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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