Faraday’s Law of InductionFaraday’s Law of Induction

Electromagnetic Induction

A changing magnetic field (intensity, movement) will induce an electromotive force (emf)

In a closed electric circuit, a changing magnetic field

will produce an electric current

Faraday’s Law of Induction

The induced emf in a circuit is proportional to the rate of changeof magnetic flux, through any surface bounded by that circuit.

Ɛ = - B / t

• In the easiest case, with a constant magnetic field B, and a flat surface of area A, the magnetic flux is

B = B A cos

• Units : 1 tesla x m2 = 1 weber

Magnetic Flux

A

B

Magnetic Flux

What is the flux of the magnetic fieldIn each of these three cases?

a)b)c)

Find the magnetic flux through the loop

Loop radius r = 2.50 cm, magnetic field B = 0.625 T.

Faraday’s Experiments

• Michael Faraday discovered induction in 1831.

• Moving the magnet induces a current I.

• Reversing the direction reverses the current.

• Moving the loop induces a current.

• The induced current is set up by an induced EMF.

N S

Iv

Faraday’s Experiments

• Changing the current in the right-hand coil induces a current in the left-hand coil.

• The induced current does not depend on the size of the current in the right-hand coil.

• The induced current depends on I/t.

I

I/t

S

EMF

(right)(left)

Faraday’s Law

• Moving the magnet changes the flux B (1).

• Changing the current changes the flux B (2).

• Faraday: changing the flux induces an emf.

i

i/t

S

EMF

N S

iv

Ɛ = - B /t

The emf induced around a loop

equals the rate of changeof the flux through that loop

Faraday’s law

1) 2)

Lenz’s Law

• Faraday’s law gives the magnitude and direction of the induced emf, and therefore the direction of any induced current.

• Lenz’s law is a simple way to get the directions straight, with less effort.

• Lenz’s Law:

The induced emf is directed so that any induced current flow

will oppose the change in magnetic flux (which causes the

induced emf).

• This is easier to use than to say ...

Decreasing magnetic flux emf creates additional magnetic field

Increasing flux emf creates opposed magnetic field

Lenz’s Law

If we move the magnet towards the loop

the flux of B will increase.

Lenz’s Law the current induced in the

loop will generate a field B opposed to B.

N S

Iv

B

B

Example of Faraday’s Law

Consider a coil of radius 5 cm with N = 250 turns. A magnetic field B, passing through it, changes at the rate of B/t = 0.6 T/s.The total resistance of the coil is 8 .What is the induced current ?

Use Lenz’s law to determine thedirection of the induced current.

Apply Faraday’s law to find theemf and then the current.

B

Example of Faraday’s Law

B

Hence the induced current must be clockwise when looked at from above.

Lenz’s law: B/t > 0

B is increasing, then the upwardflux through the coil is increasing.

So the induced current will have a magnetic field whose flux (and therefore field) is down.Induced B

I

Use Faraday’s law to get the magnitude of the induced emf and current.

B

Induced B

I

The induced EMF is: Ɛ = - B / t

In terms of B: B = N(BA) = NB (r2)

Therefore Ɛ = - N (r2)B / t Ɛ = - (250) (0.0052)(0.6T/s) = -1.18 V (1V=1Tm2 /s)

Current I = Ɛ/ R = (-1.18V) / (8 ) = - 0.147 A

B = B A cos

Ɛ= - B / t

Up until now we have considered fixed loops. The flux through them changed because the magnetic field changed with time.

Now, try moving the loop in a uniform and constant magnetic field. This changes the flux, too.

Motional EMF

B points into screen

x x x x x

x x B x x x

x x x x x

Rx

D

v

The flux is B = B A = BDx

This changes in time:

B / t = BD( x/ t) = -B D v

Hence, by Faraday’s law, there is an induced emf and current. What is the direction of the current?

Lenz’s law: there is less inward flux through the loop. Hence the induced current gives inward flux.

So the induced current is clockwise.

Motional EMF - Use Faraday’s Law

x x x x x

x x B x x x

x x x x x

Rx

D

v

Faraday’s Law B/ t = - Ɛ

gives the EMF Ɛ = BDv

In a circuit with a resistor, this gives

Ɛ = BDv = IR I = BDv/R

Motional EMFFaraday’s Law

x x x x x

x x B x x x

x x x x x

Rx

D

v

Thus, moving a circuit in a magnetic field produces an emf exactly like a battery. This is the principle of an electric generator.

A New Source of EMF

• If we have a conducting loop in a magnetic field, we can create an EMF (like a battery) by changing the value of B = B A cos

• This can be done by changing the area, by changing the magnetic field, or the angle between them.

• We can use this source of EMF in electrical circuits in the same way we used batteries.

• Remember we have to do work to move the loop or to change B, to generate the EMF (Nothing is for free!).

Motional EMF

What happens when the conductinghorizontal bar falls?

Motional EMF

What happens when the conductinghorizontal bar falls?

1. As the bar falls, the flux of B changes,so a current is induced, and the bulblights.

Motional EMF

What happens when the conductinghorizontal bar falls?

1. As the bar falls, the flux of B changes,so a current is induced, and the bulblights.

2. The current on the horizontal bar is in a magnetic field, so it experiencesa force, opposite to gravity.

Motional EMF

What happens when the conductinghorizontal bar falls?

1. As the bar falls, the flux of B changes,so a current is induced, and the bulblights.

2. The current on the horizontal bar is in a magnetic field, so it experiencesa force, opposite to gravity.

3. The bar slows down, until eventuallyit reaches constant speed (FB = mg)

Eddy Currents

A circulating current is inducedin a sheet of metal when it leavesa region with a magnetic field.

The effect of the magnetic fieldon the induced current is suchthat it creates a retarding force on the moving object.

What is the external forcenecessary to move the barwith constant speed?

The induced current isI = Ɛ / RI = B v L / R

The magnetic force onthe current is:F = B I L F = B (BvL/R) LF = B2 v L2 /R

Mechanical Power to

Electrical Power

The mechanical power usedto move the bar is:

PM = F v = B2 v2 L2 / R

The electrical powerdissipated in the circuit is:

PE = I2 R = (B v L / R)2 RPE = B2 v2 L2 / R

The mechanical power is transformed into

electrical power

Given:Bulb: R = 12 , P = 5 WRod: L = 1.25 m, v = 3.1 m/s

Find:a) Magnetic field strengthb) External force

Electric Generator

Simple Electric Motor