Lacture 7_2

CHAPTER 7 : ELECTROMAGNETIC INDUCTION
( 7 HOURS )

Terminology :

1. Electromagnetic induction : Aruhan elektromagnet.
2. Induced e.m.f. : D.g.e. aruhan.
3. Loop : Gelung.
4. Self-inductance : Swa-induktans.
5. Mutual-inductance : Induktans saling.
6. Back e.m.f : D.g.e balik.
7. Step-up transformer : Transformer menaik.
8. Step-down transformer : Transformer menurun.
9. Eddy current : Arus pusar.

Introduction :

1. In 1820, Hans Oersted conducted an experiment in which he proved that there is a
magnetic field surrounding a current-carrying conductor.

© Physics Teaching Courseware
JOTTING
SPACE

2. After Oersted’s discovery, scientists made numerous attempts to observe the
inverse effect :

CAN A MAGNETIC FIELD CAUSES A FLOW OF ELECTRIC CURRENT ?

3. In 1831, as a result of systematic studies, Michael Faraday concluded that :
“The flow of electric current in a circuit is caused by a magnetic field which is
varying in time.”

© Physics Teaching Courseware

| Chapter 7 | Electromagnetic Induction | page 1 / 18 |

If we insert a magnet into a coil that is connected to an ammeter, the indicator of
the meter will deflect, which shows us that there is a flow of current in a circuit in
which there is no cell.

The current flows only during the time when the magnet is being displaced. When
the magnet stops moving, the flow of current stops as well. As the magnet is being
quickly removed, the ammeter indicator deflects in the opposite direction.

4. Electromagnetic induction refers to phenomena which occur when there is a
change in magnetic flux, ∆φ through a conductor and thus produces an induced
e.m.f. in the conductor.

Electromagnetic Induction Phenomenon :

There are several phenomena related to the occurrence of the potential difference
(known as induced e.m.f.) between the ends of a conductor moving in a magnetic

field.

a) Phenomenon 1 :

SS

© Physics Teaching Courseware © Physics Teaching Courseware

“Relative motion between the magnet bar and the coil produces induced e.m.f in the coil
and causes induced electric current flow in the coil”.

b) Phenomenon 2 : JOTTING
SPACE

© Physics Teaching Courseware

“Adjusting resistance used a rheostat changes the current flow in circuit 1 produces
induced e.m.f in circuit 2 and causes induced electric current flow in the coil”.

| Chapter 7 | Electromagnetic Induction | page 2 / 18 |

c) Phenomenon 3 :

© Physics Teaching Courseware © Physics Teaching Courseware

“The movement of a mobile straight rod of length L in a circuit with constant velocity v
across a uniform magnetic field produces induced e.m.f in the circuit causes induced
electric current flow in the circuit”.

d) Phenomenon 4 :

JOTTING
SPACE

© Physics Teaching Courseware

“Revolution of a revolving rotor across a uniform magnetic field produces induced e.m.f in
the circuit causes induced electric current flow in the circuit”.

Conclusion :
The flow of induced electric current in a circuit is caused by a magnetic flux

which is varying in time.

| Chapter 7 | Electromagnetic Induction | page 3 / 18 |

Magnetic field flux and its changes :
1. In order to analyse the reason for flux changes, we recall the definition of flux.

© Physics Teaching Courseware

2. Magnetic field flux through an area A normal to a magnetic field B is defined as :
φ = N(B • A)

or φ = NBA cosθ …..(7.1)

where N = number of loop. JOTTING
SPACE
B = magnetic field strength.

A = area of surface.
θ = angle between magnetic field line and the normal line.

Thus, the change in the flux :
∆φ = ∆(NBA cosθ)

or dφ = d(NBA cosθ)

Note : Magnetic flux is a scalar quantity.
The SI unit for magnetic flux is T m2 or normally known as Webber (Wb)

Faraday’s law :

1. A flow of current is caused by a potential difference known as electromotive force

(e.m.f.).

I∝ξ …..Ohm’s law.

2. Faraday’s law states that :
“The magnitude of the e.m.f. induced in a circuit is directly proportional to the rate
of change of magnetic flux linkage through the circuit.

or ξ ∝ dφ …..(7.2)
dt

| Chapter 7 | Electromagnetic Induction | page 4 / 18 |

Lenz’s law :

1. The direction of the current induced in the circuit can be easily determined by using
Lenz’s law.

2. Lenz’s law states that :

“The induced current flow in such a direction that is opposes the change that

produces it.”

ξ = -  dφ  …..(7.3)
 dt 

or ξ = - d (NBA cos θ) …..(7.4)
dt

Note : Induced e.m.f, ξ = -ve (against the increment of dφ)
Induced e.m.f, ξ = +ve (against the reduction of dφ)

The direction of induced current can be determined by using ‘Right-Hand Grip’ rule.

BB JOTTING
I SPACE

I

© Mohd. Hazri @ kmph

Lenz’s law is consistent with the principle of conservation of energy. Based on the
principle of conservation of energy, any increment of energy in a closed system
should be balanced with the lost of energy in other form.

Thus, Binduced is generated in such direction which balances the change of B.

Relationship Between
Induced Current and The Rate Change of Magnetic Flux

(Zahidi, 2008)

Induced
Current

depends on… reason… Based on Ohm’s Law :
εin = Iin R
Induced
Electromagnetic Force, emf

depends on… Number of Turn, N
Magnetic Field, B
The Rate Change of Cross Sectional Area, A
Magnetic Flux Angle, θ Between B and Anormal

depends on the change of… Time Interval, t

| Chapter 7 | Electromagnetic Induction | page 5 / 18 |

Induced e.m.f. :

Let’s compute the e.m.f, ξ induced in each phenomenon.

a) Phenomenon 1 :
Relative motion between magnet bar and coil.

Based on Faraday’s and Lenz’s law, the induced

e.m.f is represented as the following expression :

ξ = -  dφ 
 dt 
© Physics Teaching Courseware B
ξ = - d (NBA cos θ)
or dt …..(i)

In this case, the change is related to the

magnitude of magnetic field within time during the

motion, thus from equation (i) :

Binduced ξ = - NA cos θ dB 
 dt 

Note :

If the magnetic field increases → + dB JOTTING
dt SPACE

If the magnetic field decreases → - dB
dt

b) Phenomenon 2 :
Adjusting resistance in circuit 1.

Based on Faraday’s and Lenz’s law, the induced

e.m.f is represented as the following expression :

© Physics Teaching Courseware ξ = -  dφ 

B  dt 

or ξ = - d (NBA cos θ) …..(i)
dt

Binduced In this case, when the resistance changes, then
the current flowing in circuit 1 is changed as well
as the magnetic field in the coil in both circuit.
Thus from equation (i) :

ξ = - NA cos θ dB 
 dt 

Note : + dB
If the magnetic field increases → dt

If the magnetic field decreases → - dB
dt

| Chapter 7 | Electromagnetic Induction | page 6 / 18 |

c) Phenomenon 3 :
A mobile rod moving with constant velocity.

Based on Faraday’s and Lenz’s law, the induced
s e.m.f is represented as the following expression :

R ξ = -  dφ 
 dt 

or ξ = - d (NBA cos θ) …..(i)
dt

© Physics Teaching Courseware In this case, when the rod is moving across a
uniform magnetic field (where θ = 0o and N = 1),

the area of surface, A changed. Thus from

equation (i) :

ξ = - NB dA  …..(ii)
 dt 

Area of surface, A = (Length of rod)(displacement)

= (L)(s) …..(iii)

Substitute (iii) into (ii) we obtain : JOTTING
SPACE
ξ = - NB d (Ls)
 dt 

= - NBL ds 
 dt 

∴ ξ = - BLv ….. v = velocity

Note :

If the magnetic field increases → + ds
dt

If the magnetic field decreases → - ds
dt

| Chapter 7 | Electromagnetic Induction | page 7 / 18 |

Another method :

When the rod (conductor) is moving in one

direction with a constant velocity across a

uniform magnetic field, free electron in the rod

are exerted by magnetic force.

FB = evBsinθ …..θ = 90o
∴ FB = evB …..(i)

The electrons are accumulated at one end

of the rod (figure) until :

FB = FE ….. FE = eE
∴ evB = eE

∴ E …..(ii) © Physics Teaching Courseware

B=
v

We know that : ξ ….. (iii)
E= V or E =
L
L

where V = potential difference between the ends. JOTTING
ξ = electromotive force (e.m.f.) SPACE

Substitute (i) and (ii) into (iii), we obtain the magnitude of induced e.m.f :

ξ = BLv ….. proven.

d) Phenomenon 4 :
Revolving rotor.

Based on Faraday’s and Lenz’s law, the induced
e.m.f is represented as the following expression :

ξ = -  dφ 
 dt 

or ξ = - d (NBA cos θ) …..(i)
dt

© Physics Teaching Courseware

In this case, when the rod is rotating across a uniform magnetic field, the angle

between magnetic field line and the normal line, θ is changing within time. Thus from

equation (i) :

ξ = - NBA d (cos θ) …..(ii)
 dt 

We know that in rotational motion : …..(iii)
θ = ωt

where ω = angular velocity.

Substitute (iii) into (ii), we obtain :

ξ = - NBA d (cos(ωt)
 dt 

or ξ = NBAωsin(ωt)

| Chapter 7 | Electromagnetic Induction | page 8 / 18 |

Example :

A circular coil of 20 turns and radius 5.0 cm is placed in a uniform magnetic field B of
magnitude 1.5 T with the normal to the coil parallel to the B-field. The flux density is
uniformly reduced to 1.0 T in a time of 0.8 s. Calculate the induced e.m.f. in the coil.

Self-Induction :

1. Consider a solenoid which is connected to a
battery and switch, forming an OFF circuit.

The magnetic field through the solenoid is zero
since there is no current in the solenoid, and so no
magnetic flux exists.

© Physics Teaching Courseware When the switch is ON, current begins to flow in
the solenoid. The rising current produces a
magnetic field, whose field lines pass through the JOTTING
solenoid and the magnitude increases with time. SPACE
Hence, the magnetic flux linkage through the
solenoid increases with time as well.

Consequently, an e.m.f. has to be induced in the solenoid itself since the flux
linkage changes with time.

Based on the Lenz’s law, since the magnetic flux increases, Binduced is
generated in opposite direction to that of B (in order to reduce the increment of dφ)

as shown in below figures.

Thus, the direction of induced e.m.f as well as induced current can be determined
using the ‘Right-Hand Grip’ rule.

S B ind N

NS

I induced

Figure 1 : ‘explicit view’. Figure 2 : ‘implicit view’ (induction).

Note : Both figures involved the same solenoid within the same time.
In this process, ξind is assigned as ‘–ve’.

On the other hand, when the switch is turned OFF, current begins to reduce. The
reduction current leads to the dissipation of magnetic field, whose field lines pass
through the solenoid and the magnitude decreases with time. Hence, the magnetic
flux linkage through the solenoid decreases with time as well.

Consequently, an e.m.f. has to be induced in the solenoid itself since the flux
linkage changes with time.

| Chapter 7 | Electromagnetic Induction | page 9 / 18 |

Based on the Lenz’s law, since the magnetic flux decreases, Binduced is generated in
the same direction to that of B (in order to increase the reduction of dφ) as shown in
below figures.

Thus, the direction of induced e.m.f as well as induced current can be determined
using the ‘Right-Hand Grip’ rule.

S B ind S

NN

I induced

Figure 3 : ‘explicit view’. Figure 4 : ‘implicit view’ (induction).

Note : Both figures involved the same solenoid within the same time.
In this process, ξind is assigned as ‘+ve’.

2. This process of producing an induced e.m.f. in the circuit due to a change of JOTTING
current flowing through the same circuit is known as ‘self-induction’. SPACE

3. The induced e.m.f produced in a circuit due to the change of current that flows
through the same circuit is known as ‘back e.m.f.’.

Self-inductance, L :

1. Self-inductance :
= The property of a circuit or component which can induce an e.m.f. in the circuit or
component itself.

2. Around every current-carrying conductor there is a magnetic field. The magnetic

field of a given circuit generates a magnetic flux, φ which passes through the

area, A.

φ=B•A …..(i)

© Physics Teaching Courseware …..(ii)
…..(iii)
Since magnetic field B and flux φ are functions of I, then :
dB ∝ dI

and dφ ∝ dI or dφ = (L)dI

where L = a constant known as ‘self-inductance’.

| Chapter 7 | Electromagnetic Induction | page 10 / 18 |

3. According to the laws of electromagnetic induction (Faraday’s and Lenz’s laws),

the so-called ‘self-induced e.m.f.’ generated in the circuit is represented by the

following expression :

ξ = -  dφ  …..(iv)
 dt 

Substitute (iii) into (iv), then we obtain : …..(v)
ξ = - L dI 
 dt 

4. Suppose current I flows through a coil with N turns. Then, we have :

ξ = -N  dφ  …..(vi)
 dt 

From equation (v) and (vi), we obtain :

- L dI  = -N  dφ 
 dt   dt 

or L = Nφ …..(7.5)
I

5. The SI unit for self-inductance is Henry (H).© Physics Teaching Courseware
1 H = 1 T m2 A-1 = 1 Wb A-1 JOTTING
SPACE
Example :

When the current in a coil is increasing at a rate of 500 A s-1, the self-induced e.m.f. is 40
V. What is the self-inductance of this coil?

Example : Self-Inductance of a solenoid.

1. Suppose current I flow through a long air-core

solenoid with length l and N turns. The self-

inductance L of the solenoid is given by :

L = Nφ …..(i)
I

The magnetic flux φ is given by : …..(ii)
φ = BA cos(0o)

The magnitude of B inside the central portion of

the solenoid is given by :

B = µonI

or B = µo  N  I …..(iii)
l

| Chapter 7 | Electromagnetic Induction | page 11 / 18 |

Assume that the magnitude of B is uniform throughout the whole length of the
solenoid. Then, by substituting (ii) and (iii) into (i), the self-inductance of the
solenoid is given by :

L = µo  N2  A
l

The expression can also be written as :

L = µo  N2  lA ….. n =  N 
l2 l

or L = µon2lA …..(7.6)

Inductor : JOTTING
SPACE
1. Inductor :
= A circuit element which possesses mainly self-inductance.

An ideal inductor is one which does not posses resistance but only
self-inductance, or the resistance is so small that it may be neglected.

Examples of inductors are coils and solenoid.

2. The symbol used to represent an inductor :

Energy stored in an Inductor :

1. An inductor is a coil that is used specifically for its property of self-inductance. In
electronics, inductors play an important role in controlling current and as stores of
energy.

2. Energy stored in an inductor refers to the work performed upon the inductor to
overcome the back e.m.f due to self-inductance.

During the process of building up the magnetic flux, self-induction would have
taken place and the applied external voltage would have done work to overcome
this self-induced e.m.f. which opposes the applied voltage.

As a result, the magnetic flux represents a storage of energy in the coil.

3. The magnitude of back e.m.f : …..(i)
ξ = L dI 
 dt 

The electrical power supplied by the battery to the inductor :

P = Iξ …..(ii)

| Chapter 7 | Electromagnetic Induction | page 12 / 18 |

Combination between (i) and (ii), we obtain : …..(iii)
P = I L dI 
 dt 

We know that, generally electrical power is defined as :

P = dW …..(iv)
dt

Substitute (iii) into (iv), we obtain : …..(v)
dW = I L dI 
dt  dt 

Hence, work performed when the current has just reached a maximum constant JOTTING
value : SPACE

∫W = dW

I

∫= L IdI
0

= 1 LI2
2

Hence, total energy stored in the inductor : …..(7.7)
U = 1 LI2
2

4. This amount of energy is stored by the inductor in the magnetic field which is
produced by the current I.

Example :

Calculate the energy stored in a coil of self-inductance 50 mH when a current of 12.0 A
flows in it.

| Chapter 7 | Electromagnetic Induction | page 13 / 18 |

Mutual-Induction :

1. Mutual-induction is the phenomenon where an e.m.f. is induced in a conductor
when the current in a neighboring conductor is changing.

2. Consider two coils (solenoids) 1
and 2 arranged coaxially closed to
each other.

When a current I flows in the coil
1, a magnetic field is produced by
the current in coil 1. As a result,
there is a magnetic flux linked with
the coil 2.

When the current in coil 1
changes, then the magnetic flux
linkage in coil 2 changes as well
and an e.m.f. induced in coil 2.

© Mohd. Hazri @ kmph JOTTING
SPACE

Mutual-Inductance, M :

1. Mutual-inductance :
= The property of an electric circuit or component that causes an e.m.f. to be
generated in it as a result of a change in the current flowing through a
neighboring circuit with which it is magnetically linked.

or ξ2 = - M21 dI1  …..(7.8)
dt

© Physics Teaching Courseware

2. In practical applications it is sometimes convenient to denote the relationship as :
N2φ2 = M21I1

or M21 = N2φ2 …..(7.9)
I1

Note : The SI unit for mutual-inductance is Henry (H).
1 H = 1 T m2 A-1 = 1 Wb A-1

| Chapter 7 | Electromagnetic Induction | page 14 / 18 |

3. When the roles are reversed, analogical reasoning will lead to relationships in
which the coils have reversed their roles :

ξ1 = - M12 dI2 
dt

or M12 = N1φ1
I2

Example : Mutual-Inductance of two coaxial solenoids.

1. Consider a long solenoid (primary) of
cross-sectional area A with N1 tightly
wound turns and carrying a current.

Outside, there is another long solenoid
secondary) with N2 turns.

© Physics Teaching Courseware

In the inner solenoid the magnetic field : JOTTING
SPACE
B1 = µ o N1I1 …..(i)
L1

The magnetic flux through the outer solenoid due to magnetic field produced by
the inner solenoid is given by :

φ2 = B1A2 …..(ii)

Substitute (i) into (ii), we obtain :

φ2 =  µ oN1I1  A2 …..(iii)
L1

Hence, mutual-inductance of coaxial solenoid is given by :

M21 = N2φ2
I1

or M21 = µ oN1N2 A 2 …..(7.10)
L1

| Chapter 7 | Electromagnetic Induction | page 15 / 18 |

Transformer :

1. In everyday life, we encounter numerous examples of the application of the
phenomenon of electromagnetic induction. One of the devices that use this
phenomenon is a transformer.

JOTTING
SPACE

© Physics Teaching Courseware © Physics Teaching Courseware

2. A transformer comprises : NP NS laminated
Ferromagnetic core. turns iron core
Primary coil. turns
Secondary coil. secondary coil

Figure : Symbol of transformer.

primary coil

© Mohd. Hazri @ kmph

3. As an alternating voltage is supplied to the primary coil, then an alternating current
flows through the coil and produces magnetic field.

Hence, back e.m.f is generated due to self-induction in the coil.

ξ = -Np  dφ  p ….. Np = number of coil turns.
 dt 

Since there is no resistance in the both coils, then :

Primary coil Secondary coil

Vp = - ξp Vs = - ξs

∴ Vp = Np  dφ  p ∴ Vs = Ns  dφ  s
 dt   dt 

4. For an ideal transformer :

 dφ  p =  dφ  s
 dt   dt 

Hence,

Vp = Np …..(7.11)
Vs Ns

Hence,

If Ns > Np, then Vs > Vp. This is the property of ‘step-up transformer’.
If Ns < Np, then Vs < Vp. This is the property of ‘step-down transformer’.

| Chapter 7 | Electromagnetic Induction | page 16 / 18 |

5. Transformation efficiency, e :
The law of conservation of energy implies that the power Pout emitted at the
secondary coil may be, at most, equal to the power Pin absorbed in the primary coil.

e = Pout x100%
Pin

For an ideal transformer :
e = Pout x100% = 100%
Pin

or Pout = Pin

or IsVs = IpVp

where Is and Ip are currents in respective coils.

Therefore,

Vp = Np = Is …..(7.12)
Vs Ns Ip

6. Energy losses in transformers : JOTTING
Although transformers are very efficient devices, small energy losses do occur SPACE
in them owing to four main causes:

i) Resistance of coils :
The wire used for the primary and secondary coils has resistance and so
ordinary heat (I2R) losses occur.
Overcome : The transformer coils are made of thick copper wire.

ii) Eddy current :
The alternating magnetic flux induces eddy currents in the iron core. This
current causes heating and dissipation of power in the core.
Overcome : The effect is reduced by using laminated core as shown in
figure below.

© Mohd. Hazri @ kmph

iii) Hysteresis :
The magnetization of the core is repeatedly reversed by the alternating
magnetic field. The resulting expenditure of energy in the core appears as
heat.
Overcome : By using a magnetic material (such as Mumetal) which has
low hysteresis loss.

iv) Flux leakage :
The flux due to the primary may not all links the secondary. Some of the
flux loss in the air.
Overcome : By designing the iron core suitably.

| Chapter 7 | Electromagnetic Induction | page 17 / 18 |

Back e.m.f. in DC Motor

1. The principle used in the DC motor is the torque on a coil carrying a current in a
magnetic field. The coil is connected to a battery by carbon brushes pressing
against a commutator which is in the form of a split-ring.

motor
εR

commutator fixed brush VI
rotates contact

© Mohd. Hazri @ kmph JOTTING
SPACE
Fig 7.9a
2. When the coil of the motor rotates in the coil, magnetic flux linkage in the coil

changes. A back e.m.f. which opposes the voltage supply is induced.
ξ = -  dφ 
 dt 

or ξ = - d (NBA cos θ)
dt

or ξ = NBAωsin(ωt)

3. Using Kirrchoff’s second law for the complete loop round the circuit :

V - ξ = IR ….. (i)

where R = total resistance in the circuit.

From (i) X I, then :

IV - Iξ = I2R

where Iξ = power output (performed by the motor).
IV = power input (supplied by the source).
I2R = rate of heat loss.

| Chapter 7 | Electromagnetic Induction | page 18 / 18 |