PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 14:

PHY 712 Spring 2014 -- Lecture 14 102/17/2014

PHY 712 Electrodynamics10-10:50 AM MWF Olin 107

Plan for Lecture 14:Start reading Chapter 6

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

2. Gauge choices and transformations3. 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

JDH

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

JEB

EMP


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

t

t

tt

AE

AE

AEBE

ABB

or

0 0

0


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

JAA

JEB

AE

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

JAA

J

JJ

JJJJJ

JAA

A

02

2

22

0002

0

022

2

22

02

1

1

0

:densitycurrent and chargefor equation Continuity0 and 0 with that Note

11

/

0 require -- form gauge Coulomb



JAA

A

JAA

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

JAA

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

JAA


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

tc

t

f rrrrr

r

rrrr

rrr



',''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

ttt

cttttG

ttttGtc

f

rrrrr

rrr

rrrr

rr

rrrr



'4 ,',~

,',~ 21',';,

:Define

21'

that note --domain timein the analysisFourier

''4',';,1

:function sGreen' theof Analysis

32

22

'

'

32

2

22

rrrr

rrrr

rrrr

Gc

GedttG

edtt

ttttGtc

tti

tti



eR

G

Gc

RdRd

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

21

'1

'1

21

,',~ 21',';,

'1,',~

:)(continuedfunction sGreen' theof Analysis

/''

/''

'

/'

rrrr

rrrr

rr

rr

rrrr

rrrr

rr

rr

rr



cttttG /'''

1',';, rrrr

rr

',''1''

1''

,, :, fieldfor Solution

3

0

tfc

ttdtrd

ttt

f

rrrrr

rrr



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

qRq(t)



3

0

1 ( ,' ')( , ) ' ' ( | | / )4 |

'|

'']

t td r dt t t c

rr r r

r rò

32

0

1 ( ', t')( , ) ] ' ' ( | ' | / ) .4 | ' |

t d r dt t t cc

J rA 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’,

3 'd r

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

1'

(

'

| ) |

' ' rq

q r q r

q r

f td f t ct t

c t

t t t t

r RR r R

r R

where the ``retarded time'' is defined to be | ( ) |

.q rr

tt t

c

Rr



Resulting scalar and vector potentials:

0

1( , ) ,4

qtR

c

v Rr ò

20

( , ) ,4

qtc R

c

vr v RA ò

Notation: ( )q rt r RR

( ),q rtv R | ( ) |

.q rr

tt t

c

Rr



In order to find the electric and magnetic fields, we need to evaluate ( , )( , ) ( , ) tt t

t

rr r AE

( , ) ( , )t tA rB r

The trick of evaluating these derivatives is that the retarded time tr depends on position r and on itself. We can show the following results using the shorthand notation:

andrtc R

c

Rv R

.rt Rt R

c

v R


Solution of Maxwell’s equations in the Lorentz gauge -- continued2

3 2 20

1( , v ) 1 ,4

q vt Rc c c c

Rc

v R Rr R Rv

vR

ò

2

3 2 2 20

( , ) 1 .4

t q R v R Rt c c Rc c c c

Rc

R

A v vr v v R v Rv R

ò

2

3 2 20

1( , ) 1 .4

q R v Rtc c c c

Rc

v v vr R R Rv

ER

ò

2

3 22 2 20

/ ( , )(

, ) 14

q v c ttc c c cR

R Rc c

R v R R R rrv

v v EBR v R

ò

Documents

PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 14: