02/19/2014phy 712 spring 2014 -- lecture 151 phy 712 electrodynamics 10-10:50 am mwf olin 107 plan...
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PHY 712 Spring 2014 -- Lecture 15 102/19/2014
PHY 712 Electrodynamics10-10:50 AM MWF Olin 107
Plan for Lecture 15:
Finish reading Chapter 61. Some details of Liénard-Wiechert results
2. Energy density and flux associated with electromagnetic fields
3. Time harmonic fields
PHY 712 Spring 2014 -- Lecture 15 502/19/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
Liénard-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 2014 -- Lecture 15 602/19/2014
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’,
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
PHY 712 Spring 2014 -- Lecture 15 702/19/2014
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 2014 -- Lecture 15 802/19/2014
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
PHY 712 Spring 2014 -- Lecture 15 902/19/2014
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 2014 -- Lecture 15 1002/19/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
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
PHY 712 Spring 2014 -- Lecture 15 1102/19/2014
Solution of Maxwell’s equations in the Lorentz gauge -- continued
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
PHY 712 Spring 2014 -- Lecture 15 1202/19/2014
Energy analysis of electromagnetic fields and sources
trd
dt
dW
rddt
dE
dt
dW
t)(
mech
DHE
JE
rJ
:producesit fields of in termscurrent source Expressing
:field neticelectromagby , sourceon done work of Rate
3
3
0 :monopoles magnetic No
0 :law sFaraday'
:law sMaxwell'-Ampere
:law sCoulomb'
B
BE
JD
H
D
t
t free
free
Maxwell’s equations
PHY 712 Spring 2014 -- Lecture 15 1302/19/2014
Energy analysis of electromagnetic fields and sources -- continued
JES
BHDE
HES
BH
DEHE
DHEJE
t
u
u
ttrd
trdrd
dt
dW
densityenergy 2
1
vector"Poynting" Let
3
33
PHY 712 Spring 2014 -- Lecture 15 1402/19/2014
Energy analysis of electromagnetic fields and sources -- continued
trdtrddt
dE
dt
dE
t
u
turdE
u
rddt
dE
fieldmech
field
mech
,ˆ ,
:analysisenergy previous theFrom
: vectorPoynting
,
2
1 :densityenergy neticElectromag
23
3
3
rSrrS
JES
HES
r
BHDE
JE
PHY 712 Spring 2014 -- Lecture 15 1502/19/2014
Momentum analysis of electromagnetic fields and sources
BBEE
PP
BEP
BvE
BJEP
220
3
30
3
2
1
: tensorstress Maxwell
:fields for vacuum Expression
:force Lorentzth analogy wiby Follows
cBBcEET
r
Tr d
dt
d
dt
d
rd
qF
rddt
d
ijjijiij
j j
ij
i
fieldmech
field
mech
PHY 712 Spring 2014 -- Lecture 15 1602/19/2014
Comment on treatment of time-harmonic fields
ti
ti
e,t)(dt) ,(
e),(d ,t) (
~
~
2
1
:domain in timeon ansformatiFourier tr
rErE
rErE
),(),(,t)( rErErE *~~ real is that Note
tititi )e,()e,()e,(,t)( rErErErE *~~
2
1~
:atmentcommon tre the tolead
principle,ion superposit theofnotion theand relations These
PHY 712 Spring 2014 -- Lecture 15 1702/19/2014
Comment on treatment of time-harmonic fields -- continued
tititi )e,()e,()e,(,t)( rErErErE *~~
2
1~
:fields harmonic for time Equations
0~
0 :monopoles magnetic No
0~~
0 :law sFaraday'
~~~ :law sMaxwell'-Ampere
~~ :law sCoulomb'
BB
BEB
E
JDHJD
H
DD
it
it freefree
freefree
Note -- in all of these, the real part is taken at the end of the calculation.
PHY 712 Spring 2014 -- Lecture 15 1802/19/2014
Comment on treatment of time-harmonic fields -- continued
tititi )e,()e,()e,(,t)( rErErErE *~~
2
1~
:fields harmonic for time Equations
),(),(,t
)e,(),()e,(),(
),(),(),(),(
)e,()e,()e,()e,(,t
,t)(,t)(,t
t
titi
titititi
rHrErS
rHrErHrE
rHrErHrE
rHrHrErErS
rHrErS
*
avg
2**2
**
**
~~
2
1
~~~~
4
1
~~~~
4
1
~~~~
4
1
: vectorPoynting
PHY 712 Spring 2014 -- Lecture 15 2002/19/2014
0 :monopoles magnetic No
0 :law sFaraday'
:law sMaxwell'-Ampere
:law sCoulomb'
B
BE
JD
H
D
t
t free
free
Maxwell’s equations
Summary and review
PHY 712 Spring 2014 -- Lecture 15 2102/19/2014
0 :monopoles magnetic No
0 :law sFaraday'
0 :law sMaxwell'-Ampere
0 :law sCoulomb'
:sources no and
; -- media isotropiclinear For
B
BE
EB
E
HBED
t
t
Maxwell’s equations
PHY 712 Spring 2014 -- Lecture 15 2202/19/2014
0 2
22
2
t
tt
BB
EB
EB
Analysis of Maxwell’s equations without sources -- continued:
0 :monopoles magnetic No
0 :law sFaraday'
0 :law sMaxwell'-Ampere
0 :law sCoulomb'
B
BE
EB
E
t
t
0 2
22
2
t
tt
EE
BE
BE
PHY 712 Spring 2014 -- Lecture 15 2302/19/2014
2
20022
2
2
22
2
2
22
where
01
01
n
ccv
tv
tv
EE
BB
Analysis of Maxwell’s equations without sources -- continued: Both E and B fields are solutions to a wave equation:
tiitii etet rkrk ErEBrB 00 , ,
:equation wave tosolutions wavePlane
PHY 712 Spring 2014 -- Lecture 15 2402/19/2014
Analysis of Maxwell’s equations without sources -- continued:
00
222
00
where
, ,
:equation wave tosolutions wavePlane
nc
n
v
etet tiitii
k
ErEBrB rkrk
Note: , e m, n, k can all be complex; for the moment we will assume that they are all real (no dissipation).
0ˆ and 0ˆ :note also
ˆ
0 :law sFaraday' from
t;independennot are and that Note
00
000
00
BkEk
EkEkB
BE
BE
c
n
t
PHY 712 Spring 2014 -- Lecture 15 2502/19/2014
Analysis of Maxwell’s equations without sources -- continued:
0ˆ and where
, ˆ
,
: wavesneticelectromag plane ofSummary
000
222
00
Ekk
ErEEk
rB rkrk
nc
n
v
etec
nt tiitii
kEkE
EkES rkrk
ˆ2
1ˆ2
ˆ1
2
1
: wavesneticelectromag planefor vector Poynting
2
0
2
0
*
00
c
n
ec
ne tiitii
avg
PHY 712 Spring 2014 -- Lecture 15 2602/19/2014
Analysis of Maxwell’s equations without sources -- continued:
0ˆ and where
, ˆ
,
: wavesneticelectromag plane ofSummary
000
222
00
Ekk
ErEEk
rB rkrk
nc
n
v
etec
nt tiitii
2
0
*
00
*
00
2
1
ˆˆ1
4
1
4
1
: wavesneticelectromag planefor density Energy
E
EkEk
EE
rkrk
rkrk
tiitii
tiitii
avg
ec
ne
c
n
eeu