2. Oersted, Ampere and a few others established the fact that electricity and magnetism are
inter-related.
whenever an electric current flows through a conductor, it produces a magnetic field around it.
This was discovered by Christian Oersted. Later, Ampere proved that a current-carrying loop
behaves like a bar magnet.
These are the magnetic effects produced by the electric current.
converse effect
A series of experiments were conducted to establish the converse effect.
These experiments were done by Michael Faraday of UK and Joseph Henry of USA, almost
simultaneously and independently.
became successful and led to the discovery of the phenomenon, called Electromagnetic
Induction.
Michael Faraday is credited with the discovery of electromagnetic induction in 1831.
Faraday discovered that whenever magnetic flux linked with a coil changes, an electric current
is induced in the circuit.
Here the flux change is the cause while the induction of current is the effect.
- A time-varying magnetic field can act as source of electric field.
- A time-varying electric field can act as source of magnetic field. Maxwell
3. Induced emf and current –Whenever there is a change in the magnetic flux linked
with a closed circuit an emf is produced. This emf is known as the induced emf
and the current that flows in the closed circuit is called induced current.
Electromagnetic induction --The phenomenon of producing an induced emf due
to the change in the magnetic flux associated with a closed circuit is known as
electromagnetic induction.
Faraday discovered the electromagnetic induction
by conducting several experiments.
A cylindrical coil C made up of several turns of insulated copper wire connected
in series to a sensitive galvanometer G. A strong bar magnet NS with its north
pole pointing towards the coil is moved up and down.
The following inferences were made by Faraday
If the magnetic flux through a circuit changes, an emf and a current are induced.
4. Electromagnetic induction (e)----When there is a change in the magnetic
field an electromotive force is induced across a conductor this phenomenon is
called electromagnetic induction.
If a conductor is in a closed circuit then you will see a flow of current inside a
conductor due to induced EMF,
whenever there will be a change in the magnetic field an EMF will be induced,
EMF will only get induced when the magnetic field is changing and as soon as
the magnetic field is stable and there is no change in it, then there will be no
induced EMF.
Change in the magnetic field can be denoted with the help of change in
Magnetic flux.
Magnetic Flux:
Magnetic flux and electric flux are the same, they both have the same concept
but the only difference is one case we are dealing with Electric field whereas on
the other one we are dealing with magnetic field.
So magnetic flux linked with an area 'A’ :
ɸB = B.A where magnetic flux is denoted by "ɸB".
5. Magnetic Flux (Φ):
i) It is defined as the number of magnetic lines of force passing normally
through a surface.
ii) Its SI unit is weber. 𝚽𝑩 = 𝑩 . 𝑨
𝚽𝑩 = BA cosθ
iii) which is denoted by symbol Wb.
iv) Dimensional formula for magnetic flux is [ M L𝟐
T−𝟐
A−𝟏
] .
v) The CGS unit of magnetic flux is maxwell.
1 weber = 1𝟎𝟖 maxwell
Special cases
(a)When 𝑩 is normal to the surface i.e., θ = 𝟎𝟎,
the magnetic flux is 𝜱𝑩 = BA (maximum).
(b) When 𝑩 is parallel to the surface i.e., θ = 9𝟎𝟎 ,
the magnetic flux is 𝚽𝑩 = 0.
6. (c) When 𝑩 is antiparallel to the surface i.e., θ = 1𝟖𝟎𝟎
,
the magnetic flux is 𝚽𝑩 = -BA. (Minimum)
Suppose the magnetic field is not uniform over the surface,
𝚽𝑩 = 𝑩 . d 𝑨
Direction of ds is along the normal to the surface A(area) vector.
Positive Flux:
Magnetic Flux is positive for 0° ≤ θ < 90° & 270°< θ ≤ 360°
Zero Flux:
Magnetic Flux is zero for θ = 90° & θ = 270°
Negative Flux:
Magnetic Flux is negative for 90°< θ < 270°
7. Magnetic Flux across a coil can be changed by changing :
1)the strength of the magnetic field B
2)the area of cross section of the coil A
3)the orientation of the coil with magnetic field θ or
4)any of the combination of the above
Magnetic flux (associated normally) per unit area is called Magnetic Flux Density
or Strength of Magnetic Field or Magnetic Induction (B).
8. First Experiment
a) Consider a closed circuit consisting of a coil C of insulated
wire and a galvanometer G . The galvanometer does not
indicate deflection as there is no electric current in the
circuit.
b) When a magnet approaches a closed circuit consisting of a
coil, it produces a current in it. This current is known as
induced current.
When a bar magnet is inserted into the stationary coil, with its
north pole facing the coil, there is a momentary deflection in the
galvanometer. This indicates that an electric current is set up in
the coil.
c) If the magnet is kept stationary inside the coil, the
galvanometer does not indicate deflection .
d) When the magnet is taken away from the closed circuit a
current is again produced but in the opposite direction The bar
magnet is now withdrawn from the coil, the galvanometer again
gives a momentary deflection but in the opposite direction.
9. e) So the electric current flows in opposite direction .
Now if the magnet is moved faster, it gives a larger
deflection due to a greater current in the circuit .
f) The bar magnet is reversed ,the south pole now faces the
coil. When the above experiment is repeated, the
deflections are opposite to that obtained in the case of
north pole .
If the magnet is kept stationary and the coil is moved
towards or away from the coil, similar results are obtained.
a) It is concluded that whenever there is a relative motion
between the coil and the magnet, there is deflection in the
galvanometer, indicating the electric current setup in the
coil.
b) If the polarity of approaching or receding magnet is
changed the direction of induced current in the coil is also
changed.
10. The circuit consisting of a coil P, a battery B and a key K is called
as primary circuit while the circuit with a coil S and a
galvanometer G is known as secondary circuit.
The coils P and S are kept at rest in close proximity with respect
to one another. If the primary circuit is closed, electric current
starts flowing in the primary circuit. At that time, the
galvanometer gives a momentary deflection .
After that, when the electric current reaches a certain steady
value, no deflection is observed in the galvanometer.
Likewise if the primary circuit is broken, the electric current starts
decreasing and there is again a sudden deflection but in the
opposite direction .
When the electric current becomes zero, the galvanometer shows
no deflection.
Note--From the above observations, it is concluded that
whenever the electric current in the primary circuit changes, the
galvanometer shows a deflection.
11. Observe:
i) the relative motion between the coil and the magnet
ii) the induced polarities of magnetism in the coil
iii) the direction of current through the galvanometer and hence the deflection in the
galvanometer
iv) that the induced current (e.m.f) is available only as long as there is relative motion
between the coil and the magnet
Note:
i)coil can be moved by fixing the magnet
ii) both the coil and magnet can be moved ( towards each other or away from each other) i.e.
there must be a relative velocity between them
iii)magnetic flux linked with the coil changes relative to the positions of the coil and the
magnet
iv)current and hence the deflection is large if the relative velocity between the coil and the
magnet and hence the rate of change of flux across the coil is more
12. When the primary circuit is open current decreases from maximum value to zero.
During this period changing current induces changing magnetic flux across the primary coil.
This changing magnetic flux is linked across the secondary coil and induces current (e.m.f) in
the secondary coil.
However, note that the direction of current in the secondary coil is reversed and hence the
deflection in the galvanometer is opposite to the previous case.
Faraday’s Laws of Electromagnetic Induction:
I.Law:
Whenever there is a change in the magnetic flux linked with a circuit, an emf and hence a current
is induced in the circuit. However, it lasts only so long as the magnetic flux is changing.
II.Law:
The magnitude of the induced emf is directly proportional to the rate of change of magnetic
flux linked with a circuit.
(where k is a constant and units are chosen such that k = 1)
E α dΦ / dt E= k dΦ / dt E= dΦ / dt E= (Φ2 – Φ1) / t
13. Faraday’s first law of induction
The magnitude of induced emf produced in the circuit is directly proportional to the rate of
change of magnetic flux linked with the circuit.
This law is known as quantitative law as it gives the magnitude of induced emf.
This law is also known as Neumann’s law .
If ∅ is the magnetic flux linked with the coil at any instant t, then the induced emf.
e 𝜶
d ∅
dt
e = 𝐊
d ∅
dt
,
K is constant of proportionality.
If e, ∅ , and t are measured in the same system of units, K = 1.
e =
d ∅
dt
If ∅ ' is the flux associated with single turn, then the total magnetic flux f for a coil consisting of
N turns, is ∅ = N ∅ ‘
e =
dN∅ ‘
dt
14. e = N
d∅ ‘
dt
This is also known as 'flux rule' according to which the emf is equal to the rate at
which the magnetic flux through a conducting circuit is changing.
SI units e is measured in volt .
d∅
dt
is measured in weber/s.
Faraday’s second law of induction ---
The direction of the induced current is such that it always opposes the cause
responsible for its production.
This law is also known as Lenz’s law .
15. change of magnetic
Close circuit open circuit
induced emf induced emf
induced current no induced current
induced charge no induced charge
16. Faraday’s Law Derivation
Consider a magnet approaching towards a coil.
Consider two-time instances T1 and T2.
Flux linkage with the coil at the time T1 is given by
T1 = NΦ1
Flux linkage with the coil at the time T2 is given by
T2 = NΦ2
Change in the flux linkage is given by
N(Φ2 – Φ1)
Let us consider this change in flux linkage as
Φ = Φ2 – Φ1
Hence, the change in flux linkage is given by
NΦ
The rate of change of flux linkage is given by
N
Φ
t
Taking the derivative of the above equation, we get
17. N
dΦ
dt
According to Faraday’s second law of electromagnetic induction, we know that the induced emf in a
coil is equal to the rate of change of flux linkage. Therefore,
e =
dΦ
dt
Considering Lenz’s law,
e = −N
dΦ
dt
From the above equation, we can conclude the following
•Increase in the number of turns in the coil increases the induced emf
•Increasing the magnetic field strength increases the induced emf
•Increasing the speed of the relative motion between the coil and the magnet, results in the
increased emf
18. Lenz’s Law and Law of Conservation of Energy:
This law was deduced in 1834 by the Russian physicist Heinrich Friedrich
Emil Lenz (1804–65).
Lenz’s law depends on the principle of conservation of energy and Newton’s third law .
Lenz’s law---- determine the direction of the induced current.
From the definition of Lenz’s law, we know that the induced current is always opposed by the cause
that produces it. Therefore, there is extra work done against the opposing force.
Law of Conservation of Energy --The work done against the opposing force results in the
change in the magnetic flux and hence the current is induced. The extra work done is known as
electrical energy which is the law of conservation of energy.
Lenz’s law is about conservation of energy applied to the electromagnetic induction whereas
Faraday’s law is about the electromagnetic force produced.
Lenz’s law is used to determine the direction of the induced current.
The negative sign in Lenz’s law indicates that the induced emf in the coil is in the opposite direction
of the magnetic flux which is linked with the coil.
19. Lenz’s law is used to explain how electromagnetic circuits obey the conservation of
energy and Newton’s third law ----- magnetic flux increases when the North
Pole of the magnet comes towards it and decreases as it is pushed
away.
20. In the first case, to oppose the cause means motion of the magnet, the
face coming towards the coil acquires North Polarity. The north pole of
the magnet and the north pole of the coil repel each other. To move the
magnet towards the coil, some sort of mechanical work has to be done
to overcome the force of repulsion.
This mechanical work is converted into electrical energy. This
electrical energy is converted into heat energy due to Joule’s Effect.
Similarly, when the magnet moves away from the coil, the nearer face
of the coil acquires south polarity. In this case, the induced emf will
oppose the outward motion of the magnet. Once again mechanical
work has to be done to overcome the force of attraction between the
North Pole of magnet and the South Pole of the coil. This work done is
converted into electrical energy.
If the magnet is not moved, no mechanical work is done and then no
emf is induced in the coil.
Thus, this proves that Lenz’s Law is in accordance with the law of
21. Fleming’s right hand rule
The forefinger, the middle finger and the thumb of the right hand are held in the
three mutually perpendicular directions.
The forefinger points along the direction of the magnetic field .
The thumb is along the direction of motion of the conductor.
The middle finger points in the direction of the induced current.
This rule is also called generator rule.
Trick to learn ---- IBM
I = current (thumb)
B = magnetic field(fore finger )
M = motion(central finger)
22. Motional emf from Lorentz force
a) Translational motion of a conductor:
AB = Consider a straight conducting rod AB .
l = length of conducting rod AB in a uniform magnetic field. The length of the rod is normal to
the magnetic field.
B = uniform magnetic field which is directed perpendicularly into the plane of the paper.
v = Let the rod move with a constant velocity towards right side.
When the rod moves, the free electrons present in it also move with same velocity v in B .
Lorentz force acts on free electrons in the direction from B to A
𝑭𝑩 = -e ( v × B )
Due to Lorentz force free electrons accumulate at the end A.
This accumulation of free electrons produces a potential difference across the rod which in
turn establishes an electric field E directed along BA
Due to the electric field E , the coulomb force starts acting on the free electrons along AB.
𝑭𝒆 = -e E
23. The magnitude of the electric field E keeps on increasing as long as accumulation of electrons
at the end A continues. The force 𝑭𝒆also increases until equilibrium is reached.
At equilibrium, the magnetic Lorentz force 𝑭𝑩 and the coulomb force 𝑭𝒆 balance each other
and no further accumulation of free electrons at the end A takes place.
𝑭𝑩 = 𝑭𝒆
-e ( v × B ) = -e E
( v × B ) = E
vB Sin𝜽 = E θ = 9𝟎𝟎
vB = E
The potential difference between two ends of the rod is
E=
𝑽
𝒍
V = Blv
Thus the Lorentz force on the free electrons is responsible to maintain this potential difference
and hence produces an emf
ε = Blv
This emf is produced due to the movement of the rod, it is often called as motional emf.
If the ends A and B are connected by an external circuit of total resistance R, then
24. current i =
ε
R
flows in it.
The direction of the current is found from right-hand thumb rule.
SI Unit of ε is volt , j/C ,m𝟐T s−𝟏
Dimension is [M𝟏
L𝟐
T−𝟑
A−𝟏
]
Note --In a region where magnetic field is changing with time, electric fields are
generated.
b) Motional emf in a rotating bar:
A rotating bar is different in nature from the sliding bar.
A copper rod of length l rotates about one of its ends with an angular velocity ω
in a magnetic field B .
The plane of rotation is perpendicular to the field.
The emf induced between the two ends of the rod.
25. Consider a small segment dx of the bar at a distance x from the pivot.
dx = small element of length at a distance x from the centre of the circle.
It is a short length dx of the conductor which is moving with velocity v in magnetic field B and
has an induced emf generated in it like a sliding bar.
This element moves perpendicular to the field with a linear velocity v = xω .
The emf developed in the element dx is
d ε = B v dx
= B xω dx
This rod is made up of many such elements, moving perpendicular to the field.
Total induced emf in rotating rod
e = 𝒅𝒆
=
𝟎
𝒍
B v dx
= Bω 𝟎
𝒍
x dx
= Bω[
𝒙𝟐
𝟐
]
= Bω
𝒍𝟐
𝟐
Compare the above result with the induced emf in sliding bar, e = Blv.
𝒍
𝟎
26. Back emf and back torque:
We know that emf can be generated in a circuit in different ways.
In a battery it is the chemical force, which gives rise to emf.
In piezoelectric crystals mechanical pressure generates the emf.
In a thermocouple it is the temperature gradient which is responsible for
producing emf in a circuit containing the junctions of two metallic wires.
In a photo electric cell, the incident light above a certain frequency is responsible
for producing the emf.
In a Van de Graaff Generator the electrons are literally loaded into a conveyor
belt and swept along to create a potential difference.
A generator utilises the movement of wire through a magnetic field to produce
motional emf/current through a circuit.
We have seen that the physical construction of a DC generator and motor is
practically the same.
If a DC generator is connected to a battery, it will run as a motor.
27. Expression for Induced emf based on both the laws:
e = -
dΦ
dt
e = -
(Φ2 – Φ1)
t
And for ‘N’ no. of turns of the coil, e = - N
dΦ
dt
e = - N
(Φ2 – Φ1)
t
Expression for Induced current --- I = -
dΦ
Rdt
Expression for Charge:
dq
dt
=
-
dΦ
Rdt
dq = -
dΦ
R
Note:
Induced emf does not depend on resistance of the circuit where as the induced current and
induced charge depend on resistance.
Methods of producing Induced emf:
1. By changing Magnetic Field B:
Magnetic flux Φ can be changed by changing the magnetic field B and hence emf can be induced
in the circuit (as done in Faraday’s Experiments).
28. 2. By changing the area of the coil A available in Magnetic Field:
Magnetic flux Φ can be changed by changing the area of the loop A which is acted upon by the
magnetic field B and hence emf can be induced in the circuit.
S R
P’ P Q’ Q
S’ R’
v
B
dA
l
v.dt
I
The loop PQRS is slided into uniform and perpendicular magnetic field. The change (increase) in area of the
coil under the influence of the field is dA in time dt. This causes an increase in magnetic flux dΦ.
dΦ = B.dA
= B.l.v.dt
e = -
dΦ
dt
e = - Blv
29. The induced emf is due to motion of the loop and so it is called ‘motional emf’.
If the loop is pulled out of the magnetic field, then e = Blv
The direction of induced current is anticlockwise in the loop. i.e. P’S’R’Q’P’ by Fleming’s Right Hand
Rule or Lenz’s Rule.
According Lenz’s Rule, the direction of induced current is such that it opposes the cause of
changing magnetic flux.
Here, the cause of changing magnetic flux is due to motion of the loop and increase in area of the
coil in the uniform magnetic field.
Therefore, this motion of the loop is to be opposed. So, the current is setting itself such that by
Fleming’s Left Hand Rule, the conductor arm PS experiences force to the right whereas the loop is
trying to move to the left.
Against this force, mechanical work is done which is converted into electrical energy (induced
current).
NOTE:
If the loop is completely inside the boundary of magnetic field, then there will not
be any change in magnetic flux and so there will not be induced current in the
loop.
30. 3.By changing the orientation of the coil (θ) in Magnetic Field:
Magnetic flux Φ can be changed by changing the relative orientation of the loop (θ) with the
magnetic field B and hence emf can be induced in the circuit.
P
Q
R
S
B
θ
ω
Φ = N B A cos θ
At time t, with angular velocity ω,
θ = ωt (at t = 0, loop is assumed to be
perpendicular to the magnetic field and θ = 0°)
Φ = N B A cos ωt Differentiating w.r.t. t,
dΦ / dt = - NBAω sin ωt E = - dΦ / dt
E = NBAω sin ωt
E = E0 sin ωt (where E0 = NBAω is
the maximum emf)
n
31. The emf changes continuously in magnitude
and periodically in direction w.r.t. time giving
rise to alternating emf.
If initial position of the coil is taken as 0°, i.e.
normal to the coil is at 90° with the magnetic
field, then
θ becomes θ + π/2 or ωt + π/2
E = E0 cos ωt
So, alternating emf and consequently
alternating current can be expressed in sin or
cos function.
This method of inducing emf is the basic principle of generators.
E
T/2 3T/4 T 5T/4 3T/2 7T/4 2T
t
E0
0
π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ =ωt
T/4
E
T/2 3T/4 T 5T/4 3T/2 7T/4 2T
t
E0
0
π/2 π 3π/2 2π 5π/2 3π 7π/2 4π θ =ωt
T/4
