Chapter 5

BP3 1 FYSL

Chapter 5: Electromagnetic Induction

5.1 Magnetic Flux

L.O 5.1.1 Define and use magnetic flux

Magnetic flux is defined as the scalar product between the magnetic flux density, B with

the vector of the area, A. It is a measure of the number of magnetic field lines that cross a

given area. Mathematically,

AB

cosBA

It is a scalar quantity.

Unit SI for Φ is T m2 or Wb ( weber )

If the coil is composed of N turns, all of the same area A, thus the magnetic flux through N

turns coil (magnetic flux linkage) is:

cosNBA

Note:

Direction of vector A always perpendicular (normal) to the surface area, A.

The magnetic flux is proportional to the number of field lines passing through the area.

Example

Question Solution

A single turn of rectangular coil of sides 10

cm × 5.0 cm is placed between north and

south poles of a permanent magnet. Initially,

the plane of the coil is parallel to the

magnetic field as shown in Figure.

If the coil is turned by 90° about its rotation

axis and the magnitude of magnetic flux

density is 1.5T, calculate the change in the

magnetic flux through coil.

where

Ф is magnetic flux

θ is the angle between and

Chapter 5

BP3 2 FYSL

5.2 Induced emf

Electromagnetic induction is the production of an induced e.m.f. (or voltage) across a

conductor or circuit situated in a changing magnetic field.

The meaning of changing in magnetic flux:

There is a relative motion of loop & magnet

field lines are ‘cut’:

The number of magnetic field lines passing

through a coil are increased or decreased:

L.O 5.2.1 Use Faraday’s experiment to explain induced emf

Faraday’s Experiment

When there is no relative

motion between the magnet

& the loop, G shows no

deflection. No induced

current.

Moving the magnet toward

the loop increases the

number of magnetic field

lines passing through loop.

The G needle is deflected

indicating an induced

current is produced.

Moving the magnet away from

the loop decreases the

number of magnetic field lines passing through the loop.

The induced current is now in

opposite direction.

Conclusion: From the experiments, it can be seen that e.m.f is induced only when the

magnetic flux through the coil change.

Chapter 5

BP3 3 FYSL

The magnitude of induced e.m.f. can be increased by:

Increasing the number of turns, N

Increasing the strength of magnet/ use stronger magnet (magnetic flux increased), B

Increasing the area of the coil or solenoid, A

Move the magnet into or out the solenoid faster

L.O 5.2.2 State Faraday’s law and use Lenz’s law to determine the direction of

induced current

Faraday’s law of induction states that “the magnitude of the induced e.m.f. is proportional to

the rate of change of the magnetic flux.”

Mathematically,

dt

d

The negative sign indicates that the direction of induced e.m.f. always oppose the change of

magnetic flux producing it (Lenz’s law). (To calculate the magnitude of induced e.m.f., the

negative sign can be ignored.)

Lenz’s law states that an induced current always flow in a direction that opposes the change

in magnetic flux that causes it.

By Lenz’s Law, when magnet is inserted into

the solenoid, a North pole will be induced on

the right side of coil to oppose the incoming

North pole. By right hand grip rule, the

induced current will flow anticlockwise so

that pointer deflects to right.

By Lenz’s Law, when magnet is withdrawn

from the solenoid, a South pole will be

induced on the right side of coil to oppose

the outgoing North pole. By right hand grip

rule, the induced current will flow clockwise

so that pointer deflects to left

Lenz’s Law is an example of principle of Conservation of Energy. Mechanical work is done

to against the opposing magnetic force experienced by the moving magnet, and this work is

converted into electrical energy as indicated by induced current flowing in the circuit.

Faraday’s Law gives the magnitude of induced e.m.f. while Lenz’s Law gives the direction

of the induced e.m.f..

where

dФ is change of magnetic flux

dt is change of time

Chapter 5

BP3 4 FYSL

Example

Question Solution

The solenoid in figure is moved at constant

velocity towards a fixed bar magnet. Using

Lenz’s law, determine the direction of the

induced current through the resistor.

Figure shows a permanent magnet

approaching a loop of wire. The external

circuit attached to the loop consists of the

resistance R. Find the direction of the induced

current and the polarity of the induced e.m.f..

Exercise

Question

A bar magnet is held above a loop of wire in a horizontal plane,

as shown in figure.

The south end of the magnet is toward the loop of the wire. The

magnet is dropped toward the loop. Find the direction of the

current through the resistor

a. while the magnet falling toward the loop and

b. after the magnet has passed through the loop and moves

away from it.

Chapter 5

BP3 5 FYSL

ε

v

B

L.O 5.2.3 Use induced emf dt

d

L.O 5.2.4 Derive and use induced e.m.f. in straight conductor/ in coil/ in rotating coil

According to Faraday’s law, the magnitude of induced e.m.f. is given by dt

d

CASE 1: Induced e.m.f. in a straight conductor

Blx

BA

cos

Blv

dt

dxBl

dt

Blxd

dt

d

In general, the magnitude of the induced e.m.f. in a linear conductor is given by

sinvBl

In vector form,

ldBvd

The direction of induced e.m.f. can be determined by

using right hand rule.

Thumb – induced e.m.f./ induced current

Other fingers – direction of motion

Palm – magnetic flux

Consider a linear (straight) conductor PQ of

length l is moved perpendicular with velocity v

across a uniform magnetic field B.

When the conductor moved through a distance x

in time t, the area swept out by the conductor is

given by

and θ = 0°

and

where θ is the angle between v and B

Chapter 5

BP3 6 FYSL

CASE 2: Induced e.m.f. in coil

Figure shows a coil of N turns each of area A in a magnetic flux density B.

By changing the magnetic flux density B

By changing the area A For a coil of N turns:

dt

dN

cosBA

dt

BAdN

cos

By changing the magnetic flux density B

dt

dBNA cos

If B is perpendicular to the plane of coil

θ = 0°

dt

dBNA

For a coil of N turns:

dt

dN

cosBA

dt

BAdN

cos

By changing the area A of the loop in uniform

magnetic field

dt

dANB cos

If B is perpendicular to the plane of coil

θ = 0°

dt

dANB

For a coil is connected in series to a resistor of resistance R

and the induced emf exist in the coil as shown in figure.

dt

dN

IR

dt

dNIR

Since, an e.m.f. can be induced in three ways:

1. by changing the magnetic flux density B

2. by changing the area A of the loop in the field

3. by changing the orientation θ with respect to the field

(rotating coil)

and and

and

Chapter 5

BP3 7 FYSL

CASE 3: Induced e.m.f. in a rotating coil

As a coil rotates in a uniform magnetic field, the magnetic flux through the area enclosed by

the coil changes with time; therefore induces an e.m.f. & current in the coil according to

Faraday’s Law.

Suppose that, coil has N turns, all of the same area A & rotates in a magnetic field B with a

constant angular velocity ω:

dt

dN

cosBA

dt

BAdN

cos

cosdt

dNAB and θ = ωt in rotational motion

tdt

dNAB cos

tNAB sin

The magnitude of the e.m.f induced in rotating coil is given by:

tNAB sin or sinNAB

From: Φ = NBA cos θ

As coil rotates, θ change, flux changes

cos θ ↓ , Φ ↓

Flux changes induces an emf or current

and

Coil perpendicular with B

θ = 0°

εmin = 0

Coil parallel with B

θ = 90°

εmax = NABω

Chapter 5

BP3 8 FYSL

Example

Question Solution

A 20 cm long metal rod CD is moved at speed of

25 m s-1

across a uniform magnetic field of flux

density 250 mT. The motion of the rod is

perpendicular to the magnetic field as shown in

figure below.

a) Calculate the motional induced e.m.f. in the

rod.

b) If the rod is connected in series to the

resistor of resistance 15 Ω, determine

i. the induced current and its direction.

ii. the total charge passing through the

resistor in two minute.

A single turn circular shaped coil has resistance of

10 Ω and area of its plane is 5.0 cm2. It moves

towards the north pole of a bar magnet as shown

in figure below.

If the average rate of change of magnetic flux

density through the plane of the coil is 0.50 T s-1

,

determine the induced current in the coil and state

the direction of the induced current observed by

the observer shown in figure above.

A narrow coil of 10 turns and diameter of 4.0 cm

is placed perpendicular to a uniform magnetic

field of 1.20 T. After 0.25 s, the diameter of the

coil is increased to 5.3 cm.

a. Calculate the change in the area of the coil.

b. If the coil has a resistance of 2.4 Ω, determine

the induced current in the coil.

Chapter 5

BP3 9 FYSL

Example

Question Solution

A rectangular coil of 200 turns has size 10 cm x

15 cm. It rotates at a constant angular velocity of

600 r.p.m. in a uniform magnetic field of flux

density 20 mT. Calculate

a. the maximum e.m.f. produced by the coil.

a. the induced e.m.f. at the instant when the

plane of the coil makes an angle of 60° with

the magnetic field.

Exercise

Question

A linear conductor of length 20 cm moves in a uniform magnetic field of flux density 20 mT

at a constant speed of 10 m s-1

. The velocity makes an angle 30° to the field but the conductor

is perpendicular to the field. Determine the induced e.m.f. across the two ends of the

conductor.

Answer: 2.0×10-2

V

A flat coil having an area of 8.0 cm2 and 50 turns lies perpendicular to a magnetic field of

0.20 T. If the flux density is steadily reduced to zero, taking 0.50 s, find

a. the initial flux through the coil.

b. the initial flux linkage.

c. the induced e.m.f.

Answer: 1.6×10-4

Wb; 8.0×10-3

Wb; 1.6×10-2

V

A circular shaped coil 3.0 cm in radius, containing 20 turns and have a resistance of 5.0 W is

placed perpendicular to a magnetic field of flux density of 5.0 x 10-3

T. If the magnetic flux

density is reduced steadily to zero in time of 2.0 ms, calculate the induced current flows in

the coil.

Answer: 2.83×10-2

A

The flexible loop has a radius of 12 cm and is in a magnetic field of strength 0.15 T. The loop

is then stretched until its area is nearly zero. If it takes 0.20 s to close the loop, find the

magnitude of the average induced e.m.f. in it during this time.

Answer: 3.4×10-2

V

A circular coil has 50 turns and diameter 1.0 cm. It rotates at a constant angular velocity of 25

rev s-1

in a uniform magnetic field of flux density 50 mT. Determine the induced e.m.f. when

the plane of the coil makes an angle 55° to the magnetic field.

Answer : 1.77 x 10-5

V

A coil of area 0.100 m2 is rotating at 60.0 rev s

-1 with the axis of rotation perpendicular to a

0.200 T magnetic field. If the coil has 1000 turns, find the maximum e.m.f. generated in it.

Answer: 7.54 kV

Chapter 5

BP3 10 FYSL

5.3 Self-inductance

Self-induction is defined as the process of producing an induced e.m.f. in the coil due to a

change of current flowing through the same coil.

• When the switch is closed, a current begin

to flow in the solenoid.

• The current produces a magnetic field

lines through the solenoid and generate

the magnetic flux linkage.

• If the resistance of the variable resistor

changes, thus the current flows in the

solenoid also changed, then so does the

magnetic flus linkage.

• According to Faraday’s law, an e.m.f has

to be induced in the solenoid itself since

the flux linkage changes.

• In accordance to the Lenz’s law, the

induced e.m.f opposes the change that has

induced it and it is therefore known as

back e.m.f.

I increases I decreases

If the current is increasing, so is the

magnetic flux.

According to the Lenz’s law, the induced

e.m.f. acts to oppose the increasing flux,

which means it acts like a source of e.m.f.

that opposes the external e.m.f.. This

induced e.m.f. is also known as back

e.m.f..

Therefore the direction of the induced

e.m.f is in the opposite direction of the

current I.

If the current is decreasing, so is the

magnetic flux.

According to the Lemz’s law, the induced

e.m.f. acts to oppose the decreaseing

flux, which means it acts to bolster the

flux, like a source of e.m.f. reinforcing

the external e.m.f..

Therefore the direction of the induced

e.m.f is in the same direction of the

current I.

I I

εinduced εinduced

N S N S

S N N S

Chapter 5

BP3 11 FYSL

L.O 5.3.1 Define self-inductance

L.O 5.3.2 Apply self-inductance equation for coil and solenoid

From the process of self-induction, we know that magnetic field B is proportional to current I,

and magnetic flux Ф is proportional to magnetic field B. Therefore

I

Mathematically,

LI

Self-inductance L is defined as the ratio set induced e.m.f. to the rate change current in the coil.

dt

dIL

It is a scalar quantity and its unit is henry (H).

Unit conversion:

For N turns of coil:

LIN

r

AN

I

NL

2

2

0

For N turns of solenoid:

l

AN

I

NL

2

0

The value of the self-inductance depends on

the size and shape of the coil,

the number of turn (N),

the permeability of the medium in the coil ().

A circuit element which possesses mainly self-inductance is known as an inductor. It is used

to store energy in the form of magnetic field.

The symbol of inductor:

Magnetic flux linkage

121 Am T1A Wb1H1

Chapter 5

BP3 12 FYSL

Example

Question Solution

At an instant, the current in an inductor increases

at the rate of 0.06 A s-1

and back e.m.f. of 0.018 V

was produced in the inductor.

a. Calculate the self-inductance of the inductor.

b. If the inductor is a solenoid with 300 turns,

find the magnetic flux through each turn

when the current of 0.80 A flows in it.

A 500 turns of solenoid is 8.0 cm long. When the

current in the solenoid is increased from 0 to 2.5

A in 0.35 s, the magnitude of the induced e.m.f. is

0.012 V. Calculate

a. the inductance of the solenoid,

b. the cross-sectional area of the solenoid,

c. the final magnetic flux linkage through the

solenoid.

(Given µ0 = 4p×10-7

H m-1

)

Exercise

Question

The coil in an electromagnet has an inductance of 1.7 mH and carries a constant direct

current of 5.6 A. A switch is suddenly opened, allowing the current to drops to zero over a

small interval of time, ∆t. If the magnitude of the e.m.f. induced during this time is 7.3 V,

what is ∆t ?

Answer: 1.3 ms

A 500 turns solenoid is 8.0 cm long. When the current in this solenoid is increased from 0 to

0.25 A in 0.35 s the magnitude of the induced e.m.f. is 0.012 V. Find

a. the inductance and

b. the cross sectional area of the solenoid.

Answer: 1.7 mH; 4.3×10-4

m2

A 40.0 mA current is carried by a uniformly wound air-core solenoid with 450 turns, a 15.0

mm diameter and 12.0 cm length. Calculate

a. the magnetic field inside the solenoid,

b. the magnetic flux through each turn,

c. the inductance of the solenoid.

Answer: 1.88´10-4

T; 3.33´10-8

Wb; 3.75´10-4

H

Chapter 5

BP3 13 FYSL

5.4 Energy stored in inductor

L.O 5.4.1 Derive and use the energy stored in an inductor

Consider a coil of self-inductance L. Suppose that at time t the current in the coil is in the

process of building up to its stable value I at a rate dI/dt. The magnitude of the back e.m.f. ε is

given by

dt

dIL

The power P in overcoming this back e.m.f. is given by

IP

dt

dIILP

ILdIPdt

ILdIdU

I

IdILdU0

2

2

1LIU

Example

Question Solution

An 8.0 cm long solenoid with an air-core

consists of 100 turns of diameter 1.2 cm. If

the current flows in it is 0.77 A, determine

a. the self-inductance of the coil

b. the energy stored in the coil

(Given µ0 = 4π x 10-7

H m-1

)

Exercise

Question

At the instant when the current in an inductor is increasing at a rate of 0.064 A s-1

, the

magnitude of the back e.m.f. is 0.016V.

a) Calculate the inductance of the inductor.

b) If the inductor is a solenoid with 400 turns and the current flows in is 0.720 A, determine

i. the magnetic flux through each turn

ii. the energy stored in the solenoid.

Answer: 0.25 H; 4.5 × 10-4

Wb, 6.48 × 10-2

J

Power × time

= Energy

Chapter 5

BP3 14 FYSL

5.5 Mutual-inductance

Mutual induction is defined as the process of producing an induced e.m.f in one coil due to

the change of current in another coil.

Consider two circular close-packed coils

near each other and sharing a common

central axis as shown in figure.

A current I1 flows in coil 1, produced by the

battery in the external circuit.

The current I1 produces a magnetic field

lines inside it and this field lines also pass

through coil 2 as shown in figure.

If the current I1 changes with time, the

magnetic flux through coils 1 and 2 will

change with time simultaneously.

Due to the change of magnetic flux through

coil 2, an e.m.f. is induced in coil 2. This is

in accordance to the Faraday’s law of

induction.

In other words, a change of current in one

coil leads to the production of an induced

e.m.f. in a second coil which is

magnetically linked to the first coil.

According to Lenz’s law, the induced

current produced in coil 2 will oppose the

change in I1.

This process is known as mutual induction.

At the same time, the self-induction occurs

in coil 1 since the magnetic flux through it

changes.

Primary

coil Secondary

coil

S N S N

S N N S

Chapter 5

BP3 15 FYSL

L.O 5.5.1 Define mutual inductance

L.O 5.5.2 Use mutual inductance question between two coaxial solenoids or a coaxial

coil and solenoid

If the current I1 in coil 1 is changes, the magnetic flux B through coil 2 will change with time t

and an induced e.m.f ε2 will occur in coil 2 where

dt

dI12

Mathematically,

dt

dIM 1

122

If vice versa, the induced e.m.f. in coil 1, ε1 is given by

dt

dIM 2

211

Conclusion,

MMM 2112

Mutual inductance is defined as the ratio of induced e.m.f in a coil to the rate of change of

current in another coil.

For a given pair of coils, the value of mutual inductance is the same and does not depend on

which coil carries the current and which coil experiences induction.

For N turns of coil:

1

2212

I

NM

or

2

1121

I

NM

Mutual inductance between two coaxial solenoids or a coaxial coil and solenoid

Lenz’s law

N1: primary coil

N2: secondary coil

l

ANNM 210

Chapter 5

BP3 16 FYSL

Example

Question Solution

A current of 2.0 A flows in coil P and produced a

magnetic flux of 0.6 Wb in it. When a coil S is

moved near to coil P coaxially, a flux of 0.2 Wb is

produced in coil S. Given that, coil P has 100

turns and coil S has 200 turns.

a. Calculate self-inductance of coil P and the

energy stored in P before S is moved near to

it.

b. Calculate the mutual inductance of the coils.

c. If the current in P decreasing uniformly from

2.0 A to zero in 0.4 s, calculate the induced

e.m.f. in coil S.

Primary coil of a cylindrical former with the

length of 50 cm and diameter 3 cm has 1000

turns. If the secondary coil has 50 turns,

calculate :

a. its mutual inductance

b. the induced e.m.f. in the secondary coil if the

current flowing in the primary coil is

changing at the rate of 4.8 A s-1

.

Exercise

Question

Two coils, X and Y are magnetically coupled. The e.m.f. induced in coil Y is 2.5 V when the

current flowing through coil X changes at the rate of 5 A s-1

. Determine:

a. the mutual inductance of the coils

b. the e.m.f. induced in coil X if there is a current flowing through coil Y which changes at

the rate of 1.5 A s-1

.

Answer : 0.5 H ; 0.75 V

Two coils, X and Y have mutual inductance of 550 mH. Determine the rate of change of

magnetic flux through coil Y at the instant when the current flowing through coil X changes

at the rate of 5.5 A s-1

. Given that, both coil X and Y has 100 turns.

Answer: 3×10-2

Wb s-1