02/17/2014phy 712 spring 2014 -- lecture 141 phy 712 electrodynamics 10-10:50 am mwf olin 107 plan...

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 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



',''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



'4 ,',~

,',~

2

1',';,

:Define

2

1'

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

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’,

3 'd r

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

( ) ( ( ))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



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

( , ) ( , )t

t tt

r

r rA

E

( , ) ( , )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:

andrt

c Rc

Rv R

.rt R

tR

c

v R



2

3 2 20

1( ,

v ) 1 ,

4

q vt R

c c c cR

c

v R Rr R R

v

v

R

ò

2

3 2 2 20

( , ) 1.

4

t q R v RR

t c c Rc c c cR

c

R

A v vr v v R v R

v R

ò

2

3 2 20

1( , ) 1 .

4

q R v Rt

c c c cR

c

v v vr R R R

vE

R

ò

2

3 22 2 20

/ ( , )(

, ) 1

4

q v c tt

c c c cRR R

c c

R v R R R rr

v

v v EB

R v R

ò

02/17/2014phy 712 spring 2014 -- lecture 141 phy 712 electrodynamics 10-10:50 am mwf olin 107 plan...

Documents

solution of maxwells

lorentz gauge slide

maxwells discovery

vector potentials

scalar potentials

terms of vector

gauge choices

q r q t