid with an air core has an average radius of 15 cm, area of cross-section 12 cm2 and
1200 turns. Ignoring the field variation across the cross-section of the toroid, the self-inductance of
the toroid is
(a)
3.
1H
4.6 mH
(b)
6.9 mH
(c)
2.3 mH
(d)
9.2 mH
A coil is wound on a frame of rectangular cross-section. If all the linear dimensions of the frame are
increased by a factor 2 and the number of turns per unit length of the coil remains the same, selfinductance 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 U 
C7
1 2
Li
2
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  L
(b)
dI
.
dt
If the direction of assumed current is opposite to the diretion of motion the voltage across the inductor rises
and is given by  L
dI
.
dt
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 the
switch S is thrown in contact at t = 0, current i in the circuit tends to grow. Hence an emf is induced across
the coil in such a direction as to oppose this current.
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PEMI – 5
By Kirchoff’s voltage law, we have
di
E  iR  L  0
dt

di
dt

E  iR L
Rt
[log(E – iR)]0i = 
Here

E 
Rt
1e L
 i(t) =

R
L

i


0
 Rdi

E  iR
t

0
 Rdt
L




L
is known as time constant of the circuit.
R
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  , i 
3.
The rate at which the source or battery will supply energy = Ei, rate at which the energy is dissipated in
E
, we can say at t  , the inductor behaves like a connecting wire.
R
 di 
resistor = i2R and the rate at which the energy stored in the inductor = i L  . From conservation of
 dt 
 di 
energy Ei  i 2 R  i L  .
 dt 
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 :
i
dI
R
dI
  dt 
Using KVL, iR  L
0 
I
L
dt

0
t
dI
R

dt  I  I e  t /  L
0
I
L

0
Practice Problems :
1.
In the following circuit initially there is no current through the inductor. Find the current passing
through the battery at any time t. Also find the current through the battery at t = 0 and t = .
(a)
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(b)
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PEMI – 6
2.
A solenoid has an inductance of 53 mH and a resistance of 0.37 . If it is connected to a battery, how
long 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 a
14.0 V battery is switched across the pair, how long will it take for the current through the resistor to
reach 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 and
R = 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
C9
uB 
B2 .
2µ 0
Mutual Induction
When a changing current i1 in one circuit causes a changing magnetic flux in a second circuit, an emf e2 is
induced in the second circuit; likewise, a changing current i2 in the second circuit induced an emf e1 in the
first circuit. This is called mutual induction.
e 2  M
di1
di
and e1  M 2
dt
dt
The constant M, called the mutual inductance, depends on the geometry of the two coils and on the material
between them. If the circuits are coils of wire with N1 and N2 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
:
M
N 2  B 2 N1 B1

i1
i2
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 µ0npnsAl where np and ns are
number of terms per unit length for primary and secondary coils and A is the cross-sectional area of primary
coil 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 with
angular frequency  :

1
LC
Such a circuit is analogous to a mechanical harmonic oscillator, with inductance L analogous to mass m, the
reciprocal 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 of
10 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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PEMI – 7
2.
A capacitor of 1 µ F initially charged to 10 V is connected across an ideal inductor of 0.1 mH. The
maximum current in the circuit is
(a)
0.5 A
(b)
1A
(c)
1.5 A
(d)
2A
[Answers : (1) d (2) b)
C11
Back EMF in D.C. Motor : A motor is the reverse of generator – it converts electrical energy into
mechanical energy. When currents is passed through a coil placed in a magnetic field, it rotates. As the coil
rotates, the magnetic flux linked with changes, giving rise to an induced emf. This emf opposes the applied
emf () and is, therefore, called back emf (e). If R is the resistance of the coil, the current through it is given
by I 
e
.
R
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e
E
(b)
e
E
(c)
Ee


 E 
2
(d)
e
 
E
2
[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 is
placed in a changing magnetic field, induced currents circulate throughout the volume of the body. These
are 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, and
area A is rotated in a uniform magnetic field B with angular velocity . As the coil rotates, the flux through
it 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 with
constant angular velocity  equal to the angular frequency of the sinusoidal quantity. Its projection on the
horizontal 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 current
amplitude I0 by
Iav 
2
I0  0.637I0 , I  I 0 .
rms

2
In the same way, the rms value of the snusoidal voltage is related to the voltage amplitude V0 by
V rms 
V0
2
The voltage v in an ac circuit is represented by v = v0sint 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 are
related by VR = IR
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PEMI – 8
Pure inductive circuit
The voltage across an inductor L leads the current by 900, the voltage and current amplitude are related by
VL = IXL,
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 = IXC,
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,
Z  R 2  ( X L  X C ) 2  R 2  (L  (1 / C)]2 ,
and the phase angle  of the voltage relative to the current is
tan  
L  1 / C
R
Practice Problems :
1.
A 40 electric heater is connected to 200 V, 50 Hz main supply. The peak value of the electric current
flowing in the circuit is approximately
(a)
2.
(b)
5.0 A
(c)
7A
(d)
10 A
An alternating voltage V = 2002 sin 100 t, where V in volt and t seconds, is connected to a series
combination of 1 µF capacitor and 10 k resistor through an ac ammeter. The reading of the
ammeter will be
(a)
3.
2.5 A
2 mA
(b)
102 mA
(c)
2 mA
(d)
20 mA
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
Pav 
1
VI cos   Vrms I rms cos 
2
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 = E0 sin t
has been applied, then the power consumed in the circuit depends on
(a)
2.
E0
(b)
I0
(c)
both
(d)
none
In circuit 1, an alternating current of 2 A flows for 10 minutes. In another similar circuit 2, a direct
current of 2 A flows for the same time. If the heat produced in circuit 1 is X then the heat produced
in circuit 2 is
(a)
0.5 X
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(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)
1 2
IpR
2
(c)
4 2
IpR

(d)
1 2
IpR

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 impedance
becomes 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 is
equal to the resistance R.
Practice Problems :
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]
1.
C18
Quality Factor
0L
where 0 is the resonance angular
R
frequency. It is an indicator of the sharpness of the current peak – higher the value of Q, sharper is the peak.
Transformer
A transformer converts a low aleternating voltage to a high voltage and vice-versa. It is based on the
principle of mutual induction. It consists of two coils wound on a soft iron core. The primary coil is
connected to an a.c. source.The secondary coil is connected to the load which may be a resistor or any other
electrical 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
The Quality factor of an LCR series circuit is defined as Q 
C19
Es
N
 s .
Ep N p
If Ns > Np, then Es > Ep and the transformer is called a step-up transformer.
If Ns < Np, then Es < Ep and the transformer is called a step-down transformer.
For an ideal transformer, Input power = Output power  EpIp = EsIs 
1.
2.
3.
4.
1.
2.
Ip
Is

Es
N
 s .
Ep N p
In actual transformers, there is some power loss. The main sources of power loss are :
I2R loss due to Joule heat in copper windings.
Heating produced due to Eddy currents in the iron core. This is reduced by using laminated core.
Hysteresis loss due to repeated magnetisation of the iron core.
Loss due to flux leakage.
When all the losses are minimized, the efficiency of the transformer becomes very high (90-99%).
Practice Problems :
In a step-down tranformer the input voltage is 22 kV and the output voltage is 550 V. The ratio of the
number of turns in the secondary to that in the primary is
(a)
1 : 20
(b)
20 : 1
(c)
1 : 40
(d)
40 : 1
In a noiseless transformer an alternating current of 2 A is flowing in the primary coil. The number of
turns in the primary and secondary coils are 100 and 20 respectively. The value of the current in the
secondary coil is
(a)
0.08 A
(b)
0.4 A
(c)
5A
(d)
10 A
[Answers : (1) c (2) d]
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 10
SINGLE CORRECT CHOICE TYPE
1.
In the following figure there is a magnetic f i e l d
of 0.2 Tesla along the positive x-axis.
5.
The magnetic flux through the face BCDG is
2.
3.
(a)
0
(b)
1.8mWb
(c)
1.0mWb
(d)
0.8mWb
A copper disc of radius R is rotated about its centre
with n revolutions per second in a uniform
magnetic field B. Choose the incorrect statement :
(a)
If the field is in the plane of the disc then
the induced emf between the centre and
the edge of the disc is zero
(b)
If the field is in the plane of the disc then
the induced emf between the centre and
the edge of the disc is non-zero
(c)
If the field is perpendicular to the disc
then the induced emf between the centre
and the edge of the disc is BnR2
(d)
both (a) and (c) are correct
A player with 3 metre long iron rod runs towards
east with a speed of 30 km/hr. Horizontal
component of earth’s magnetic field is
4 × 10 –5 Wb/m 2 . If he runs with the rod in
horizontal and vertical positions, then the
potential difference induced between the two ends
of the rod in the two cases will be
(a)
zero in vertical position, 1 × 10–3 V in
horizontal position
(b)
1 × 10–3 V in vertical position, zero in
horizontal position
(c)
zero in both positions
(d)
4.
–3
6.
(a)
0.5 cm/s
(b)
1 cm/s
(c)
2 cm/s
(d)
4 cm/s
Two inductors, each of inductance L, are connected
in series then effective self inductance (Leff) is given
by
(a)
Leff = 2L
(b)
L  Leff  3L
(c)
0  Leff  4L
(d)
2L  Leff  4L
A rectangular loop with a sliding connector of length
l is located in a uniform magnetic field
perpendicular to the loop plane as shown in figure.
The magnetic induction is equal to B. The
connector has an electric resistance R, the sides AB
and CD have resistance R1 and R2 respectively.
Neglecting the self inductance of the loop, find the
current flowing in the connector during its motion
with a constant velocity.
(a)
(b)
(c)
1 × 10 V in both positions
A square metal loop of side 10 cm and resistance
1 ohm is moved with a constant velocity partly
inside a uniform magnetic field of 2 Wb/m2, directed
into the paper, as shown in the figure. The loop is
connected to a network of five resistors each of value
3. If a steady current of 1 mA flows in the loop,
then the speed of the loop is
Einstein Classes,
(d)
BlV
R 1R 2
R
R1  R 2
2BlV
R 1R 2
R
R1  R 2
3BlV
R 1R 2
R
R1  R 2
4BlV
R 1R 2
R
R1  R 2
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 11
7.
8.
9.
10.
11.
An emf of 15 V is applied in a circuit containing 5
H inductance and 10 resistance. The ratio of the
currents at time t =  and t = 1 s is
(a)
e1/ 2
e1/ 2  1
(b)
e2
e2  1
(c)
1 – e–1
(d)
e–1
A rectangular loop of sides 8 cm and 2 cm is lying
in a uniform magnetic field of magnitude 0.5 T with
its plane normal to the field. The field is now
gradually reduced at the rate of 0.02 T/s. If the
resistance of the loop is 1.6 , then the power
dissipated 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
A torodial solenoid with an air core has an average
radius of 15 cm, area of cross-section 12 cm2 and
1200 turns. Ignoring the field variation across the
cross-section of the toroid, the self-inductance of
the toroid is
(a)
4.6 mH
(b)
6.9 mH
(c)
2.3 mH
(d)
9.2 mH
A coil is wound on a frame of rectangular
cross-section. If all the linear dimensions of the
frame are increased 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
13.
12.
(b)
both B1 and B2 die out with some delay
(c)
B1 dies out promptly but B2 with some
delay
(d)
B2 dies out promptly but B1 with some
delay
Two resistors of 10 and 20 and an ideal
inductor of 10 H are connected to a 2 V battery as
shown. The key K is inserted at time t = 0. The
initial (at t = 0) and final (at t = ) currents through
the battery are
Einstein Classes,
(b)
1
1
A; A
10 15
(c)
2
1
A; A
25 10
(d)
1
2
A; A
15 25
A tranformer is used to light a 140 W, 24 V bulb
from a 240 V A.C. mains. The current in the main
cable is 0.7 A. The efficiency of the transformer is
(a)
63.8%
(b)
83.3%
(c)
16.7%
(d)
36.2%
Two conducting rings of radii r and 2r move in
opposite directions with velocities 2v and v
respectively on a conducting surface S. There is a
uniform magnetic field of magnitude B
perpendicular to the plane of the rings. The
potential difference between the highest points of
the two rings is
15.
both B1 and B2 die out promptly
1
1
A; A
15 10
14.
In the given circuit R is a resistor, L is an inductor
and B1 and B2 are two bulbs. If the switch S is turned
off
(a)
(a)
(a)
zero
(b)
2rvB
(c)
4rvB
(d)
8rvB
A magnet is moved with a high speed towards a
coil at rest. Due to this, the induced emf, the
induced current and the induced charge in the coil
are E, I and Q respectively. If the speed of the
magnet 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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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 12
16.
17.
The number of turns of primary and secondary
coils of a transformer are 5 and 10 respectively and
the mutual inductance of the transformes is 25 H.
If the number of turns in the primary and
secondary are made 10 and 5 respectively, then the
mutual inductance of the transformer will be
(a)
6.25 H
(b)
12.5 H
(c)
25 H
(d)
50 H
22.
In an LCR circuit, choose the correct statement
(a)
An ac source of angular frequency  is fed across a
resistor R and a capacitor C in series. The current
registered is I. If now the frequency of source is
changed 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 original
frequency 
(a)
1
5
(b)
2
5
(c)
3
5
(d)
4
5
current and voltage are always in phase
1
if  
LC
(b)
23.
current leads the voltage if
A 200 km long telegraph wire has capacity of
0.014 µF/km. If it carries an alternating current of
frequency 5 kHz, the value of an inductance
required to be connected in series so that the
impedance is minimum.
  1 / LC
(a)
0.36 mH
(b)
0.18 mH
All are correct
(c)
0.9 mH
(d)
0.3 mH
current lags behind the voltage if
  1 / LC
(c)
(d)
18.
19.
20.
21.
A small square loop of wire of side l is placed inside
a large square loop of wire of side L (L >> l). The
loops are coplanar and their centres coincide. The
mutual inductance of the system is
µ0 l 2
 L
(a)
2 2
(c)
µ l2
2 0
 L
(b)
(d)
2
µ0 l2
 L
25.
µ l2
4 2 0
 L
Two circular loops of radii a and b (b >> a) are
placed coaxially a distance r ( r >> b) apart. The
mutual inductance between the loops is
(a)
2µ0a2b2/(2r3)
(c)
2 2
3
3µ0a b /(2r )
(b)
(d)
24.
µ0a2b2/(2r3)
2 2
3
4µ0a b /(2r )
A 750 hertz, 20 V source is connected to a
resistance of 100 ohm, an inductance of 0.1803
henry and a capacitance of 10 microfarad all in
series. The time in which the resistance (thermal
capacity 2 J/0C) will get heated by 100C.
(a)
2.8 min.
(b)
3.8 min.
(c)
4.8 min.
(d)
5.8 min.
An LCR series circuit with 100  resistance is
connected to an ac source of 200 V and angular
frequency 300 rad/s. When only the capacitance is
removed, the current lags behind the voltage by 600.
When only the inductance is removed, the current
leads the voltage by 600. The power dissipated in
the LCR circuit is
(a)
300 W
(b)
400 W
(c)
500 W
(d)
600 W
A magnetic flux through a stationary loop with a
resistance 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 loop
during that time
(a)
a23/R
(b)
1/2 a23/R
(c)
1/3 a23/R
(d)
1/4 a23/R
A current I = 3.36(1 + 2t) × 10–2 A increase at steady
rate in a long straight wire. A small circular loop of
radius 10–3 m has its plane parallel to the wire and
is placed at a distance of 1 m from the wire. The
resistance of the loop is 8.4 × 10 –4 . The
magnitude 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
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 13
EXCERCISE BASED ON NEW PATTERN
1.
2.
3.
4.
5.
COMPREHENSION TYPE
Comprehension-1
A 10 ohm coil of mean area 500 cm2 and having
1000 turns is held perpendicular to a uniform field
of 0.4 gauss. The coil is turned through 1800 in
(1/10)s.
The change in flux is
(a)
4 mWb
(b)
6 mWb
(c)
8 mWb
(d)
10 mWb
The average induced emf is
(a)
10 mV
(b)
(c)
30 mV
(d)
Average induced current is
(a)
2 mA
(b)
(c)
6 mA
(d)
7.
20 mV
40 mV
The maximum power delivered to the disk is
(a)
R 2 B 2 max .  2 b

(b)
R 2 B 2 max .  2 b
2
(c)
R 2 B 2 max .  2 b
3
(d)
R 2 B 2 max .  2 b
4
The average power delivered to the disk is
(a)
R 2 B 2 max .  2 b

(b)
R 2 B 2 max .  2 b
2
(c)
R 2 B 2 max .  2 b
3
(d)
R 2 B 2 max .  2 b
4
By what factor does the power change when the
amplitude 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 area
A and mean radius r is closely wound with N turns
of wire. The wire is carrying a current ‘i’. The field
of an idealized toroidal solenoid is confined
completely to the space enclosed by the windings.
Assume that the magnetic field inside the solenoid
is uniform across cross-section that is, neglect the
variation of magnetic field with distance from the
toroidal axis.
4 mA
8 mA
The total induced charge is
(a)
100 µC
(b)
200 µC
(c)
300 µC
(d)
400 µC
Comprehension-2
An induction furnace uses electromagnetic
induction to set up eddy currents in a conductor,
thereby heating the conductor. Commercial units
operate at frequencies ranging from 60 Hz to 1 MHz
and deliver powers from a few watts to several
magawatts. Induction heating can be used for
welding in a vacuum chamber, to avoid oxidation
or contamination of the metal. At high frequencies,
induced currents appear only near the suface of the
conductor — this is the “skin effect”. By creating
an induced current for a short time at an
appropriate high frequency, a sample can be heated
down to a controlled depth. For example, the
surface of a farm tiller can be tempered to make it
hard and brittle for effective cutting while keeping
the metal’s interior soft and ductile to resist
breakage.
To explore induction heating. Consider a flat
conducting disk of radius R, thickness b and
resistivity . A magnetic field Bmax cos t is applied
perpendicular to the disk. Assume that the
frequency is so low that the skin effect is not
important. Assume the eddy currents flow in circles
concentric with the disk.
Einstein Classes,
6.
8.
9.
10.
11.
The magnetic field inside the solenoid is
(a)
µ 0 NI
2r
(b)
µ 0 NI
4r
(c)
µ 0 NI
r
(d)
2µ 0 NI
r
Suppose N = 200 turns, A = 5.0 cm2 and r = 0.10 m
then its self inductance is
(a)
10 µH
(b)
20 µH
(c)
30 µH
(d)
40 µH
If the current in the toroidal solenoid increases
uniformly from zero to 6.0 amp in 3.0 µs. The value
of self induced emf is
(a)
80 V
(b)
60 V
(c)
40 V
(d)
20 V
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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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 14
Comprehension-4
Superconducting power transmission
The use of superconductors has been proposed for
power transmission lines. A single coaxial cable
could carry 1.00 × 103 MW (the output of a large
power plant) at 200 kV, dc, over a distance of 1000
km without loss. An inner wire of radius 2.00 cm,
made from the superconductor Nb3Sn, carries the
current I in one direction. A surrounding
superconducting cylinder, of radius 5.00 cm, would
carry the return current I.
12.
13.
14.
15.
Comprehension-6
Choke Coil
A choke coil is an electrical instrument used for
controlling current in an a.c. circuit. Choke coil
consists of an inductor with very small resistance.
Choke coils are used with fluorescent mercury-tube
fittings in houses. Let the inductance of the
inductor is L and resistance is R. Let the voltage
applied is V = V0 sin t.
19.
In such a system, the magnetic field at the surface
of the inner conductor is
(a)
50.0 mT
(b)
40.0 mT
(c)
30.0 mT
(d)
20.0 mT
In such a system, the magnetic field at the inner
surface of the outer conductor is
(a)
50.0 mT
(b)
40.0 mT
(c)
30.0 mT
(d)
20.0 mT
The energy that would be stored in the space
between the conductors in a 1000 km superconducting line is
(a)
1.29 MJ
(b)
2.29 MJ
(c)
3.29 MJ
(d)
4.29 MJ
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 closed
at time t = 0. Thereafter, the constant current source,
by varying its emf, maintains a constant current i
out of its upper terminal.
(a)
17.
18.
The time constant of the circuit is
(a)
L/R
(b)
zero
(c)
L/2R
(d)
none
The current through the inductor as a function of
time 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)
The current through the resistor equals the current
through the inductor at time
(a)
L
ln 2
2R
(b)
L
ln 2
R
(c)
2L
ln 2
R
(d)
none
Einstein Classes,
V0
2
2( R   2 L2 )
V0
(b)
R 2   2 L2
(c)
V0
L
(d)
V0
R
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
Iron cored chokes are used for reducing
(a)
low frequency a.c.
(b)
high frequency a.c.
22.
23.
16.
The rms current through the choke coil is
24.
(c)
all types of frequencies
(d)
none
Air cored chokes are used for reducing
(a)
low frequency a.c.
(b)
high frequency a.c.
(c)
all types of frequencies
(d)
none
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 from
a battery into mechanical energy. It is based on :
torque will act on a current carrying coil placed in
the magnetic field.
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 15
We know that, when a coil will be rotated in a
magnetic field then there will be the induced emf.
This is known as back emf.
(C)
(D)
The voltage across P in
volt
The voltage across Q in
(R)
7.7
(S)
9.76
volt
MULTIPLE CORRECT CHOICE TYPE
1.
25.
Let the resistance of the coil is R, voltage of D.C.
source is V and back emf is E.
The maximum current is
(a)
V/R
(b)
E/R
(c)
26.
27.
28.
VE
R
(d)
2V  E
R
A small magnet M is allowed to fall through a fixed
horizontal conducting ring R. Let g be the
acceleration due to gravity. The acceleration of M
will be
(a)
< g when it is above R and moving
towards R
(b)
> g when it is above R and moving
towards R
(c)
< g when it is below R and moving away
from R
The current at any time is
(a)
V/R
(b)
E/R
(c)
VE
R
(d)
2V  E
R
The efficiency of the motor is
(a)
E
V
(b)
V
E
(c)
2V
E
(d)
V
2E
A motor having an amature of resistance 2.0 ohm
operates on 220 V mains. At its full speed, it
developes a back e.m.f. of 210 V.
The current when the motor is switched
on is 110 amp
(b)
The current when the motor is at full
speed is 5 amp
(c)
The efficiency of the motor is 95.5 %
(d)
All the above
MATRIX-MATCH TYPE
Matching-1
(d)
> g when it is below R and moving away
from R
2.
(a)
(A)
(B)
A box P and a coil Q are connected in series with an
ac source of variable frequency. The emf of source
is constant at 10 V. Box P contains a capacitance of
1 µF in series with a resistance of 32. Coil Q has a
self-inductance 4.9 mH and a resistance of 68  in
series. The frequency is adjusted so that the
maximum current flows in P and Q.
Column - A
Column - B
The impedance of P at
(P)
77
this frequency in ohm
The impedance of Q at
(Q)
97.6
this frequency in ohm
Einstein Classes,
A square loop ABCD of edge a moves to the right
with a velocity v, parallel to AB. There is a uniform
magnetic field of magnitude B, directed into the
paper, in the region between PQ and RS only. I, II
and III are three positions of the loop.
(a)
the emf induced in the loop has
magnitude Bav in all three positions.
(b)
The induced emf is zero in position II.
(c)
The induced emf is anticlockwise in
position I.
(d)
The induced emf is clockwise in position
III.
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 16
for the second coil at the same instant are I2, V2 and
W2 respectively. Then :
3.
A conducting disc of radius r spins about its axis
with an angular velocity . There is a uniform
magnetic field of magnitude B perpendicular to the
plane of the disc. C is the centre of the ring.
(a)
No emf is induced in the disc.
(b)
I1 1

I2 4
(b)
I1
4
I2
(c)
W2
4
W1
(d)
V2 1

V1 4
An inductance L, resistance R, battery B and switch
S are connected in series. Voltmeters VL and VR are
connected across L and R respectively. When ‘S’ is
closed
The potential difference between C and
the rim is
(c)
(d)
7.
(a)
1 2
Br .
2
C is at a higher potential than the rim.
Current flows between C and the rim.
(a)
(b)
(c)
4.
(d)
A flat coil, C, of n turns, area A and resistance R is
placed 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 axis
XY, charge of amount Q flows through it.
8.
BAn
R
(a)
If  = 900, Q 
(b)
2BAn
If  = 1800, Q 
R
9.
5.
(c)
If  = 1800, Q = 0
(d)
If  = 3600, Q = 0
The SI unit of inductance, the henry, can be
written as
(a)
weber/ampere
(b)
volt second/ampere
2
6.
(c)
joule/ampere
(d)
ohm second
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 also
increased at the same constant rate. At a certain
instant of time, the power given to the two coils is
the same. At that time the current, the induced
voltage and the energy stored in the first coil are
I1, V1 and W1 respectively. Corresponding values
Einstein Classes,
10.
The initial reading in VL will be greater
than in VR
The initial reading in VL will be lesser
than VR
The initial readings in VL and VR will be
the same
The reading in VL will decrease as time
increases while that in VR will increase to
a maximum value
If L, Q, R represent inductance, charge and
resistance 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
An AC voltage of angular frequency  is applied to
a circuit which consists of an inductor of inductance
L and a capacitor of capacitance C in parallel. Then
across 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
A capacitor is charged to a potential of V0. It is
connected with an inductor through a switch S. The
switch is closed at time t = 0. Which of the
following statements are correct
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
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(a)
the maximum current in the circuit is
V0
(b)
5.
C
L
potential across capacitor becomes zero
for the first time at time t   LC
(c)
energy stored in the inductor at time
t
(d)

1
LC is CV02
2
4
6.
maximum energy stored in the inductor
1
2
is CV0
2
7.
Assertion-Reason Type
Each question contains STATEMENT-1 (Assertion)
and STATEMENT-2 (Reason). Each question has
4 choices (A), (B), (C) and (D) out of which ONLY
ONE is correct.
(A)
Statement-1 is True, Statement-2 is True;
Statement-2 is a correct explanation
for Statement-1
STATEMENT-2 : emf will be induced only when
the flux will change with time.
STATEMENT-1 : The figure shows an inductor L
and a resistor R connected in parallel to a battery
through a switch. The resistance of R is the same
as that of the coil that makes L. Two identical
bulbs B1 and B2 are put in series with L and R
respectively. When S is closed B1 lights up earlier
than B2.
(B)
1.
2.
3.
4.
Statement-1 is True, Statement-2 is True;
Statement-2 is NOT a correct
explanation for Statement-1
(C)
Statement-1 is True, Statement-2 is False
(D)
Statement-1 is False, Statement-2 is True
STATEMENT-1 : A cylindrical magnet is placed
near a circular coil. If the magnet is rotated about
its own axis, no current is induced in the coil.
STATEMENT-1 : If an aluminium plate is moved
rapidly through the region between the poles of
an electromagnet, it experiences a strong
retarding force. However, if slots are cut into it,
the force is greatly diminished.
STATEMENT-2 : It is due to the eddy current
flowing through the aluminium plate in larger
amount, when the slots are not made.
STATEMENT-1 : A metal coil is kept stationary
in a non-uniform magnetic field. An emf is induced
in the coil.
STATEMENT-2 : There is no change in magnetic
flux through the loop
STATEMENT-1 : Inserting an iron core in a coil
increases its self-inductance.
STATEMENT-2 : The self-inductance of the coil
depends on the relative permeability.
STATEMENT-1 : Electric field lines produced by
time varying magnetic field is closed curves.
STATEMENT-2 : Electric field produced by time
varying magnetic field is non-conservative.
STATEMENT-1 : A transformer works on a.c.
only and not on d.c.
STATEMENT-2 : In case of d.c., flux will be
constant and so no emf will be induced in the
secondary.
Einstein Classes,
8.
9.
10.
11.
STATEMENT-2 : The bulbs will be equally bright
after some time when the steady state is reached.
STATEMENT-1 : A light aluminium disc is
suspended on a long string in front of the pole of
an electromagnet. When an alternating current
is passed through the winding of the
electromagnet, the disc is repelled
STATEMENT-2 : Due to change of magnetic flux
induced (eddy) currents are set up in the disc in
the opposite direction. Hence the disc is repelled.
STATEMENT-1 : A capacitor of suitable
capacitance can be used in an a.c. circuit in place
of the choke coils.
STATEMENT-2 : Average power consume per
cycle in an ideal capacitor is zero.
STATEMENT-1 : A bulb connected in series with
a solenoid is lit by a.c. source. If a soft iron core is
introduced in the solenoid then the bulb will glow
dimmer.
STATEMENT-2 : On introduction of soft iron
code in the solenoid, its inductance increases.
STATEMENT-1 : Using the ordinary ammeter,
we can measure the value of alternating current.
STATEMENT-2 : The average value of a.c. in
complete cycle is zero.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 18
12.
STATEMENT-1 : Using hot wire instrument we
can measure the a.c. and d.c. both.
STATEMENT-2 : Both a.c. and d.c. produce heat
which 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, a
distance s from a very long straight wire, which
carries a current I, as shown in figure.
(b)
(a)
(b)
2.
Find the flux of B through the loop.
If someone now pulls the loop directly
away from the wire, at speed v, what emf
is generated ? In what direction
(clockwise or anticlockwise) does
the current flow ?
A square loop of wire with resistance R is moved at
constant speed v across a region whose sides are
twice the length of those of the square loop.
(a)
Sketch a graph of the external force
F needed to move the loop at
constant speed, as function of the
coordinate x, from x = –2L to
x = +2L. Take positive force to
be the right.
Einstein Classes,
3.
Sketch a graph of the induced current in
the loop as a function f(x) ? Take
anticlockwise current to be positive.
The current in an ideal solenoid of radius R varies
as a function of time. Find the magnitude of induced
electric field at points (a) inside, and (b) outside the
solenoid. Express the result in terms of
dB . Also
dt
draw the variation of electric field with ‘r’. Here
‘r’ is the distance of a point from the axis of the
solenoid.
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 19
4.
Figure shows a square loop of side L perpendicular
to the uniform field of a solenoid
Show that at any point on a side the component of
9.
10.
1 dB .
the induced electric field along the side is L
4 dt
5.
6.
7.
8.
A long coaxial cable consists of two thin-walled
concentric conducting cylinders with radii a and b.
The inner cylinder carries a steady current i, the
other cylinder providing the return path for that
current. The current sets up a magnetic field
between the two cylinders. Calculate the energy
stored in the magnetic field for a length l of the cable.
In an L C circuit Qmax = 100 µC; L = 40 mH;
C = 100 µF. Find : (i) the equation for instant charge
on the capacitor; (ii) the equation for instant
current in the circuit; (iii) Plot the following graphs
(a)
q versus t,
(b)
i versus t,
(c)
UE versus t,
(d)
UB versus t
A pair of parallel horizontal conducting rails of
negligible resistance shorted at one end on a table.
The distance between the rails is l. A conducting
massless rod of resistance R can slide on the rails
frictionlessly. The rod is tied to a massless string
which passes over a pulley fixed to the edge of the
table. A mass m, tied to the other end of the string,
hangs vertically. A constant magnetic field B exists
perpendicular to the table. If the system is released
from rest, calculate
(a)
the terminal velocity achieved by the rod,
and
(b)
the acceleration of the mass at the instant
when the velocity of the rod is half the
terminal velocity.
A coil A-C-D of radius R and number of turns n
carries a current i amp, and is placed in the plane
of paper. A small conducting loop P of radius r is
placed at a distance y0 from the centre and above
the coil A C D. Calculate the induced emf produced
in the ring when the ring is allowed to fall freely.
Express induced emf in terms of speed of the ring.
Einstein Classes,
11.
12.
An inductor of inductance 2.0 mH is connected
across a charged capacitor of capacitance 5.0 µF
and the resulting L – C circuit is set oscillating at
its natural frequency. Let q denote the
instantaneous charge on the capacitor, and I the
current in the circuit. It is found that the maximum
value of q is 200µC. (a) When q = 100 µC what is
the value of |dI/dt| ? (b) When q = 200 µC, what is
the value of I ? (c) Find the maximum value of I
(d) When I is equal to one half its maximum value
what is the value of |q| ?
Calculate the steady – state velocity with which a
conducting wire of length l, mass m, and resistance
R will slide down without friction on two parallel
conducting rails of negligible resistance. The
bottom ends of the rails and connected as shown in
figure.
The inclination of the rails to the horizontal is ,
and a uniform magnetic field of induction B is
assumed to exist in the vertical direction.
A square frame with side a and a long straight wire
carrying a current I are located in the same plane
as shown in the figure. The frame translates to the
right with a constant velocity v.
Find the emf induced in the frame as function of
distance x.
A long straight wire carrying a current I and a
U-shaped conductor with sliding connector are
located in the same plane as shown in the figure.
The connector of length l, and resistance R slides to
the right with a constant velocity v.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 20
17.
18.
13.
14.
Find the current induced in the loop as a function
of separation x between the connector and the
straight wire.
A 3.56 H inductor is placed in series with a 12.8 
resistor, and an emf of 3.24 V is then suddenly
applied across the RL combination.
(a)
At 0.278 s after the emf is applied what is
the rate at which energy is being
delivered by the battery ?
(b)
At 0.278 s, at what rate is energy
appearing as thermal energy in the
resistor ?
(c)
At 0.278 s, at what rate is energy being
stored in the magnetic field ?
A long, straight wire has a constant current I. A
metal rod of length l moves at velocity v relative to
the wire, as shown in figure.
19.
Find the inductance of a unit length of a cable
consisting of two thin-walled coaxial metallic cylinders if the radius of the outside cylinder is  = 3.6
times that of the inside one. The permeability of a
medium between the cylinders is assumed to be
equal to unity.
A superconducting round ring of radius a and
inductance L was located in a uniform magnetic
field of induction B. The ring plane was parallel to
the vector B, and the current in the ring was equal
to zero. Then the ring was turned through 900 so
that its plane became perpendicular to the field.
Find :
(a)
the current induced in the ring after the
turn;
(b)
the work performed during the turn.
Two straight conducting rails form a right angle
where their ends are joined. A conducting bar in
contact with the rails starts at the vertex at time
t = 0 and moves with a constant velocity of
5.20 m/s along them, as shown in fig. A magnetic
field with B = 0.350 T is directed out of the page.
Calculate (a) the flux through the triangle formed
by the rails and bar at t = 3.00 s and (b) the emf
around the triangle at that time. (c) If we write the
20.
15.
16.
What is the potential difference between the ends
of the rod
A metal disc of radius a = 25 cm rotates with a
constant angular velocity  = 130 rad/s about its
axis. Find the potential difference between the
centre and the rim of the disc if
(a)
the external magnetic field is absent;
(b)
the external uniform magnetic field of
induction B = 5.0 mT is directed
perpendicular to the disc.
Find the inductance of a solenoid of length l whose
winding is made of copper wire of mass m. The
winding resistance is equal to R. The solenoid
diameter is considerably less than its length. The
resistivity and density of copper is  and  0
respectively.
Einstein Classes,
emf as  = atn, where a and n are constants, what
is the value of n ?
A conducting ring of radius a is rotated in a
uniform magnetic field B about P in the plane of
the paper as shown in the figure.
(a)
(b)
Find the induced emf between P and Q
and indicate the polarity of the points P
and Q
If a resistance R is connected between P
and Q determine the current through the
resistor.
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
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21.
A plane loop as shown in figure is shaped as two
squares with sides a = 20 cm and b = 10 cm and is
introduced into a uniform magnetic field at right
angles to the loop’s plane. The magnetic induction
varies with time as B = B0 sin t, where B0 = 10 mT
and  = 100 s–1.
26.
22.
23.
24.
25.
Find the amplitude of the current induced in the
loop if its resistance per unit length is equal to
 = 50 m m–1.
An inductor 20 × 10–3 henry, a capacitor 100 µF
and a resistor 50  are connected in series across a
source of emf V = 10 sin 314t. Find the energy
dissipated in the circuit in 20 minutes. If resistance
is removed from the circuit and the value of inductance is doubled, then find the variation of current
with time in the new circuit.
A current of 4 A flows in a coil when connected to a
12 V dc source. If the same coil is connected to a 12
V, 50 rad/s ac source a current of 2.4 A flows in the
circuit. Determine the inductance of the coil. Also
find the power developed in the circuit if a 2500 µF
capacitor is connected in series with the coil.
A choke coil is needed to operate an arc lamp at
160 V (rms) and 50 Hz. The arc lamp has an
effective resistance of 5  when running at
10 A(rms). Calculate the inductance of the choke
coil. 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.
An ac source is connected to two circuits as shown
in fig. (A) and (B).
27.
28.
29.
30.
Einstein Classes,
Obtain current through the resistance R at resonance in both the circuits.
For the circuit shown in fig. current in inductance
is 0.8 A while in capacitance is 0.6 A.
What is the current drawn from the source ?
For the circuit shown in figure
Find the expressions for the impendence of the
circuit and phase of current.
A box contains L, C and R. When 250 V dc is
applied to the terminals of the box, a current of 1.0
flows in the circuit. When an ac source of 250 V
rms at 2250 rad/s is connected, a current of 1.25 A
rms flows. It is observed that the current rises with
frequency and becomes maximum at 4500 rad/s.
Find the values of L, C and R. Draw the circuit
diagram.
An LCR circuit has L = 10 mH, R = 3 and
C = 1 µF connected in series to a source of 15 cos t
V. Calculate the current amplitude and the
average power dissipates per cycle at a frequency
that is 10% lower than the resonance frequency.
A series LCR circuit containing a resistance of 120
 has angular resonance frequency 4 × 105 rad s–1.
At resonance the voltages across resistance and
inductance are 60 V and 40 V respectively. Find
the values of L and C. At what frequency the
current in the circuit lags the voltage by 450 ?
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 22
FINAL STEP EXERCISE
(SUBJECTIVE)
1.
A square loop of wire, with sides of length a, lies in
the first quadrant of the xy plane, with one corner
at 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 the
loop.
2.
3.
A wire frame of area 3.92 × 10–4m2 and resistance
20 ohm is suspended freely from a 0.392 m long
thread. There is a uniform horizontal magnetic field
of 0.784 tesla and the plane of the wire frame is
perpendicular to the magnetic field. The frame is
made to oscillate under the force of gravity by
displacing it through 2 × 10–2 m from its initial
position along the direction of the magnetic field.
The plane of the frame is always along the
direction of thread and does not rotate about it.
What is the induced emf in the wire frame as a
function of time ? Also find the maximum current
in the wire frame.
5.
Two long parallel horizontal rails, a distance d apart
and each having a resistance  per unit length, are
joined at one end by a resistance R. A perfectly
conducting rod MN of mass m is free to slide along
the rails without friction. There is a uniform
magnetic field of induction B normal to the plane
of the paper and directed into the paper. When a
variable force F is applied, a constant current I flows
through it
(b)
Find the force required to maintain the
connector’s velocity constant.
In the figure shown i l = 10e –2tA, i 2 = 4A and
vC = 3e–2tV. Determine
(a)
iL and vL
(b)
vac, vab and vcd
(c)
the energy stored in L and C, all as
functions of time.
A wire shaped as a semi-circle of radius a rotates
about its diamatric axis with an angular velocity 
in a uniform magnetic field of induction B as shown
in the figure. The rotation axis is perpendicular to
the field direction. The total resistance of the
circuit is equal to R. Neglecting the magnetic field
of the induced current, find the mean amount of
thermal power being generated in the loop during
a rotation period.
7.
A very long conductor and an isosceles triangular
conductor lie in a plane and separated from each
other as shown in the figure, a = 10 cm; b = 20 cm;
h = 10 cm
Find the velocity of the rod and the applied force F
as a function of the distance x from R.
(b)
4.
Find the magnitude and the direction of
the current induced in the connector
6.
R.
(a)
(a)
What fraction of the work done per sec
by F is converted into heat ?
A long straight wire carries a current I0. At distances
a and b from it there are two other wires, parallel
to the former one, which are interconnected by a
resistance R (figure). A connector slides without
friction along the wires with a constant velocity v.
Assuming the resistances of the wires, the
connector, the sliding contacts, and the
self-inductance of the frame to negligible,
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 23
8.
(a)
Find the coefficient of mutual induction.
(b)
If current in the straight wire is
increasing at a rate of 2 A/s, find the
direction and magnitude of current in the
triangular wire. Diameter of the wire
cross-section d = 1 mm. Resistivity of the
wire,  = 1.8 × 10–8 m.
A very small loop of radius a is placed at the centre
of a very large loop of radius b as shown in the
figure. The large loop carries a constant current I0
and is kept fixed in space. The small loop is rotated
about its diametric axis with angular velocity . If
the resistance of the small loop is R and the self
inductance is negligible.
10.
A metal rod of mass m can rotate about a
horizontal axis O, sliding along a circular
conductor of radius a. The arrangement is located
in a uniform magnetic field of induction B directed
perpendicular to the ring plane. The axis and the
ring are connected to an emf source to form a
circuit of resistance R. Neglecting the friction,
circuit inductance, and ring resistance, find the law
according to which the source emf must vary to
make the rod rotate with a constant angular
velocity .
11.
A  - shaped conductor is located in a uniform
magnetic field perpendicular to the plane of the
conductor and varying with time at the rate
B = 0.10 T/s. A conducting connector starts moving
with an acceleration w = 10 cm/s2 along the parallel
bars of the conductor. The length of the connector
is equal to l = 20 cm. Find the emf induced in the
loop t = 2.0 s after the beginning of the motion, if at
the moment t = 0 the loop area and the magnetic
induction are equal to zero. The inductance of the
loop is to be neglected.
12.
A closed circuit consists of a source of constant emf
 and a choke coil of inductance L connected in
9.
(a)
Calculate the current in the small loop
as a function of time.
(b)
Find the torque required to rotate the
small loop.
A wire loop enclosing a semi-circle of radius a is
located on the boundary of a uniform magnetic field
of induction B. At the moment t = 0 the loop is set
into rotation with a constant angular acceleration
 about an axis O coinciding with a line of vector B
on the boundary. Find the emf induced in the loop
as a function of time t. Draw the approximate plot
of this function. The arrow in the figure shows the
emf direction taken to be positive.
Einstein Classes,
series. The active resistance of the whole circuit is
equal to R. At the moment t = 0 the choke coil
inductance was decreased abruptly  times. Find
the current in the circuit as a function of time t.
13.
A copper bar of the mass m and length l slides along
the rails . The two upper ends of the rails are
connected by the capacitor of capacitance C. The
distance between the rails is l. The entire system is
placed in a homogeneous magnetic field with
induction B directed vertically upward as shown
in the figure. The self induction of the loop is
assumed negligible.
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PEMI – 24
Find the acceleration of the bar if coefficient of friction between the buses and the bar is µ.
14.
16.
Two parallel vertical metallic rails AB and CD are
separated by 1 m. They are connected at the two
ends by resistances R1 and R2 as shown in figure. A
horizontal metallic bar of mass 0.2 kg slides
without friction, vertically down the rails under the
action of gravity. There is a uniform horizontal
magnetic field of 0.6 T perpendicular to the plane
of the rails. It is observed that when the terminal
velocity is attained, the power dissipated in R1 and
R2 are 0.76 W and 1.2 W respectively. Find the
terminal velocity of the bar and the value of R1 and
R 2.
17.
A capacitor with capacitance of 10µF is periodically
charged from a battery which produces a potential
difference of 120V and is discharged through a
solenoid 10cm long and with 200 turns. The mean
magnetic field induction inside the solenoid is
3 × 10–4 T. How many times is the capacitor switched
over a second.
18.
A circuit containing a two-position switch is shown
in figure.
A metal rod OA of mass m and length l is kept
rotating with a constant angular speed  in a
vertical plane about a horizontal axis at the end O.
The free end A is arranged to slide without friction
along a fixed conducting circular ring in the same
plane as that of rotation. A uniform and constant
magnetic induction B is applied perpendicular and
into the plane of rotation as shown in figure.
An inductor L and an external resistance R are
connected through a switch S between the point O
and a point C on the ring to form an electrical
circuit. Neglect the resistance of the ring and the
rod. Initially, the switch is open.
15.
(a)
What is the induced emf across the
terminals of the switch ?
(b)
The switch S is closed at time t = 0
(i)
Obtain an expression for the
current as a function of time.
(ii)
In the steady state obtain the
time dependence of the torque
required to maintain the
constant angular speed, given
that the rod OA was along the
positive x - axis at t = 0.
 B0 y 
 k̂ is into the paper
 a 
A magnetic filed B  
in the +z direction. B0 and ‘a’ are positive constant.
A square loop EFGH of side ‘a’, mass m and
resistance R in x – y plane, starts falling under the
influence of gravity. Note the directions of x and y
axes in the figure. Find
(a)
the induced current in the loop and
indicate its direction, and
(b)
the total Lorentz force acting on the loop
and indicate its direction, and
(c)
an expression for the speed of the loop
v(t) and its terminal value.
Einstein Classes,
(a)
The switch S is in position 1. Find the
potential difference VA – VB and the rate
of production of joule heat in R1.
(b)
If now the switch is put in position 2 at
t = 0 find
(i)
steady current is R4 and
(ii)
The time when the current in R4
is half the steady value. Also
calculate the energy stored in the
inductor L at that time.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 25
19.
A metal bar AB can slide on two parallel thick
metallic rails separated by a distance l. A resistance
R and an inductance L are connected to the rails as
shown in figure. A long straight wire carrying a
constant current I 0 is placed in the plane of the rails
and perpendicular to them as shown. The bar AB
is held at rest at a distance x0 from the long wire. At
t = 0, it is made to slide on the rails away from the
wire. 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 the
circuit.
(b)
it is observed that at time t = T, the
metalbar AB is at a distance of 2x0 from
the long wire and the resistance R
carries a current i1. Obtain an expression
for the net charge that has flown through
resistance R from t = 0 to t = T.
(c)
The bar is suddenly stopped at time T.
The current through resistance R is found
to be i1/4 at time 2T. Find the value of
L/R in terms of the other given
quantities.
20(a). A thin non-conducting ring of mass m carrying a
charge q can freely rotate about its axis. At the
initial moment the ring was at rest and no
magnetic field was present. Then a practically
uniform magnetic field was switched on, which was
perpendicular 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)
B (t).
A thin wire ring of radius a and resistance r is
located inside a long solenoid so that their axes
coincide. The length of the solenoid is equal to l, its
cross-sectional radius, to b. At a certain moment
the solenoid was connected to a source of a
constant voltage V. The total resistance of the
circuit is equal to R. Assuming the inductance of
the ring to be negligible, find the maximum value
of the radial force acting per unit length of the ring.
Einstein Classes,
(c)
The arrangement shown is placed in vertical
uniform magnetic field. Two metal rod of length l
and mass m1 and m2 are pulled apart from rest by
a constant force F. Find the current in the resistor
as a function of time ?
(d)
A L-C circuit (inductance 0.01 H, capacity 1 µF) is
connected to a variable frequency ac source. Draw
a rough sketch of the current variation as the
frequency is changed from 1 kHz to 2 kHz.
ANSWERS (SINGLE CORRECT
CHOICE TYPE)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
c
b
b
c
c
a
b
a
c
b
c
a
b
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
d
d
c
d
a
b
c
d
c
a
d
b
PEMI – 26
ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE)
(b)
µ 0 Ia 2 v
2s(s  a)
1.
(a)
µ 0 Ia s  a
ln
2
s
3.
(a)
r dB
2 dt
(b)
R 2 dB
2r dt
5.
µ 0 i 2l b
ln
4
a
6.
(i)
100cos(500t),
(ii)
–50sin(500t)
7.
(a)
mgR
(Bl ) 2
(b)
1
g
2
8.
3 µ0 niR 2 r 2 yv
2 (R 2  y 2 ) 5 / 2
9.
(a)
(b)
0
104As–1
2 2
2A
11.
µ0 2Ia 2 v
4 x ( x  a )
328 mW
(c)
191 mW
15.
3.0 nV;
2
10.
mgR sin /(B l cos )]
12.
I ind 
13.
(a)
14.
µ 0 Iv l  d
ln
2
d
16.
L
17.
18.
20.
0.26 µH/m.
(a)
I = a2B/L;
(a)
Ba2
(d)
173 µC

1 µ 0 l vI
, where  
x
2 R
518 mW
(b)
(a)
(b)
20 mV
µ 0 mR
4 l 0 , where  and 0 are the resistivity and the density of copper..
22.
24.
0.52 cos 314 t
r = 11 
27.
2
 1 
1  



C



 R2 
L  


29.
(c)
5.16 × 10–4
Einstein Classes,
J
cycle
(b)
(b)
25.
A = ½2a4B2/L
Ba2/R
V/R, 0
19.
21.
(c)
1
0.05 A
23.
26.
17.28 W
0.2 A
28.
1 µF, 0.049 H
30.
0.2 mH,
1 / 2
1
µF , 8 × 105 Hz.
32
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PEMI – 27
ANSWERS SUBJECTIVE (FINAL STEP EXERCISE)
1.
1
ta 5
2
2. e = 2 × 10–6 sin (10t) volt, 0.1 µA
(a)
2 mI 2
( R  2 x )
BId

( R  2 x )
I,
(Bd ) 2
Bd
4.
(a)
µ0 I0
b
v ln
2 R
a
5.
(a)
(b)
(c)
iL = 4 – 2e–2t A; vL = 16e–2t V
vac = 20e–2t –12; vab = 17e–2t; vcd = 12 + 16e–2t
UL = 8(2 – e–2t)2 ; UC = 9e–4t
3.
6.
Pavg

1 a 2 B

8
R
(b)
 2mIR  2x  
1 

B3 d 3


1
2
 µ0 I b  v
 2 ln a  R


(b)

2
7.
1.22 × 10–8 H; 2.2 µA
2
8.
9.
 µ a 2 I 0  
 µ 0 a 2 I 0      2
I 0
sin

t
;




   sin t
 2bR 
 2b   R 
½(–1)n Bat, where n = 1, 2, .... is the number of the half-revolution that the loop performs
at the given moment t.
11.
14.
12 mV
12.
2
10.
I

[1  (  1)e  tR / L ]
R
½ Bl
15.
(a)
B0 av
(anticlockwise) (b)
R
16.
1 ms–1, R 1 
(b)
(i)
20.
13.
d
dt

(a)
q 

B( t )
2m
Einstein Classes,
mg sin   µmg cos 
Cl 2 B2 cos 2   m
(c)
mR
log e mg
B02 a 2
18.
(a)
2
B0 a 2 v
(upward)
R
9
, R 2  0.3
17.
100
19
0.6A
(ii)
1.386 ms, 4.5 × 10–4 J
(a)
a
Mgl
B 2 l 4
cos

t

(b)
2
4R
(a)
19.
½ (a3B3 + 2mg sin t)/aB
(b)
1  µ 0 Il

 Li l 

R  2

(c)
(b)
µ0a 2 V 2
4rRlb 2

F 
I  1  e µR
Bl 

T
2 ln 2
B2l 2
(c)
5V, 24.5 W
t

 µ  m1m 2
m1  m 2

Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111