Chapter 23

Faraday’s Law

Moving charges make magnetic fields

Magnetic fields affect moving charges

Electric and magnetic fields depend on reference frame

Change reference frame:

magnetic field is converted into electric

Ex' Ex Ey

'Ey vBz

1 v2 / c2 Ez

'Ez vBy

1 v2 / c2

Time varying magnetic field can produce electric field!

(magnetic induction)

Electric and Magnetic Fields

Electric and magnetic fields are interconnected

Solenoid: insided

NIB 0

outside 0B

Constant current: there will be no forces

on charges outside (B=0, E=0)

What if current is not constant in time? Let B increase in time

E~1/r E~dB/dt

Non-Coulomb ENC !

Changing Magnetic Field

1. Coulomb electric field: produced by charges rr

qE ˆ

4

12

1

0

1

2. Non-Coulomb electric field:

using changing magnetic fieldrdt

BdE

1~ 1

1

Same effect on charges: 1221 EqF

Two Ways to Produce Electric Field

Field outside of solenoid

Right hand rule:

Thumb in direction of

fingers: ENCdt

Bd 1

Direction of the Curly Electric Field

ENC causes current to

run along the ring

What is emf and I?

ldEemf NC

22 rENC

R

rEI NC 22

Driving Current by Changing B

Ring has resistance, R

ldEemf NC

22 rENC

1. Change radius r2 by a factor of 2.

2/1~ rENC

22 rLemf does not depend on radius of the ring!

2. One can easily show that emf will be the same for any circuit

surrounding the solenoid

Effect of the Ring Geometry

ldEemf NC

A

D

NC

D

C

NC

C

B

NC

B

A

NC ldEldEldEldEemf

=0 =0

D

C

NC

B

A

NC ldEldEemf

0emf for any path which does not

enclose the solenoid!

Round-Trip Not Encircling the Solenoid

Is there current in these circuits?

Exercise

Can observe experimentally:

I=emf/R ENC~emf

1. ENC~dB/dt

2. ENC~ cross-section of a solenoid

emfd

dtB1 r1

2

Quantitative Relationship Between B and EMF

emfd

dtB1 r1

2

2

11 rB - magnetic flux mag on the area encircled by the circuit

Magnetic flux on a small area A:

ABAnB ˆ

Definition of magnetic flux:

dABdAnBmagˆ

Magnetic Flux

dt

demf

mag

Faraday’s law cannot be derived from the

other fundamental principles we have studied

Formal version of Faraday’s law:

Sign: given by right hand rule Michael Faraday

(1791 - 1867)

Faraday’s Law

dt

demf

mag

mag B1 r12

Example:

B1 changes from 0.1 to 0.7 T

in 0.2 seconds; area=3 cm2.

emft

0.6 T 3 10 4 m2

0.2 s9 10 4 V

What is the ammeter reading? (resistance of ammeter+wire is 0.5 )

A 3108.1/ RemfI

A Circuit Surrounding a Solenoid

Each loop is subject to similar

magnetic field emf of loops

in series:

dt

dNemf

mag

The EMF for a Coil With Multiple Loops

Time varying B can be

produced by moving coil:

by moving magnet:

by rotating magnet:

(or coil)

Moving Coils or Magnets

1. A bar magnet is moved toward a coil.

What is the ammeter reading?

a) Positive

b) Negative

dt

demf

mag