Electromagnetic Induction
Chapter - 6
Electromagnetic Induction
When a current carrying conductor is kept in a magnetic field, it experiences a force and
it moves. If a closed circuit is moved in magnetic field then emf and current are induced. The
phenomenon in which electric current is induced by varying magnetic fields is called
Electromagnetic Induction.
This was observed by Faraday in 1831.
Experiments of Faraday and Henry
Experiment 1: Current Induced by a Magnet
1. When the North pole of a bar magnet is
pushed towards the coil, the galvanometer
shows a sudden deflection indicating that
the current is induced in the coil.
2. When the magnet is brought to rest, the
deflection reduces to zero. There is no
deflection when the bar magnet is held
stationary anywhere.
3. When the magnet is moved away from the coil, the galvanometer shows deflection in the
opposite direction which indicates the reversal in the direction of induced current.
4. The galvanometer deflections are found to be larger when the magnet is moved faster
towards or away from the coil.
5. When the bar magnet is held fixed and the coil C is moved towards or away from the
magnet, the same effects are observed. This shows that the relative motion between the
magnet and the coil is responsible for the generation of electric current in the coil.
Experiment 2: Current Induced by a Current
1. When the coil C2 is moved towards the coil C1 the
galvanometer shows a deflection which indicates
that an electric current is induced in the coil C1.
2. When C2 is moved away from the coil C1, the
galvanometer shows deflection in the opposite
direction. This indicates that the direction of
induced current in C1 is reversed.
Electromagnetic Induction
3. If the coil C2 is held fixed and coil C1 is moved the same effects are observed.
4. The deflection is observed as long as there is a relative motion between the coils C1 and C2.
No deflections is observed if two coils are held stationary or moved with zero relative
velocity.
5. The galvanometer deflection is found to be larger when coils are moved faster towards or
away from each other.
Experiment 3: Current Induced by a Changing Current
1. When the key K is pressed the galvanometer
shows momentary deflection and returns to zero
immediately.
2. When the key K is kept pressed continuously,
there is no deflection in the galvanometer.
3. When the key K is released a galvanometer again
shows momentary deflection, but in the opposite direction and returns to zero immediately.
4. The galvanometer deflection increases dramatically when the iron rod is inserted into the
coils along the axis and the key is pressed or released.
Magnetic Flux
The magnetic flux ϕ through any surface held in a magnetic field ⃗ is
defined as “The total number of magnetic lines of force crossing the
surface”.
ϕ = ⃗ . ⃗ = B A Cos θ
Units of magnetic flux:
The SI unit of magnetic flux is Weber (Wb) or tesla metre square (Tm2).
One Weber is the amount of magnetic flux over an area of 1 m2 held normal to a uniform
magnetic field of one tesla.
1 Weber = 1 tesla × 1 m2
The C.G.S unit of ϕ is Maxwell (Mx).
1 Weber = 108 Maxwell
Electromagnetic Induction
Faraday’s Law of Electromagnetic Induction
First law: Whenever there is a change in magnetic flux linked with a circuit changes, an emf is
induced in the circuit. The induced emf lasts as long as the change in magnetic flux continues.
Second law: The magnitude of induced emf is directly proportional to the rate of change of
magnetic flux linked with the circuit.
e∝
e=k
k is a constant of proportionality = 1
∴e=
If dϕ is the small change in magnetic flux in a small time dt.
e=The –ve sign indicates the induced emf always opposes any change in magnetic flux with the
circuit.
For N turns
e=-N
Lenz Law and Conservation of Energy
In 1834, German physicist Heinrich Friedrich Lenz deduced a rule, known as Lenz’s law
which gives the polarity of the induced emf clearly and precisely.
It states that “the polarity of the induced emf is such that it tends to produce a current
which opposes the change in magnetic flux that produced it”.
When North Pole of a bar magnet is pushed towards a closed coil, magnetic flux through
the coil increases. Hence an emf is induced because of which induced current flows in the coil
the direction of the induced
current is such that it opposed the
increase in magnetic flux. This is
possible only if the current in the
coil is anti-clockwise direction with respect to an observer on the side of the magnet. Now the
magnetic dipole moment associated with this current has north polarity towards the north pole of
the approaching magnet. This north polarity repels in the north pole of the approaching magnet.
Similarly if the north pole of the magnet is moved away from the coil, the magnetic flux through
the coil decreases. Hence an emf is induced in the coil because of which an induced current
flows. The direction of the induced current is such that it opposes the decrease in magnetic flux.
Electromagnetic Induction
This if possible only if the current in the coil is in clockwise direction with respect to an observer
on the side of the magnet. Now the magnetic dipole moment associated with the current has
south polarity towards the north pole of the magnet moving away. This south polarity attracts the
north pole of the magnet moving away.
Lenz law is in accordance with the law of
conservation of energy. In the above experiment, when
North Pole of magnet is moved towards the coil, the right
face of the coil acquires North polarity. Thus, work has to
be done against the force of repulsion in bringing the magnet closer to the coil.
When N pole of magnet is moved away, South Pole develops on the right face of the coil.
Therefore, work has to be done against the force of attraction in taking the magnet away from the
coil.
This mechanical work in moving the magnet with respect to the coil changes into
electrical energy producing induced current. Hence, energy transformation takes place.
Motional Electromotive Force (emf)
Consider a long straight conductor PQ of length
l placed in a uniform magnetic field B such that
the direction of B is perpendicular to the length
of the conductor and perpendicular to the plane
as shown in the figure. The conductor is moved
with a velocity v in the direction perpendicular
to the magnetic field B. Let the conductor PQ
moves by a distance dx in a small interval dt.
The area covered by conductor is ds = l dx
The change in magnetic flux with the conductor is dϕ = B ds = B l dx
The magnitude of the emf induced in the conductor
e=
=
e=Blv
(
)
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Electromagnetic Induction
Energy Consideration in Motional emf
When a straight conductor PQ of length l is
moved with a velocity v in the direction
perpendicular to the magnetic field B, the
motional emf is produced in the conductor is
e=Blv
Let r be the resistance of movable arm PQ. The current through the conductor is
I= =
The magnetic force on the conductor PQ moving perpendicular to the magnetic field B is
F=BIl
F=B
l=
The conductor PQ is being paused with a constant speed v. The power required to push the
conductor against the force is
P=Fv=
P=
v
... (1)
As the conductor is pushed mechanically, the mechanical energy dissipated per second is given
by
PJ = I2r
PJ = (
P=
) r
... (2)
This equation is same as equation (1). Hence the mechanical energy required to move the
conductor is converted into electrical energy and then to thermal energy.
Eddy Currents
“Eddy currents are the currents induced in a conductor when placed in a changing magnetic
field”.
Experiment to Demonstrate Eddy Currents
A flat metallic plate is suspended between
the pole pieces N and S of an electromagnet
as shown in the diagram. It is allowed to
S
N
come to rest with its plane perpendicular
to the magnetic field.
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Electromagnetic Induction
When the magnetic field is off, the metallic plate is allowed to oscillate. It is found to
oscillate for a long time. When the electromagnet is switched on and the plate is set into
oscillations, it is found that the plate stops oscillating and comes to rest quickly.
When the electromagnet is switched on, the plate oscillates in a strong magnetic field.
Due to the continuous change of magnetic flux linked with the plate, eddy currents are setup in
the body of the metal plate. Since the eddy currents are set up due to the oscillations of the metal
plate, the eddy currents tends to oppose the motion of the metal plate and finally bring it to rest.
This kind of opposition offered by eddy currents to the motion of a system is called
Electromagnetic Damping.
In a thick conductor, resistance is low. Therefore eddy currents are quite large in
magnitude and produce heat and hence there is a loss of power. The heating effect of eddy
currents is undesirable in devices like dynamos, motors and transformers. In these devices eddy
currents are reduced considerably by using laminated core. The laminations are separated by
insulating material and the plane of laminations is arranged parallel to the magnetic field so that
they cut across the eddy current paths. This arrangement reduces the strength of the eddy
currents and minimises heating of the core.
Applications of Eddy Currents
1.
Electromagnetic damping: Some galvanometers have a fixed core, which is made of
non-magnetic metallic materials. When the coil oscillates, the eddy currents generated in
the core oppose the motion and bring the coil to rest quickly.
2.
Induction furnace: In an induction furnace, very high temperature can be produced by
producing large eddy currents.
3.
Magnetic braking in trains: Strong electromagnets are situated above the rails in some
electrically powered trains. When the electromagnets are activated, the eddy currents
induced in the rails oppose the motion of the train.
4.
Speedometer: A tiny magnet rotates according to the speed of the vehicle and produces
the required changing magnetic field. The magnet rotates in an aluminium drum. The
eddy currents set up increases as the speed increases. Using calibrated scale, the speed of
the vehicle can be obtained.
5.
Energy meters: Meters used to record the consumption of electricity makes use of eddy
currents.
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Electromagnetic Induction
Self-Induction
growth
current
0
decay
t1
time
t2
Consider a coil connected in series with a cell and a plug key. When the circuit is
completed the current in the circuit does not rise from zero to maximum suddenly instead it takes
a finite time to do so. When current starts increasing, the magnetic flux linked with the circuit
also increases and induces an emf which opposes the growth of current. The time taken by
current to become maximum form zero is called time of growth and the emf which opposes the
growth of current is called back emf. When current becomes steady there will be no change in
the magnetic flux and hence no emf is induced. When the key is released current starts
decreasing, due to this the magnetic flux linked with the coil also decreases and hence an emf is
induced which opposes the decay of current. The time taken by the current to become zero from
maximum is called time of decay and the emf which opposes the decay of current is called
forward emf.
The magnitude of forward emf is always more than the back emf. Thus the induced emf
at the break of the circuit is always greater than emf during the make of the circuit.
“The phenomenon in which an emf is induced in the coil due to the change in
magnetic flux in the coil itself is called Self-Induction”. Self-Induction is also called inertia of
electricity.
Coefficient of Self-Induction or Self-Inductance
We have
e=-
We know that ϕ ∝ I
ϕ=LI
∴ e=If
Where L = coefficient of self-induction
=-L
= 1 A s-1 then L = e
The self inductance of a coil is numerically equal to the emf induced in the coil, when the current
in it changes at a rate of 1 A s-1.
The S.I unit is henry (H).
If
= 1 A s-1 and e = 1 V then L = 1 H
Self inductance of a coil is said to be 1 H if 1 V of emf is induced in the coil when current
changes at a rate of 1 A s-1.
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Electromagnetic Induction
Self-Inductance of a solenoid
Consider a long solenoid of cross-sectional area A and length l having n turns per unit length.
The magnetic field due to a current I flowing in the solenoid is
B = μ0nI
We have
Where N = nl
ϕ = NBA
i.e., the total number of turns
ϕ = (nl)( μ0nI)A
ϕ = μ0n2lIA
but ϕ = L I
∴ L I = μ0n2l I A
L = μ 0 n2 l A
If the solenoid has a core of some magnetic material of relative permeability μr then
L = μ0 μr n2l A
Energy Stored in a Self Inductance
In self inductance the induced emf always opposes any change in current in the circuit called
back emf. Work has to be done against this back emf in establishing the current. This work done
is stored as magnetic potential energy. Let dw be the work done in establishing a current I in the
coil in a time dt. Then
dw = - e I dt
But
e=-L
∴ dw = L
I dt
dw = L IdI
the total work done is
W=∫
W = ∫ IdI = L∫ dI
W=L
W = LI2
But work done is potential energy
∴ U = LI2
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Electromagnetic Induction
Mutual Induction
The phenomenon in which a change of current in one coil induces an emf in another
neighbouring coil is called mutual induction.
The emf produced in secondary coil is directly proportional to rate of change of current through
the primary coil
We have
e=
We know that ϕ ∝ I
ϕ=MI
Where M = coefficient of mutual induction
e=
If
=M
= 1 A s-1 then M = e
The mutual induction of coils is numerically equal to the emf induced in the secondary coil when
the current through primary coil changes at a rate of 1 A s-1.
Mutual Inductance of Two Long Solenoids
Consider two long solenoids S1 and S2 of same length l, such that
solenoid S2 surrounds solenoid S1 completely. Let n1 and n2 be the
number of turns per unit length of S1 and S2, I be the current passed
through S1.
We know that ϕ21 ∝ I1
ϕ21 = M21 I1
M21 = co-efficient of mutual induction of the two solenoids
B1 = μ0n1I1
w.k.t
total magnetic flux linked with the solenoid S2
ϕ21 = N2B1A
ϕ21 = (n2l)( μ0n1I1)(πr2)
ϕ21 = μ0n1 n2πr2l I1
but
ϕ21 = M21 I1
M21 I1= μ0n1 n2πr2l I1
M21 = μ0n1 n2πr2l
Similarly
M21 = M12 = M
∴ M = μ0n1 n2πr2l
For a relative permeability present inside a solenoid
M = μ0 μr n1 n2πr2l
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Electromagnetic Induction
AC Generator
Principle: Based on the phenomenon of Electromagnetic Induction
Construction
Main parts of an ac generator:

Armature − Rectangular coil ABCD

Filed Magnets − Two pole pieces of a strong electromagnet

Slip Rings − The ends of coil ABCD are connected to two hollow metallic rings R1 and
R2 .

Brushes − B1 and B2 are two flexible metal plates or carbon rods. They are fixed and are
kept in tight contact with R1 and R2 respectively.
Theory and Working
As the armature coil is rotated in the magnetic field, angle θ between the field and normal to the
coil changes continuously. Therefore, magnetic flux linked with the coil changes. An emf is
induced in the coil. According to Fleming’s right hand rule, current induced in AB is from A to
B and it is from C to D in CD. In the external circuit, current flows from B2 to B1.
Let n be the number of turns and A be the area of
cross section placed in a magnetic field of flux density
B. θ be the angle made by the coil placed in an magnetic
field.
Magnetic flux linked with the coil is given by
ϕ = NBA Cos θ
But θ = wt
∴ ϕ = NBA Cos wt
We have
e=e=Page 10
Electromagnetic Induction
e = - NBA
e = - NBA (- Sin wt) w
∴ e = NBA w Sin wt
e = e0 Sin wt
Where NBAw = e0 called peak or maximum value of emf.
We have w = 2πυ
υ = frequency of A.C
∴ e = e0 Sin 2πυt
Note: frequency of AC in India is 50 Hz.
List of formulas
ϕ = B A Cos θ
e=Blv
Questions
Page 11
P=
L = μ0 n2l A
U = LI2
M = μ0n1 n2πr2l
e = e0 Sin wt
w = 2πυ
e0 = NBAw