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Chapter 31 Faraday’s Law

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- Faraday’s Law of Induction
- Motional emf
- Lenz’s Law
- Induced emf and electric fields

* This chapter explores the effects produced by magnetic fields that vary in time.

* Michael Faraday (1831) showed that an emf can

be induced in a circuit by a changing magnetic field.

when a magnet is moved near a wire loop of area A, current flows through that wire without any batteries! ;

moving a magnet means changing the B-field through the loop area the magnetic flux changing

Variable ΦB (increasing)

Constant ΦB

Variable ΦB (decreasing)

The needle deflects momentarily when the switch is closed; initially (the instant of closing switch), the magnetic field increases during a certain very small period of time (the needle deflects) until B-field becomes steady (the needle goes back to zero)

The emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit.

where,

For N loops,

- To induce an emf we can change over time if the following:
- the magnitude of B
- the area enclosed by the loop
- the angle between B and the normal to the area
- any combination of the above

Ex: 31.1: a Way to Induce an emf in a Coil

B-field ┴ to the plane.

B-field changes linearly from 0 to 0.50 T in 0.80 s.

What is the magnitude of the induced emf in the coil while the field is changing?

square coil

N=200 turns

18 cm

The induced emf is

B-field

A=(0.18 m)² = 0.0324 m2

At t = 0 s, ΦB = 0 (no magnetic field)

At t = 0.8 s, ΦB= B.A = BAcos0

= BA= (0.50 T)(0.0324 m²) = 0.0162 T.m²

I

B

A wire loop of area A in a magnetic field B

B┴ A

Find the induced emf in the loop

as a function of time.

Electrons moves through the conductor because of FB

* When B changes and the conductor is stationary induced emf (examples befor)

*When a conductor moves through a constant magnetic field motional emf,

charges moves in the direction of FB and leaves positive charges behind.

As the wire moves,

* When B changes and the conductor is stationary induced emf (examples befor)

*When a conductor moves through a constant magnetic field motional emf,

charges moves in the direction of FB and leaves positive charges behind.

As the wire moves,

As they accumulate on the bottom, an electric field is set up inside.

In equilibrium,

Voltage drop across the conductor

a potential difference is maintained between the ends of the conductor as long as the conductor continues to move through the uniform magnetic field.

If the moving conductor is part of a closed conducting

path (closed circuit of resistance R).

The area enclosed by the circuit is A = lx

If the bar is moved with constant velocity,

conducting bar moves on two frictionless parallel rails in a uniform magneticfield directed into the page.

Using Newton’s laws, find the velocity of the bar as a function of time.

, but

The bar has a mass, m, and an initial velocity vi

where

- Sign indicate the force is to left

The polarity of the induced emf is such that it tends to produce a current that creates a magnetic flux to oppose(يعاكس)the change in magnetic flux through the area enclosed by the current loop.

As the bar is slid to the right, the flux through the loop increases.

This induces an emf that will result in an opposing flux.

Since the external field is into the screen, the induced field has to be out of the screen.

Which means a counterclockwise current

Suppose, instead of flowing counterclockwise, the induced current flows clockwise:

Then the force will be towards the right

which will accelerate the bar to the right

which will increase the magnetic flux

which will cause more induced current to flow

which will increase the force on the bar

… and so on

the system wouldacquire energy with no input of energy.

All this is inconsistent with the conservation of energy

- Left moving magnet decreases flux through the loop.
- It induces a current that creates it own magnetic field to oppose the flux decrease.

- Right moving magnet increases flux through the loop.
- It induces a current that creates it own magnetic field to oppose the flux increase.

When the switch is closed, the flux goes from zero to a finite value in the direction shown.

To counteract this flux, the induced current in the ring has to create a field in the opposite direction.

After a few seconds, since there is no change in the flux, no current flows.

When the switch is opened again, this time flux decreases, so a current in the opposite direction will be induced to counter act this decrease.

Find

(A) the magnetic flux through the area enclosed by the loop,

(B) the induced motional emf, and

(C) the external applied force necessary to counter the

magnetic force and keep v constant.

Read carefully the text of the example from the book

Induced Electric Field is created Inside a Conductor

Changing Magnetic Flux

EMF

This induced electric field is non-conservative and time-varying

For any closed path, we find E by

Hence, If the B changes with time

Induced E-field

General Form of Faraday’s Law

Work for moving a charge q one cycle

E is not conservative, because if it is conservative we will have

A conducting loop of radius r with B changing with time

For a circular loop

A long solenoid of radius R has n turns of wire per unit length and time varying current

r>R

Find E outside the solenoid at a distance r > R and inside the solenoid at r < R

r<R

Summary

*Faraday’s law of induction

induced emf

*conducting bar of length l moves at a velocity v through a magnetic field B

motional emf induced in the bar

*The applied force to keep constant v is

Where R is resistance connected to moving bar

*Lenz’s law states that the induced current and induced emf in a conductor are in such a direction as to set up a magnetic field that opposes the change in the magnetic flux

*A general form of Faraday’s law of induction is

Which implies induced E-field is not conservative

But,

=

First we need to find the change in the ΦB of the solenoid

Which is the same flux that change in the coil

A

A at P1

E