02/16/2015phy 712 spring 2015 -- lecture 141 phy 712 electrodynamics 9-9:50 am mwf olin 103 plan for...

PHY 712 Spring 2015 -- Lecture 14 102/16/2015

PHY 712 Electrodynamics9-9:50 AM MWF Olin 103

Plan for Lecture 14:

Start reading Chapter 6

1. Maxwell’s full equations; effects of time varying fields and sources

2. Gauge choices and transformations

3. Green’s function for vector and scalar potentials


Full electrodynamics with time varying fields and sources

Maxwell’s equations

http://www.clerkmaxwellfoundation.org/

Image of statue of James Clerk-Maxwell in Edinburgh

"From a long view of the history of mankind - seen from, say, ten thousand years from now - there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics"

Richard P Feynman

http://www.clerkmaxwellfoundation.org/



0 :monopoles magnetic No

0 :law sFaraday'

:law sMaxwell'-Ampere

:law sCoulomb'

B

BE

JD

H

D

t

t free

free



00

2

02

0

1

0 :monopoles magnetic No

0 :law sFaraday'

1 :law sMaxwell'-Ampere

/ :law sCoulomb'

:0) 0;( form or vacuum cMicroscopi

c

t

tc

B

BE

JE

B

E

MP


Formulation of Maxwell’s equations in terms of vector and scalar potentials

t

t

tt

AE

AE

AE

BE

ABB

or

0 0

0


Formulation of Maxwell’s equations in terms of vector and scalar potentials -- continued

J

AA

JE

B

A

E

02

2

2

02

02

0

1

1

/

:/

ttc

tc

t



20

2

02 2

20

22

02 2 2

General form for the scalar and vector potential equations:

/

1

Coulomb gauge form -- require 0

/

1 1

Note that

C

C

CCC

l

t

c t t

c t c t

A

AA J

A

AA J

J J J with 0 and 0t l t J J



tC

C

lCC

Cll

tltl

CCC

C

C

tc

ttc

ttt

tctc

JA

A

J

JJ

JJJJJ

JA

A

A

02

2

22

0002

0

022

2

22

02

1

1

0

:densitycurrent and chargefor equation Continuity

0 and 0 with that Note

11

/

0 require -- form gauge Coulomb



JA

A

A

JA

A

A

02

2

22

02

2

22

2

02

2

2

02

1

/1

01

require -- form gauge Lorentz

1

/

:equations potential vector andscalar theof Analysis

tc

tc

tc

ttc

t

LL

LL

LL



01

: thatprovided physics same Yields

' and ' :potentials Alternate

1

/1

01

require -- form gauge Lorentz

2

2

22

02

2

22

02

2

22

2

tc

t

tc

tc

tc

LLLL

LL

LL

LL

AA

JA

A

A


Solution of Maxwell’s equations in the Lorentz gauge

source , field wave,

41

:equation waveldimensiona-3 theof form general heConsider t

1

/1

2

2

22

02

2

22

02

2

22

tft

ftc

tc

tc

LL

LL

rr

JA

A


Solution of Maxwell’s equations in the Lorentz gauge -- continued

','',';,'',,

:, fieldfor solution Formal

''4',';,1

:function sGreen'

,4,1

,

30

32

2

22

2

2

22

tfttGdtrdtt

t

ttttGtc

tft

t

ct

f rrrrr

r

rrrr

rr

r

represent ,Let ,,x y zA AA represent ,Let ,,x y zf J J J



',''1

''

1''

,,

:, fieldfor solution Formal

'1

''

1',';,

:infinityat aluesboundary v isotropic of case For the

''4',';,1

:function sGreen' for the form theofion Determinat

3

0

32

2

22

tfc

ttdtrd

tt

t

cttttG

ttttGtc

f

rrrrr

rr

r

rrrr

rr

rrrr



22 3

2 2

'

Analysis of the Green's function:

1 , ; ', ' 4 ' '

"Proof" -- Fourier analysis in the time domain -- note that

1 '

2

Define:

1 , ; ', '

2

i t t

G t t t tc t

t t d e

G t t

r r r r

r r

'

22 3

2

, ',

, ', 4 '

i t td e G

Gc

r r

r r r r



eR

G

Gc

RdR

d

R

RG

GG

Gc

cRi /

32

2

2

2

32

22

1,',

~ :Solution

'4 ,',~1

:'in isotropic is ,'~

that assumingFurther

,'~

,',~

:infinityat aluesboundary v isotropic of case For the

'4 ,',~

:)(continuedfunction sGreen' theof Analysis

rr

rrrr

rrrr

rrrr

rrrr



cttctt

ed

eed

GedttG

eG

ctti

citti

tti

ci

/'''

1/''

'

1

2

1

'

1

'

1

2

1

,',~

2

1',';,

'

1,',

~

:)(continuedfunction sGreen' theof Analysis

/''

/''

'

/'

rrrr

rrrr

rr

rr

rrrr

rrrr

rr

rr

rr



cttttG /'''

1',';, rr

rrrr

',''1

''

1''

,,

:, fieldfor Solution

3

0

tfc

ttdtrd

tt

t

f

rrrrr

rr

r



Liènard-Wiechert potentials and fields --Determination of the scalar and vector potentials for a moving point particle (also see Landau and Lifshitz The Classical Theory of Fields, Chapter 8.)

Consider the fields produced by the following source: a point charge q moving on a trajectory Rq(t).

3(Charge density: , ) ( ( )) qt q t r r R.

3 ( )Current density: ( , ) ( ) ( ( )), where ( ) .q

q q q

d tt q t t t

dt J r r

RR R R

q

Rq(t)



3

0

1 ( ,' ')( , ) ' ' ( | | / )

4 |'

|'

']t

td r dt t t c

r

r r rr rò

32

0

1 ( ', t')( , ) ' ' ' ( | ' | / ) .

4 | ' |t d r dt t t c

c

J r

A r r rr rò

We performing the integrations over first d3r’ and then dt’ making use of the fact that for any function of t’,

( )( ) ( | ( ) | / ) ,

( ) ( ( ))1

'

(

'

| ) |

' ' rq

q r q r

q r

f td f t c

t t

c t

t t t t

r RR r R

r R

where the ``retarded time'' is defined to be

| ( ) |.q r

r

tt t

c

Rr



Resulting scalar and vector potentials:

0

1( , ) ,

4

qt

Rc

v Rr

ò

20

( ,

) ,4

qt

c Rc

vr

v RA

ò

Notation: ( )q rt r RR

( ),q rtv R | ( ) |

.q rr

tt t

c

Rr


Comment on Lienard-Wiechert potential results

i

x

i

i xi

i

i

i

dxdp

xF

dxdx

dpxxxFdxxpxF

i)p(xp(x)

xFdxxxxF

F(x)

)(

)())(()(

,2,1for 0for which ,function aconsider Now

)()()(

:function any for that Note

--

-

00


Comment on Lienard-Wiechert potential results -- continued

-

In this case we have: ( ') ( ') ''

1

' where: ( ') '

'' ' '( ') ' 1 1

' ' '

r

q r q

q r

q

qq q q

q q

f tf t p t dt

t t

c t

tp t t t

c

d tt t tdp t dt

dt c t c t

R r R

r R

r R

Rr R R r R

r R r R


Summary of results for fields due to moving charge – Liénard Wiechert potentials

Resulting scalar and vector potentials:

0

1( , ) ,

4

qt

Rc

v Rr

ò

20

( ,

) ,4

qt

c Rc

vr

v RA

ò

Notation: ( )q rt r RR

( ),q rtv R | ( ) |

.q rr

tt t

c

Rr

02/16/2015phy 712 spring 2015 -- lecture 141 phy 712 electrodynamics 9-9:50 am mwf olin 103 plan for...

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