Electromagnetic
Induction
Magnetic Flux
The magnetic flux is proportional to the number of magnetic flux lines passing
through the area.
  BA cos
The unit of flux density is the weber per square meter.
EXAMPLE : Magnetic Flux
A rectangular coil of wire is situated in a constant magnetic field whose magnitude is
0.50 T. The coil has an area of 2.0 m2 . Determine the magnetic flux for the three
orientations, f = 0°, 60.0°, and 90.0°, shown below.
Faraday's Law of Electromagnetic Induction
Michael Faraday found experimentally that the magnitude of the induced emf is proportional to the rate at
which the magnetic flux changed.
“Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced.
If the conductor circuit is closed, a current is induced which is called induced current.”
Faraday's Law of Electromagnetic Induction
“The induced emf in a coil is equal to the rate of change of flux linkage.”
Faraday’s law can be written as,
∆Φ
𝜀 = −𝑁
Δ𝑡
where N is the number of turns in
the loops, A is the area of one
loop, 𝜀 is the induced emf.
The SI unit for the induced emf is the Volt (V). The minus sign in the above Faraday’s
law of induction is due to the fact that the induced emf will always oppose the change.
Lenz Law
The induced emf resulting from a changing magnetic flux has a polarity that leads to
an induced current whose direction is such that the induced magnetic field opposes
the original flux change.
EXAMPLE : Faraday’s Law
Consider a flat square coil with 5 turns. The coil is 0,50 m on each side and has a
magnetic field of 0,5 T passing through it. The plane of the coil is perpendicular to the
magnetic field. Use Faraday’s Law to calculate the induced emf, if the magnetic field is
increases uniformly from 0,5 T to 1 T in 10 s.
EXAMPLE : Faraday’s Law
The magnetic flux linked with a coil is changing with time t(second)
according to 𝜙 = 6𝑡 2 − 5𝑡 + 1, where 𝜙 is in Wb. The resistance of the
circuit is 10 Ω. At t = 0.5 s, determine the induced current in the coil!
EXAMPLE : Faraday’s Law
Consider a solenoid of 9 turns with unknown radius, r. the solenoid is
subjected to a magnetic field od 0,12 T. the axis of solenoid is parallel to
the magnetic field. When the field is uniformly switched to 12 T over a
period of 2 minutes an emf with a magnitude of -0,3 V is induced.
Determine the radius of the solenoid.
Motion EMF
Motion is one of the major causes of induction. Consider the situation shown in Figure 1.
A rod is moved at a speed v along a pair of conducting rails separated by a distance ℓ in a
uniform magnetic field B. The rails are stationary relative to B and are connected to a
stationary resistor R.
Consider the area enclosed by the moving rod,
rails, and resistor. The area is increasing as the rod
moves. Thus the magnetic flux enclosed by the
rails, rod, and resistor is increasing. When flux
changes, an emf is induced according to Faraday’s
law of induction.
𝜀 = −𝐵𝑙𝑣
EXAMPLE : Motion EMF
A movable conducting rod is traveling to the right with an unknown
velocity, v. if the magnetic field is uniform and equal to 0,3 T, the length
of the rod is 20 cm and the induced emf is 60 mV, calculate the speed of
the conductor.
EXAMPLE : Motion EMF
A conducting rod is moving to the right with velocity of 2 m/s. the
resistance in the wire is 5 Ω while the length of the rod is 20 cm. if the
external magnetic field is 0,8 T :
a. Calculate the motional emf
b. Calculate the induced current inside the wire
Self Induction
Self Induction is the phenomenon of inducing emf in the self coil due to change in
current and hence the change in magnetic flux in the coil.
∆𝐼
𝜀 = −𝐿
∆𝑡
or
𝑑𝐼
𝜀 = −𝐿
𝑑𝑡
L is self inductance. SI unit of self inductance is henry (H).
Self inductance is said to be 1 henry when 1 A current in a coil links magnetic flux of 1
weber.
EXAMPLE : Self Induction
The inductance of a coil is 3 mH. The current flow though the coil changes
from 0,26 A to 1,3 A in 0,18 s. Calculate the magnitude of the average
induced EMF in the coil within the period.
EXAMPLE : Self Induction
An induced emf of 2 V is measured across a coil of 50 closely wound turns
while the current through it increases uniformly from 0 to 5 A in 0.1 s.
(a) What is the self-inductance of the coil?
(b) With the current at 5.0 A, what is the flux through each turn of the coil?
A Simple Generator
Faraday’s law dictates that if a coil of N turns experiences a change in
magnetic flux, then the induced voltage V is given by
𝑑Φ
𝜀 = −𝑁
𝑑𝑡
If a coil of area A rotates with respect to a field B, and if at a particular
time it is at an angle  to the field, then the flux linking the coil is BAcos,
and the rate of change of flux is given by
𝑑Φ
𝑑(cos 𝜃)
𝑑𝜃
=BA
= −𝐵𝐴 sin 𝜃
= −𝐵𝐴𝜔𝑠𝑖𝑛𝜃
𝑑𝑡
𝑑𝑡
𝑑𝑡
Thus for the arrangement shown below
𝑑Φ
𝑑(cos 𝜃)
𝜀 = −𝑁
= −NB A
= 𝑁𝐵𝐴𝜔 𝑠𝑖𝑛𝜃
𝑑𝑡
𝑑𝑡
EXAMPLE : Generator
A certain electric (AC) generator is used to produce a peak emf of 48 V.
The square loops of wire have a side length of 10 cm and rotate within
a magnetic field of 0,64 T. If the frequency of rotation is 50 Hz, find the
number of loops of wire found on the generators.
EXAMPLE : Generator
An ac generator has 12 turns of wire of area 0.08 m2. The loop rotates in a
magnetic field of 0.3 T at a frequency of 60 Hz. Find the maximum
induced emf
Transformers
A transformer is a device for increasing or decreasing an ac voltage.
A transformer consists of a primary coil and a secondary coil, both wound on an iron core.
The changing magnetic flux produced by the current in the primary coil induces an emf in the
secondary coil. At the far right is the symbol for a transformer.
When an alternating voltage, Vp is applied to the primary coil of Np turns it causes
an alternating to flow in this coil. This current causes a changing magnetic field in
the laminated iron core which cuts across the secondary coil of Ns turns.
Electromagnetic induction occurs in this coil which produces an alternating voltage,
Vs.
The voltages or potential differences across the primary and secondary coils of a
transformer are related by the equation:
𝑉𝑝 𝑁𝑝
=
𝑉𝑠 𝑁𝑠
If a transformer is 100% efficient then the power input to the primary coil is
equalled by the power output from the secondary coil.
𝑉𝑝 × 𝐼𝑝 = 𝑉𝑠 × 𝐼𝑠
𝑉𝑝 𝑁𝑝 𝐼𝑠
=
=
𝑉𝑠 𝑁𝑠 𝐼𝑝
EXAMPLE : Transformer
Calculate the number of turns required for the primary coil of a transformer
if secondary has 400 turns and the primary voltage is stepped up from 12V
to a secondary voltage of 48V.
EXAMPLE : Transformer
Complete:
PRIMARY
SECONDARY
Voltage
Turns
Voltage
Turns
230 V
1000
11.5 V
50
230 V
500
46 V
100
230 V
200
920 V
800
9V
120
72 V
960
EXAMPLE : Transformer
Calculate the primary current if when a transformer is supplied with 230V the
secondary provides 4A at a voltage of 13V. Assume that the transformer is
100% efficient.
EXAMPLE : Transformer
Complete:
PRIMARY
SECONDARY
Np
Vp
Ip
Ns
600
200V
0.4 A
30
100
12V
8A
300
72V
Vp
480V
50
10 A
Is
250
2.4 A
125V
2A
Step-down and Step-up Transformers
In a step-down transformer the voltage across the secondary
coil is smaller than the voltage across the primary coil.
Use: To decrease the voltage output from the mains supply
from 230V to 18V to power and recharge a lap-top computer.
In a step-up transformer the voltage across the secondary coil
is greater than the voltage across the primary coil.
Use: To increase the voltage output from a power station
from 25 kV (25 000 V) to up to 400 kV.