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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
PHY 712 Spring 2015 -- Lecture 14 302/16/2015
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
PHY 712 Spring 2015 -- Lecture 14 402/16/2015
Maxwell’s equations
0 :monopoles magnetic No
0 :law sFaraday'
:law sMaxwell'-Ampere
:law sCoulomb'
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BE
JD
H
D
t
t free
free
PHY 712 Spring 2015 -- Lecture 14 502/16/2015
Maxwell’s equations
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
PHY 712 Spring 2015 -- Lecture 14 602/16/2015
Formulation of Maxwell’s equations in terms of vector and scalar potentials
t
t
tt
AE
AE
AE
BE
ABB
or
0 0
0
PHY 712 Spring 2015 -- Lecture 14 702/16/2015
Formulation of Maxwell’s equations in terms of vector and scalar potentials -- continued
J
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JE
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A
E
02
2
2
02
02
0
1
1
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tc
t
PHY 712 Spring 2015 -- Lecture 14 802/16/2015
Formulation of Maxwell’s equations in terms of vector and scalar potentials -- continued
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
PHY 712 Spring 2015 -- Lecture 14 902/16/2015
Formulation of Maxwell’s equations in terms of vector and scalar potentials -- continued
tC
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lCC
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:densitycurrent and chargefor equation Continuity
0 and 0 with that Note
11
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0 require -- form gauge Coulomb
PHY 712 Spring 2015 -- Lecture 14 1002/16/2015
Formulation of Maxwell’s equations in terms of vector and scalar potentials -- continued
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require -- form gauge Lorentz
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PHY 712 Spring 2015 -- Lecture 14 1102/16/2015
Formulation of Maxwell’s equations in terms of vector and scalar potentials -- continued
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require -- form gauge Lorentz
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PHY 712 Spring 2015 -- Lecture 14 1202/16/2015
Solution of Maxwell’s equations in the Lorentz gauge
source , field wave,
41
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1
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2
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22
02
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02
2
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tft
ftc
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PHY 712 Spring 2015 -- Lecture 14 1302/16/2015
Solution of Maxwell’s equations in the Lorentz gauge -- continued
','',';,'',,
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30
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represent ,Let ,,x y zA AA represent ,Let ,,x y zf J J J
PHY 712 Spring 2015 -- Lecture 14 1402/16/2015
Solution of Maxwell’s equations in the Lorentz gauge -- continued
',''1
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,,
:, fieldfor solution Formal
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PHY 712 Spring 2015 -- Lecture 14 1502/16/2015
Solution of Maxwell’s equations in the Lorentz gauge -- continued
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
PHY 712 Spring 2015 -- Lecture 14 1602/16/2015
Solution of Maxwell’s equations in the Lorentz gauge -- continued
eR
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Gc
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R
RG
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32
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1,',
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:'in isotropic is ,'~
that assumingFurther
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:infinityat aluesboundary v isotropic of case For the
'4 ,',~
:)(continuedfunction sGreen' theof Analysis
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rrrr
rrrr
rrrr
rrrr
PHY 712 Spring 2015 -- Lecture 14 1702/16/2015
Solution of Maxwell’s equations in the Lorentz gauge -- continued
cttctt
ed
eed
GedttG
eG
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citti
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ci
/'''
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rrrr
rrrr
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rr
PHY 712 Spring 2015 -- Lecture 14 1802/16/2015
Solution of Maxwell’s equations in the Lorentz gauge -- continued
cttttG /'''
1',';, rr
rrrr
',''1
''
1''
,,
:, fieldfor Solution
3
0
tfc
ttdtrd
tt
t
f
rrrrr
rr
r
PHY 712 Spring 2015 -- Lecture 14 1902/16/2015
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).
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)
PHY 712 Spring 2015 -- Lecture 14 2002/16/2015
Solution of Maxwell’s equations in the Lorentz gauge -- continued
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
PHY 712 Spring 2015 -- Lecture 14 2102/16/2015
Solution of Maxwell’s equations in the Lorentz gauge -- continued
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
PHY 712 Spring 2015 -- Lecture 14 2202/16/2015
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
PHY 712 Spring 2015 -- Lecture 14 2302/16/2015
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
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