Note 9: Faraday’s Law

In 1831, Faraday discovered that EMF (electromotive force, i.e., voltage)

was induced by time varying magnetic flux. This was a monumental

discovery in the physics history. Before Faraday, various attempts had

been made to produce electric field from magnetic filed since it was

known that magnetic field was produced by electric field. Power

generation at large power plants and wind power generation all

depends on the Faraday's law.

Photocell produces electricity directly from solar radiation. This is

quantum mechanical effect.

(a) North pole of bar magnet is approaching a coil. The magnetic

flux in the coil is increasing with time. The current driven by the

induced EMF tends to oppose the increasing ma .

(

gne

b)

tic fl

No

ux

motion no EMF

(c) When the north pole is moving away from the loop,

EMF opposite to the case (a) is induced. The direction of the current

is reversed.

EMF induced by Time Varying Magnetic Flux

Faraday's law can be expressed as

Electromotive force =

where is the magentic flux (Wb).

Change in the magnetic flux is due to either timevarying

magnetic field , change in th

A

dEMF

dt

d

t

B A

Be area , or change in

the angle between and .

The magnetic flux through an area is

= cos (Wb)

where is the angle between the B field

and the vector normal to . Change in the

flux can

dA

dt

A

AB

A

B A

be due to that in , , or .B A

2 . When a magnetic field normal to a square coil of 18 cm

having 200 winding turns increases from 0 to 0.5 T in 0.8 sec,

what is the induced EMF?

Sol. The magnetic flux linked to the coil is

B

Example 1

2 4 2

The mi

0.5 T cos 200 18 10 (m ) 1 4.0 (Wb)

0.8 sec

4.0 V nus sig. Note cos 0 1.

A r

n is w.r.t. the di

ectangula

rection of

the current to produce the B field.

NAB t t

dEMF

dt

Example 2.

r coil of area is rotating in a magnetic field

at angular frequency rad/sec. Find the EMF to be induced.

Sol. The magnetic flux through the coil is changing with time as

= cos (Wb)

wh

A ab

B

abB t

ere 0 is chosen when the coild plane is normal to B (maximum flux).

Induced EMF is ( cos ) sin

This is the principle of ac generator.

t

dEMF abB t abB t

dt

A conductor loop of radius ia approaching a current carrying

wire at a velocity . Determine the direction of current induced in the loop.

Sol. The current is the loop will flow so as to o

a

v

Example 3.

ppose the change in the

magnetic flux enclosd by the loop. The magnetic flux out of the page increases

as the loop approaches the wire. Therefore, the current will flow clock wise.

If the coil is moving away from the current, the EMF is reversed.

v

I'

I

0

A rectangular loop of area is moving away from a

current carrying wire at velocity . Find the EMF to be indued.

Sol. The magnetic field due to the current carrying wire is

2

Th

ab

v

IB

r

Example 4.

0 0

0

e magntic flux through the loop is

1 = ln

2 2

Then the induced EMF is

1

2

The secondary current ' in the loop would

flow clockwise. (Its magni

vt a

vt

Ib Ib vt adr

r vt

Ibd aEMF

dt a vt t

I

tude depends on the resistance

in the loop.)

b

a

v

vt

a + vt

I

I '

In the figure, a conducting bar is

moving across the magentic field

at a vlocity . The magnetic flux

is increasing and

the induced EMF is

(V)

The minus sign indicates that the

curre

v

lxB lvtB

EMF lvB

nt fill flow so as to oppose

the change in the magnetic flux,

i.e., counterclockwise in this

case. This is known as Lenz's law.

Motional EMF

5. Electromagnetic Braking. A conducting bar of mass moves across

a magnetic field at an initial velocity on frictionless rails. The total resistance of

the circuit is . When the extern

i

m

B v

R

Example

al force is removed at 0, show that the

velocity decreases expoenentially with time.

Sol. The EMF is ( ) and the current can be found from

0

The equation of motion for the bar gives

t

Blv t I

Blv t RI

2

2

Solution is exp , (sec)i

lBdvm IlB v

dt R

t mRv t v

lB

If the velocity is reversed, so is EMF.

Motional Electric Field: E = v x B (V/m)

.

A conductor rod of length is

rotating in a magnetic field at an agular

velocity (rad/sec). Calculate the EMF

induced.

Sol. At radius , the electric field is

(V/m)

Then, the

r

l

r

E vB rB

Example 6

2

0

2

EMF is

1 (V)

2

If a conducting disk of radius is rotating

in a magnetic field, an EMF is induced

between the center and the edge,

1 (V)

2

This is the principle of homopol

l

rEMF E dr Bl

a

EMF Ba

ar generator.

. Problem 25. 0.5T, 1.2 m, 1 , 25 , 0.25kg.

Find the terminal velocity .

cosSol. EMF = cos

The source of motion is the gravitational potential

energy which is converted

B l R m

v

lBvlBv RI I

R

Example 7

2

2

2 2

to heat in the resistor .

sin

cossin

sin 1 0.2 9.8 sin 252.8 m/s

cos 1.2 0.5 cos 25

R

dRI mgh mgv

dt

lBvR mgv

R

Rmgv

lB

. Faraday tried to measure motional emf across the River Thames.

The sensitivity of instruments was not high enough and he could not confirm.

If the earth magentic field (nearly downward) is

Example 7

4

4

5 10 T and water flow velocity

is 3 m/sec, what emf is expected across the river width of 150 m?

Ans. emf = 3 5 10 150 0.225 V

. Having learned that an object moving across a magnetic fie

vBl

Example ld should induce

an emf, a student tried to measure it in a car which goes practically normal to the earth

magnetic field. He used a volt meter to measure voltage across the width of the car.

Explain why he failed.

Ans. The magnetic flux enclosed by the car is constant and there should be no emf.

0

9. The current of a long solenoid having winding

density (turns/m) is given by cos . Find the electric field

at distance from the axis of the solenoid.

Sol. The magnetic field in hte

n I t

r

Example

0 0

2 2

0 0 0 0

2

0 0

2 2

0 0 0 0

0 0

solenoid is cos .

Outside the sollenoid

2 cos sin

1sin

2

Inside

2 cos sin

1sin

2

B nI t

r R

drE EMF R nI t R I t

dt

RE I t

r

r R

drE EMF R nI t R I t

dt

E r I t

Generators and Motors

When a coudctor loop rotates in a magetic field, a sinusoidal

voltage is induced. The magnetic flux is = cos .

Induced EMF is

sin

AB t

dEMF AB t

dt

AC Generator

If a commutator is installed, the voltage becomes unipolar.

DC Generator

Practical dc generatorfor smooth dc EMF

Eddy Current

2

In a conductor, electric field drives current through the Ohm's law

(A/m )

where (S/m) is the conductivity. When a conductor plate is placed

near a coil carrying time varying current, electri

J E

c field is induced

and electric current is driven. The current heats the conductor

(Joule heating). Induction kitchen stove uses this principle.

In the pendulum shown, the one without

slits damps faster because eddy current

can flow without being impeded. The slits

prevents eddy current and oscillation

lasts longer.