Chapter 7 Electrodynamics

7.0 Introduction

7.1 Electromotive Force

7.2 Electromagnetic Induction

7.3 Maxwell’s Equations

?Et

? 0t

7.0 Introduction

electrostatic static

0

1E

magnetostatic

0B J

conservation of charge

? B

E

00?

0

0 ?B Jt

0

=

0

0E

0B

0Jt

?Et

7.0 (2)

Maxwell’s equations:

0E

0B

BE

t

0 0 0B J Et

dJ displacement current

7.0 (3)

dE

Magnetic flux

Induced electric field (force)

)(tB

induce

EB

E

•

=

BE

t

E da B dat t

B da

7.0 (4)

E,B fields propagate in vacuum e.g. , BE

, ~ )( wtkxie

• E Bt

0 0B Et

aB

a aE induced by B

b aB induced by E

b bE induced by B

c bB induced by E

wave

7.0 (5)

•

A.C. current can generate electromagnetic waveantennacyclotron massfree electron laser …..

E Bt

0 0 0B J Et

0( , )J x x t

aB

aE

bB

bE

7.1 Electromotive Force

7.1.1 Ohm’s Law

7.1.2 Electromotive Force

7.1.3 Motional emf

7.1.1 Ohm’s Law

Current density conductivity force per unit charge of the medium

resistivity

0 for perfect conductors

for vk

usually true

but not in plasma; especially, hot.

Ohm’s Law

•

•

( a formula based on experience)

J f

1

for f E v B

( )J E v B

J E

7.1.1 (2)

Total current flowing from one electrode to the other

V=I R Ohm’s Law (based on experience)

Potential current resistance [ in ohm (Ω) ]

Note : for steady current and uniform conductivity

•

10E J

7.1.1 (3)

Ex. 7.1

sol:

LV

AEAJAI

parallelin

seriesin

AL

R

I=?R=?

uniform

uniform

V

1 2 1 2,L L R R R

211 2

1 1 1,A A

R R R

7.1.1 (4)

Ex. 7.3 Prove the field is uniform E

i.e.,

V=0 V=V0A=const =const

ˆ0 0 at the surfaces on the two endsJ J n

ˆ 0E n

0V

n

2 0 Laplace equationV

0( )V z

V zL

0 ˆV

E V zL

7.1.1 (5)

L2

)ab(ln

R

Ex. 7.2 V ?Is

0ˆ

2E s

s

: line charge density

0ln ( )

2

a

b

bV E d

a

E V

10

0 02 [ ln ]

bI J da E da L V L

a

2

ln ( )

LV

ba

7.1.1 (6)

The physics of Ohm’s Law and estimation of microscopic

the charge will be accelerated by before a collision

time interval of the acceleration is

E

a

2,

vmint mfp

thermal

mfp

mean free path

2

21

tamfp typical casefor very strong field and long mean free path

2 thermalave

nfq FJ n f q v

v m

7.1.1 (7)The net drift velocity caused by the directional acceleration is

molecule density e charge

free electrons per molecule

Eq

=

mass of the molecule

RIVIP 2Power is dissipated by collision

Joule heating law

1

2 2 thermalave

av at

v

2

2 thermal

nf qJ E

mv

b bab s sa aV E d f d f d

sf d f d

7.1.2 Electromotive Force

The current is the same all the way around the loop.

force electrostatic

electromotive force

0dE )0( E

outside the source

Produced by the charge accumulationdue to Iin > Iout

sourcef f E

E V

0f

sE f

7.1.3 motional emf

,,

mag vmag v

Ff vB

q

B

,mag vF qvB

, causesmag vf d vBh u

( ) ( )dx d d

vBh Bh Bhxdt dt dt

7.1.3 (2)

h

cossin

dd

=

sin)

cos)((

huBdf pull

cossin

uv

vBh

Work is done by the pull force, not . B

magnetic flux

pullf uB for equilibrium

7.1.3 (3)

magnetic flux

for the loop

flux rule for motional emf

B da Bhx

d dxBh vBh

dt dt

d

dt

( ) magw B d f d

7.1.3 (4)

a general proof

dtd

•

ribbon

( ) ( )d t dt t

ribbonB da

( )da v d dt

( ) ( )d

B v d B w ddt

magf

7.1.3 (5)

Ex. 7.4

=?

0

a

magf ds

0

awsB ds

2

2

wBa

2

R 2

wBaI

R

ˆ( )magf v B ws w B

ˆwsB s

7.2 Electromagnetic Induction

7.2.1 Faraday’s Law

7.2.2 The Induced Electric Field 7.2.3 Inductance

7.2.4 Energy in Magnetic Fields

7.2.1 Faraday’s Law M. Faraday’s experiments

Induce induce induce

Faraday’s Law (integral form)

Faraday’s Law (differential form)

loop moves B moves B Area ,

[ ]I v B

[ ]I E

[ ]I E

( )emfd

E d E dadt

d

B da B dadt t

BE

t

Lenz’s law : Nature abhors a change in flux ( the induced current will flow in such a direction that the flux it produces tends to cancel the change. )

7.2.1 (2)

A changing magnetic field induces an electric field.

(a) (b) & (c) induce that causesE

I

drive I

, notv B E

7.2.1 (3)

sol:ˆ MnMKb

MB

0

at center , spread out near the ends

2

0max aM

Ex. 7.5

Induced ? )(t

ˆz r

loop

7.2.1 (4) Ex. 7.6

Plug in, why ring jump?rI

Plug in, induces

B

B F

F

F

ring jump.

sI

I

B

induces rB I

v B

v B

7.2.2 The Induced Electric Field

0 encB d I

dtd

dE

BE

t

0B J

0 ( 0)E

0B

7.2.2 (2)

induced = ?E

sol:

dtBd

stBsdtd

dtd

dE

22 )]([ sE 2

=

2 dtBds

E

E

B

Ex. 7.7

7.2.2 (3)

dtdB

adtd

dE 2

0BB

The charge ring is at rest

0B

What happens?sol:

torque on d ˆ( ) ( )dN r F b d E z b Ed

2 2ˆ ˆ [ ]dB dB

N dN zb E d zb a b adt dt

the angular momentum on the wheel

zbBaBdabdtNB ˆ0

202

0

Ex. 7.8. z

7.2.2 (4)

sol:

Induced ?)( sE

quasistatic

z

B

( )I t

0 ˆ2

IB

s

7.2.2 (5)

=

Constant K( s , t )

0 '2 '

Id dE d B da ds

dt dt s

0( ) ( )E s E s

0

0 1'

2 '

s

s

dIds

dt s

00 (ln ln )

2

dIs s

dt

0 ˆ( ) [ ln ]2

dIE s s K z

dt

s << c = I / (dI/dt)

7.2.3 Inductance

121212 IMadB

21)( adA

mutual inductance

1 2A d

0 11 1 12

ˆ

4

d RB I I

R

0 1 124

I dd

R

0 1 11 4

I dA

R

7.2.3 (2)

Neumann formula

The mutual inductance is a purely geometrical quantity

0 1 221 4

d dM

R

M21 = M12 = M 1 = M12 I2

1 = 2 if I1 = I2

7.2.3 (3)

Ex. 7.10

sol:B1 is too complicated… 2 = ?

Instead, assume I running through solenoid 2

20 1 2M a n n

III 12

?

?2

M

n2 turns per unit length

n1 turns per unit length

2

1 I given

assume I too.

1 1 1, per turmn 21 2

20 1 2 2

20 1 2

2 2 1( )

n a B

a n n I

a n n I

I I I

2 0 2 2B n I

7.2.3 (4)

• )(1 tI

dtdI

Mdtd 12

2

changing current in loop1, induces current in loop21I

• self inductance

)(tI

self-inductance (or inductance )

[ unit: henries (H) ]A

VoltH

sec11

• back emf

L I

will reduce it.dI

L Idt

7.2.3 (5)

Ex. 7.11

sol: adBN

sNI

B

20

b

adss

hNI

N1

20

20 ln ( )2

N h bL

a

L(self-inductance)=?

b

a

N turns

20 ln ( )2

N Ih b

a

7.2.3 (6)

Ex. 7.12

sol:

IRdtdI

L 0

0( )Rt

LI t keR

particular solution

)1()1()( 00 tt

LR

eR

eR

tI

R0

( ) ?I t

0if (0) 0 ,I kR

time constantL

R

general solution

7.2.4 Energy in Magnetic Fields

From the work done, we find the energy

in , E

dEdVWe20

2)(

21

But, does no work.B

In back emf

In E.S.

test charge

q

21( )2B

d dI dW I L I LI

dt dt dt

21 1

2 2BW LI I 21

( )2kW mv

( )s s loopB da A da A d

1 1

( )2 2B loop loop

W I A d A I d

WB = ?

7.2.4 (2)In volume

1( )

2B VW A J d

dBAV )(

21

0

dBAdBVV )(

21

21

0

2

0

)()()( BAABBA

B

2B

s

adBA )(

s0

dBWspaceallB 2

021

dEdVWelec20

2)(

21

dBdJAWmag2

021

)(21

7.2.4 (3)

Ex. 7.13

sol:

bsasI

B ˆ2

0

＜ ＜ 0B

20

0

1( ) (2 )

2 2B BI

W dW sdss

)length(

?BW

s as b

20 ln( )4

I b

a

21

2BW L I

0 ln ( )2

bL

a

20

4

b

a

I ds

s

7.3 Maxwell’s Equations

7.3.1 Electrodynamics before Maxwell 7.3.2 How to fix Ampere’s Law 7.3.3 Maxwell’s Equations

7.3.4 Magnetic Charge

7.3.5 Maxwell’s Equation in Matter

7.3.6 Boundary Conditions

7.3.1 Electrodynamics before Maxwell

0)()()(

B

ttB

E

but

?)()( 0 JB

=0

Ampere’s Law fails because 0 J

0E

0B

BE

t

0B J

(Gauss Law)

(no name)

(Faraday’s Law)

(Ampere’s Law)

7.3.1

an other way to see that Ampere’s Law fails for nonsteady current

encIdB 0

they are not the same.

loop 1

2

For loop 1, Ienc = 0For loop 2, Ienc = I

7.3.2 How to fix Ampere’s Law

)(][ 00 tE

Ett

J

continuity equations, charge conservation

such that, Ampere’s law shall be changed to

tE

JB

000

A changing electric field induces a magnetic field.

Jd displacement current

7.3.2

adtE

JadB

)( 000

adtE

IdB enc

000

=

for the problem in 7.3.1

between capacitorsAQ

E00

11

IAdt

dQAt

E

00

11

IIdBloop 0

01 00

10

IIdBloop 02 0 0

loop 1

2

7.3.3 Maxwell’s equations

0 B

Et

JB

000

tB

E

0 E

Gauss’s law

Faraday’s law

Ampere’s law with Maxwell’s correction

Force law

continuity equationt

J

( the continuity equation can be obtained from Maxwell’s equation )

( )F q E v B

7.3.3

0 B

JEt

B

000

0tB

E

0 E

Since , produce , J

E

B

),( trJ

E

B

7.3.4 Magnetic Charge

Maxwell equations in free space ( i.e., , )0e 0eJ

symmetric

BE

EB

00

With and , the symmetry is broken.If there were ,and .

e eJ

m mJ

mB 0

tB

JE m

0

tE

JB e

000 symmetric

tJ ee

t

J mm

and

So far, there is no experimental evidence of magnetic monopole.

0E

0B

Et

0B

0 0B Et

0

eE

7.3.5 Maxwell’s Equation in Matter

bound charge bound current

Pb MJb

0 no correspondingbJ

tP

tb

polarization currentPJ

0

Pb Jt

da

tda

tdI b )(

daJadtP

P

Pb

Q

surface charge

7.3.5 (2)

Pfbf

Pt

MJJJJJ fPbf

0

1Gauss's law ( )fE P

fDor

PED

0

Et

Pt

MJB f

000 )(

Ampere’s law ( with Maxwell’s term )

)()( 0000 PEt

JMB f

Dt

JH f

MBH

0

1

7.3.5 (3)

In terms of free charges and currents, Maxwell’s equationsbecome

fD

Dt

JH f

0 B

tB

E

displacement current, and , are mixed.D H E B

one needs constitutive relations: ( , ) and ( , )D E B H E B

for linear dielectric.

7.3.5 (4)

orExP e

0

ED

HxM m

BH

1

)1(0 ex

)1(0 mx

0 B

fE

tB

E

tE

JB f

7.3.6 Boundary Condition

Maxwell’s equations in integral form

Over any closed surface S

for any surface bounded by the S closed loop L

L s

dE d B da

dt

,f encsD da Q

0

sB da

fencL s

dH d I D da

dt

1 1,D B

2 2,D B

7.3.6

aaDaD f 21

0S 021 adB

dtd

EE

fDD 21

021 BB

021 EE

= =

)nK()n(KHH ff

21

nKHH f ˆ21 = =

nKBB f ˆ11

22

11

= =

fEE 2211