02/17/2014 PHY 712 Spring 2014 -- Lecture 14 1 PHY 712 Electrodynamics 10-10:50 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
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PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 14:
PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 Plan for Lecture 14: Start reading Chap ter 6 Maxwell’s full equations; effects of time varying fields and sources Gauge choices and transformations Green’s function for vector and scalar potentials. - PowerPoint PPT Presentation
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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
PHY 712 Spring 2014 -- Lecture 14 202/17/2014
PHY 712 Spring 2014 -- Lecture 14 302/17/2014
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"
Formulation of Maxwell’s equations in terms of vector and scalar potentials -- continued
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
PHY 712 Spring 2014 -- Lecture 14 1202/17/2014
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
PHY 712 Spring 2014 -- Lecture 14 1302/17/2014
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
PHY 712 Spring 2014 -- Lecture 14 1402/17/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
',''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
PHY 712 Spring 2014 -- Lecture 14 1502/17/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
'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
PHY 712 Spring 2014 -- Lecture 14 1602/17/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
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
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rrrr
rrrr
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PHY 712 Spring 2014 -- Lecture 14 1702/17/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
cttctt
ed
eed
GedttG
eG
ctti
citti
tti
ci
/'''
1/'''
1
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'1
'1
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:)(continuedfunction sGreen' theof Analysis
/''
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/'
rrrr
rrrr
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rr
rrrr
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rr
PHY 712 Spring 2014 -- Lecture 14 1802/17/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
cttttG /'''
1',';, rrrr
rr
',''1''
1''
,, :, fieldfor Solution
3
0
tfc
ttdtrd
ttt
f
rrrrr
rrr
PHY 712 Spring 2014 -- Lecture 14 1902/17/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
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).
Solution of Maxwell’s equations in the Lorentz gauge -- continued
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
PHY 712 Spring 2014 -- Lecture 14 2102/17/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
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
PHY 712 Spring 2014 -- Lecture 14 2202/17/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
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
PHY 712 Spring 2014 -- Lecture 14 2302/17/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued2