Clicker Question: Non-uniform B field

A current loop is oriented so that its magnetic dipole is orientedalong direction “2”. In which direction is the force on the loop?

A) In direction 1B) In direction 2C) In direction 3D) In direction 4E) No force

0

ˆ

insideqdAnE

0 ldE

pathinsideIldB _0

Gauss’s law for electricity

Gauss’s law for magnetism

Incomplete version of Faraday’s law

Ampere’s law(Incomplete Ampere-Maxwell law)

First two: integrals over a surfaceSecond two: integrals along a path

Incomplete: no time dependence

Maxwell’s Equations (incomplete)

0ˆ AnB

Chapter 23

Faraday’s Law

Solenoid: insided

NIB 0

outside 0B

Constant current: there will be no forceson 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

121

01

2. Non-Coulomb electric field:using changing magnetic field

Same effect on charges: 1221 EqF

Two Ways to Produce Electric Field

Field outside of solenoid

𝐸1

𝑑𝐵1

𝑑𝑡1𝑟

Right hand rule:

Thumb in direction offingers: ENC

dt

Bd 1

Exercise:

Magnetic field points downfrom the ceiling and is increasing.

What is the direction of E?

Direction of the Curly Electric Field

The direction of the induced magnetic field due to a change in flux is such it attempts to keep the flux constant.

Lenz’s Rule

𝐵

𝐵𝑜𝑢𝑡𝑎𝑛𝑑𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔

𝐸𝑁𝐶

𝐵

𝐵𝑜𝑢𝑡𝑎𝑛𝑑𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔

𝐸𝑁𝐶

• The direction of the induced current (and is what is needed to produce the induced magnetic field.

𝐵𝑖𝑛𝑑

X 𝐵𝑖𝑛𝑑

ENC causes current torun along the ring

What is the surface chargedistribution?

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 rL emf 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 notEnclosing 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/dt2. ENC~ cross-section of a solenoid

emf d

dtB1r1

2

Quantitative Relationship Between B and EMF

emf d

dtB1r1

2

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

Magnetic flux on a small area A:

ABAnB ˆ

Definition of magnetic flux:

dABdAnBmag ˆ

Magnetic Flux

This area does not enclose a volume!

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 ruleMichael Faraday

(1791 - 1867)

Faraday’s Law

∮𝐸𝑁𝐶 ∙𝑑 𝑙=−𝑑𝑑𝑡 [𝐵 ∙ �̂�𝑑 𝐴 ]

Can we use total E in Faraday’s law?

ldEEldE CNCtotal

ldEldE CNC

=0

Including Coulomb Electric Field

∮𝐸𝑁𝐶 ∙𝑑 𝑙=−𝑑𝑑𝑡 [𝐵 ∙ �̂�𝑑 𝐴 ]

∮𝐸 ∙𝑑 𝑙=−𝑑𝑑𝑡 [𝐵 ∙ �̂�𝑑 𝐴 ]

dt

demf mag

mag B1 r12

Example: B1 changes from 0.1 to 0.7 T in 0.2 seconds; area=3 cm2.

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

A Circuit Surrounding a Solenoid

V

V downwards

𝐸𝑁𝐶

If we increase current through solenoid what will be ammeter reading?

A Circuit Not Surrounding a Solenoid

Each loop is subject to similarmagnetic field emf of loops in series:

dt

dNemf mag

The EMF for a Coil With Multiple Loops

Time varying B can beproduced 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 (+/-)?

2. The bar magnet is moved away from the coil.What will ammeter read?

3. The bar magnet is rotated. What will ammeter read?

dt

demf mag

Exercise

+

_

_

𝐵𝐵

?

‘Magnetic force’ approach:

BvqEqFtot

vBE LvBemf I

Use Faraday law:

dt

demf mag

ABmag tLvB

LvBt

emf mag

t

0lim I

Faraday’s Law and Motional EMF

𝑣 ∆ 𝑡

𝐿

𝐿

B1 B2

vL

R

dt

demf mag

mag B2A B1A B2 B1 A

Lvt tLvBBmag 12

120

lim BBvLt

emf mag

t

R

BBvL

R

emfI 12

I

Example

dt

demf mag

AnB ˆ

cosBwh

t Bwh cos t

tdt

dBwh

dt

demf mag cos

tBwhemf sin

I

Faraday’s Law and Generator

dt

demf mag

Two ways to produce curly electric field:1. Changing B2. Changing A

ABdt

d

dt

d mag

dt

dABA

dt

dB

Two Ways to Produce Changing