PHYSICS FOR YOU Electromagnetic Induction 1

Chapter-6
Electromagnetic Induction
6.1 Introduction
Devices such as generators, transformers, microphones, sensitive chemical balances, pickup coils in
guitars etc., work on the principle of electromagnetic induction; a phenomenon of inducing current
by time varying magnetic fields. The main credit for the development of this phenomena shall go to
“symmetry” of nature.
Nature’s display of symmetry in art as well in physics are beautiful and alluring (look at the wings of
butterfly, sun flower, Oliva palm and peacock feathers!!). These symmetries are a source of wonder
and support us when we explore nature.

Fig.1 Symmetry in nature

We have many examples for symmetry in physics and one such is the interrelation between electricity
and magnetism. When Oerstead, In 1820, showed that an electric current generates a magnetic field,
the immediate question was can a magnetic field generate an electric current?. Physicists were tempted
to ask this question believing that nature loves symmetry. Anyway, this was answered in 1831 by
Joseph Henry of United States and Michael Faraday of Great Britain and the answer is YES. They did
the work almost simultaneously and independently1
The phenomena of generating electric current by a variable magnetic field is known as electromagnetic
induction (EMI). Faraday and Henry demonstrated this phenomena by conducting two experiments.
They are (i) Coil-Magnet experiment (ii) Coil – coil experiment

6.2 Coil-Magnet experiment

Coil - magnet experiment consists of a bar magnet and a coil connected to sensitive galvanometer. The
bar magnet moves to and fro near the coil. The following results are observed

1
[Very interesting information]:
Another curious symmetry at human level- in case of Faraday and Henry. It is like this
Symmetry-1: Self-educated Faraday was apprenticed at the age of 14 to a London bookbinder. Henry was apprenticed
at the age of 13 to a watch maker in New York.
Symmetry-2: Faraday was appointed as Director of the Royal Institution in London funded by an American, Benjamin
Thomson. Henry became secretary of the Smithsonian Institution in Washington DC, funded by an
Englishman, Smithson.
Source: Fundamentals of Physics-4th Ed-David Halliday, Robert Resnick, Jearl Walker, Asian Books Pvt.Ltd.-page-874

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 When magnet moves towards the coil momentary deflections are produced in the galvanometer.
 Deflections are reversed when the magnet is pulled away from the coil
 More deflections are produced when the magnet moves fast
 No deflections are noticed when the magnet is stationary (ie, not moving)
 No deflections are noticed when both coil and magnet move with same velocity
These observations clearly indicates that an emf and hence current is induced in the coil.

6.3 Coil-Coil Experiment

In this experiment there are two coils namely primary (C1) and secondary (C2) coils. C1 is connected
to galvanometer (G) and C2 is connected to battery along with tapping key. When the tapping key is
closed momentary deflections are produced in G and when tapping key is released deflections are
reversed. Further, if C1 and C2 are moved towards or away from each other keeping key closed, then
also deflections are produced. This indicates that the relative motion between the coils induces current.
Deflection increases
 When the magnet (or the coil) is moved very fast
 When the gap between the coils is too small (or too large)
 When an iron rod is inserted into the coils
 When number of turns in the coil increases

Fig.2.Coil-Magnet experiment Fig.3 Coil-Coil experiment

6.4 Faraday’ laws

Faraday gave two statements based on the experimental observations. They are now called as Faraday’s
laws of EMI
Law-I: An emf and hence current is induced in a coil due to change in magnetic flux linked with the
coil and this emf exists as long as the ‘flux change’ exists.
Law-II: The magnitude of induced emf is directly proportional to rate of change of magnetic flux
linked with the coil.
𝑑𝜙
𝑒=−
𝑑𝑡
If there are N number of turns in the loop then

𝑑𝜙
𝑒 = −𝑁
𝑑𝑡
Here e is induced emf and  is magnetic flux. Negative sign represents the Lenz law
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6.5 Lenz law

Faraday’s laws can give only the reason for induced emf and its magnitude but there is no clue about
the direction of flow of induced current. However Lenz law gives the direction of induced current flow.
It is as follows
Statement: The polarity of induced emf is such that it tends to produce a current which opposes the
change in magnetic flux that produces it. Or in simple words; the effect is against the cause that
produces it! Here the effect is ‘induced emf’ and the cause is ‘flux change’
 Significance: This law is in accordance with the laws of conservation of energy
 Explanation: When S-pole approaches the coil, induced current flows in clockwise direction
and the coil behaves as S-pole. Conversely, when N pole of the bar magnet approaches the coil
induced current starts flowing in anticlockwise direction so that the center of the coil behaves
as N-pole.
 Conclusion: The direction of induced current is anticlockwise when N-pole approaches the coil
and vice versa.

S N

Fig.4. Lenz law explained

6.6 Magnetic flux ()

The number of magnetic field lines passing through an imaginary loop of area dS placed in the field is
known as magnetic flux and it is given by the dot product of magnetic
field and the surface area of the loop
𝜙 = 𝐵. 𝑑𝐴 = 𝐵𝑑𝐴 cos 𝜃
Here  is the angle between direction of B and the normal drawn to the
surface of the loop.
The SI unit of magnetic flux is weber (Wb). It is maximum when the
surface is held perpendicular to the direction of B ( = 0) and zero when
Fig.5 Magnetic flux
parallel ( = 90). The maximum flux is given by
𝜙 = 𝐵𝐴
If the magnetic field has different magnitude and direction at different places then

𝜙 = 𝐵1 . 𝑑𝐴1 + 𝐵2 . 𝑑𝐴2 + 𝐵3 . 𝑑𝐴3 + − − −= ∑ 𝐵𝑖 . 𝑑𝐴𝑖

𝑖=𝑛

The change in magnetic flux linked with the coil is the main reason for induced emf. This change can
be initiated by
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i. Relative motion between the coil and magnet

ii. Changing the area of the coil
iii. Changing the shape of the coil
iv. Changing the orientation of the coil with the magnetic field direction
v. Changing the number of turns in the coil
vi. Introducing magnetic material (as a medium) in the coil

6.7 Types of induced emf

The variation of flux linked with the coil can happen in two ways as follows.
i. When a bar magnet is moved towards or away from the coil, the magnitude of the flux changes and
emf is induced. Emf is induced due to time varying magnetic field. It is known as stationary emf
or transformer emf (‘stationary’ because coil is at rest)
ii. When a coil (or conductor) is moved with certain velocity across a stationary magnetic field, flux
linked with the coil changes and emf is induced. It is known as motional or generator emf
(‘motional’ because coil is moving)

6.8 Motional emf

An emf induced in a coil when it is moved across the stationary magnetic field with a velocity v is
known as motional emf.
Consider a straight conductor of length l sliding on two conducting railings with a velocity v. These
railings are connected at one end by high resistance voltmeter and at the other end by a sliding rod.
Hence a closed circuit is formed. This closed circuit is placed in a static magnetic field B directed
towards the plane of the paper.

Fig.7. Equivalent circuit

Fig.6 Motional emf

Let dx is the distance covered by the conductor in dt seconds. The area it sweeps during this interval
is
𝐴 = 𝑙𝑑𝑥
But the change in flux linked with the conductor is
dΦ = 𝐵𝐴
Hence
dΦ = 𝐵𝑙𝑑𝑥
But the emf induced is given by

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𝑑Φ
𝑒=−
𝑑𝑡
𝑑 𝑑𝑥
𝑒 = − (𝐵𝑙𝑑𝑥) = −𝐵𝑙
𝑑𝑡 𝑑𝑡
𝒆 = −𝑩𝒍𝒗
This is the expression for motional emf
The equivalent circuit is as shown in Fig.7. The total resistance of the circuit is taken as R and induced
emf is e. The induced current is calculated as follows
𝑒 −𝐵𝑙𝑣
𝑖= =
𝑅 𝑅
It is also given by
1 𝑑𝜙
𝑖=− ( )
𝑅 𝑑𝑡
The direction of induced current can be determined from this equation. According to this equation, if
flux is increasing then d is positive and i is negative (means anticlockwise) and if flux is decreasing
then d is negative and i is positive (means clockwise)
NOTE-1:
1 𝑑𝜙
𝑖=− ( ) . This equation leads to following conclusions
𝑅 𝑑𝑡
 If a coil is moving towards the region of increasing magnetic flux the induced current flows in the
anticlockwise direction
 If a coil is moving towards the region of decreasing magnetic flux the induced current flows in the
clockwise direction
 If a coil is moving in the region of uniform magnetic flux there is no induced current

In the fig

 Coil (i) is moving towards increasing magnetic flux. Hence

current flows in bcda direction (anti)
 Coil (ii) moves towards decreasing flux. Hence current
flows in bac direction
 Coil (iii) also moves away from the field (towards
decreasing flux). Hence current flows in dabc direction
 Coil (iv) moves in uniform field. Hence no induced effect

NOTE-2:
Charge induced during the interval dt is

1 𝑑𝜙
𝑑𝑞 = 𝑖𝑑𝑡 = − ( ) 𝑑𝑡
𝑅 𝑑𝑡

𝑑𝜙
𝑑𝑞 = − ( )
𝑅

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It may be noted that the induced emf does not depend on resistance of the coil but induced current and induced
charge depend on the resistance R

NOTE-3:
As the conductor moves in a magnetic field the free charge carriers present in it will experience a force of 𝐹⃗ =
⃗⃗). As a result they move towards the ends and hence potential difference is developed across the conductor.
𝑞(𝑣⃗ × 𝐵
The resultant electric field is known as motional electric field (E) and it is given by
𝐹⃗
𝐸⃗⃗ =
𝑞
⃗⃗)
𝑞(𝑣⃗ × 𝐵
𝐸⃗⃗ = = (𝑣⃗ × 𝐵⃗⃗)
𝑞
6.9 Eddy currents
If a solid conducting plate is moved across the magnetic field current is induced. It cannot flow in a
definite direction because there is no conducting path and hence it swirls/rotates in a localized region.
This current is known as eddy current.

Fig.8 Eddy currents

Consider a lamina (metal plate) free to rotate about a pivot in a fixed magnetic field. It is allowed to swing down
through a magnetic field like a pendulum. When each time the plate enters and leaves the field, a portion of its
mechanical energy is transferred to its thermal energy. After several swings, no mechanical energy remains and
the warmed-up plate just hangs from its pivot. This is nothing but loss of energy due to eddy currents

Fig.9 Example for eddy current loss

Eddy current has both advantage as well as disadvantage

Advantages:
Eddy currents are useful in
1. Induction heating
2. Levitating (MAGLEVS)
3. Electromagnetic damping
4. Induction motor
5. Material surface defect testing
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6. Electromagnetic braking
7. Electric power meters
Disadvantages:
Some of the disadvantages of eddy currents are
1. Eddy currents produce heat in the core of transformers and hence there is power loss
2. Material testing can be done using eddy currents only for metals. It is not possible in case of
nonmetals because eddy currents are not induced in non-metals
3. The value of eddy current is highly sensitive to the value of permeability. This causes difficulty in
the functioning of detectors.

6.10 Inductance
Inductor is a device used to produce desired magnetic field. It is similar to capacitor (which produces
electric field).
Consider a solenoid having N turns and carrying current I. The magnetic flux linked with each turn is
 and the total flux linkage is N. It is given by
𝑁𝜙 = 𝐿𝐼
Here L is known as inductance. It is given by
𝑁𝜙
𝐿=
𝐼
Inductance is a measure of total flux linkage produced by the inductor (or coil) per unit current.
Inductance is the property of an inductor. Like capacitance is for capacitor, inductance is for inductor.
It is a scalar quantity and the SI unit is henry (H).
There are two types in inductance namely self-inductance and mutual inductance

6.11 Self-inductance (L)

The phenomena in which an emf is induced in a coil due to change in the magnetic flux in the same
coil is known as self-inductance. Solenoid is the best example for self-inductance. The coefficient of
self-inductance of solenoid is calculated as follows

Inductance of solenoid
Consider a solenoid of length l and number of turns per unit length n carrying current I. The flux linkage
is given by
𝑁𝜙 = (𝑛𝑙)𝐵𝐴
Put 𝐵 = 𝜇0 𝑛𝐼
𝑁𝜙 = 𝑛𝑙(𝜇0 𝑛𝐼)𝐴
𝑁𝜙 = 𝜇0 𝑛2 𝑙𝐴𝐼
𝑁𝜙
= 𝜇0 𝑛2 𝑙𝐴
𝐼
𝐿 = 𝜇0 𝑛2 𝑙𝐴

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This is the expression for inductance of solenoid. According to this equation inductance depends on
the geometry of the coil and number of turns.
The induced emf in the coil due to flux change in it is
𝑑(𝑁𝜙)
𝑒=−
𝑑𝑡
𝑑(𝐿𝐼)
𝑒=−
𝑑𝑡
𝑑𝐼
𝑒 = −𝐿
𝑑𝑡
i.e, in an inductor a self induced emf appears whenever the current in it changes with time. The negative
sign indicates that the induced emf is opposing the rate of change of current (Lenz law).
It may be noted here that if the magnitude of the current is constant (ie, DC) then there is no self induced
emf. Only the rate of change of the current counts.

6.12 Mutual Inductance (M)

Consider two coils namely primary coil and secondary coil. When current in the primary coil changes
an emf is induced in the secondary coil. This is known as mutual inductance.

Fig.10 Mutual inductance

The induced emf due to change in primary current I is given by

𝑑𝐼
𝑒 = −𝑀
𝑑𝑡
Here M is known as co efficient of mutual inductance. Coefficient of mutual inductance gives the
measure of flux linkage with the secondary coil due to current flowing in primary coil. It also depends
on the geometry of the coils. The expression for M is derived as follows

Expression for mutual inductance (M):

Consider two coils of same length l and cross sectional area A. Let
I = current in primary coil then M is given by
N1 = number of turns in secondary coil
N2 = number of turns in primary coil
l = length of each coil
A = cross sectional area of each coil

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The flux linked with secondary coil is given by

𝑁1 𝜙 = 𝑀𝐼
Put N1 = n1l 𝑛1 𝑙𝜙 = 𝑀𝐼
Put 𝜙 = 𝐵𝐴 𝑛1 𝑙𝐵𝐴 = 𝑀𝐼
Put B = 0n2I 𝑛1 𝑙𝜇0 𝑛2 𝐼𝐴 = 𝑀𝐼
∴ 𝑀 = 𝜇0 𝑛1 𝑛2 𝐴𝑙
This is the expression for mutual inductance. If a medium of relative permeability r is used then
∴ 𝑀 = 𝜇0 𝜇𝑟 𝑛1 𝑛2 𝐴𝑙
6.13 Energy stored in inductor
The self induced emf in inductor opposes the growth of current and hence it is also called as back emf.
It plays the role of inertia and hence can be treated as electromagnetic analogue of mass in mechanics.
Work is required to be done against the back emf and this work done is stored in the form of magnetic
energy. It is calculated as follows
The rate of doing work
𝑑𝑊
= |𝑒|𝐼 (it is similar to P = VI)
𝑑𝑡
𝑑𝐼
Put |𝑒| = 𝐿
𝑑𝑡
𝑑𝑊 𝑑𝐼
=𝐿 𝐼
𝑑𝑡 𝑑𝑡
𝑑𝑊 = 𝐿𝐼𝑑𝐼
The total work done in growing current from zero to I is
𝐼

∫ 𝑑𝑊 = ∫ 𝐿𝐼𝑑𝐼
0

1
𝑊 = 𝐿𝐼 2
2
This is stored in the form of energy. Hence
1
𝑈 = 𝐿𝐼 2
2
6.14 AC generator

AC generator converts mechanical energy into electrical energy. It consists of a coil of N turns each of
area A rotating between the two fixed magnetic pole pieces. As it rotates about its axis, the flux linkage
will change and emf is induced. It is calculated as follows

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Let B be the magnetic field and  be the angle between magnetic field direction and normal drawn to
the surface of the coil. Then magnetic flux is
𝜙 = 𝑁𝐵𝐴 cos 𝜃
If  is the angular velocity of the coil and  is the angular displacement in t seconds then
𝜃 = 𝜔𝑡
∴ 𝜙 = 𝑁𝐵𝐴 cos 𝜔𝑡
According to Faraday’s law
𝑑𝜙
𝑒=−
𝑑𝑡
𝑑(𝑁𝐵𝐴 cos 𝜔𝑡)
∴𝑒=−
𝑑𝑡
𝑒 = −𝑁𝐵𝐴𝜔 (−sin 𝜔𝑡)
𝑒 = 𝑁𝐵𝐴𝜔 sin 𝜔𝑡
𝑃𝑢𝑡 𝑒0 = 𝑁𝐵𝐴𝜔 called ′peak voltage′
Fig.11. AC generator
𝑒 = 𝑒0 sin 𝜔𝑡 − − − (1)
This is the expression for AC voltage. Current flowing through external resistor is
𝐼 = 𝐼0 sin 𝜔𝑡 − − − (2)
According equation (1) the variation of e with time t is periodic. A graph of e v/s t is sinusoidal as
shown in fig

Fig.12. Alternating voltage

As the coil rotates the angle  varies with time t and hence the emf value also changes.

i. At t = 0,  = 0, e = 0
ii. At t = T/4,  = 900, e = +emax
iii. At t = T/2,  = 1800, e = 0
iv. At t = 3T/4,  = 2700, e = -emax
v. At t = T,  = 3600, e = 0

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During one rotation the coil covers an angle of 3600 and takes time T (called time period). Induced emf
e changes its value from 0 to +emax and to 0 during positive half cycle, 0 to –emax to 0 during negative
half cycle. Both positive and negative cycle put together is known as one AC cycle.
Time taken for complete AC cycle is known as time period (T). The number of AC cycles completed
in one second is known as frequency (f). The value of AC frequency varies from country to country. It
is 50Hz in India

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