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03/20/2015PHY 712 Spring Lecture 223
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PHY 712 Spring 2015 -- Lecture 22 103/20/2015
PHY 712 Electrodynamics9-9:50 AM MWF Olin 103
Plan for Lecture 22:Start reading Chap. 9
A. Electromagnetic waves due to specific sources
B. Dipole radiation patterns
PHY 712 Spring 2015 -- Lecture 22 203/20/2015
PHY 712 Spring 2015 -- Lecture 22 303/20/2015
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
JEB
EMP
PHY 712 Spring 2015 -- Lecture 22 403/20/2015
Formulation of Maxwell’s equations in terms of vector and scalar potentials
t
t
tt
AE
AE
AEBE
ABB
or
0 0
0
PHY 712 Spring 2015 -- Lecture 22 503/20/2015
Formulation of Maxwell’s equations in terms of vector and scalar potentials -- continued
41
:formequation General
1
/1
01 :require -- form gauge Lorentz
2
2
22
02
2
22
02
2
22
2
ftc
tc
tc
tc
LL
LL
LL
JAA
A
PHY 712 Spring 2015 -- Lecture 22 603/20/2015
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 2015 -- Lecture 22 703/20/2015
Electromagnetic waves from time harmonic sources
0,~,~ 0,,:condition continuity that theNote
,~, :densityCurrent
,~, :density Charge
rJrrJr
rJrJ
rr
itt
t
et
etti
ti
,~4
,~ or
,~4
1,~ For
,~, :source General
0
0
rr
rr
rr
i
ti
Jf
f
eftf
PHY 712 Spring 2015 -- Lecture 22 803/20/2015
,'~
'' ,~
,'~'1''
1''
,~,~
',''1''
1''
,,
'3
0
'3
0
3
0
tii
tif
ti
tif
ti
f
eferde
efc
ttdtrd
ee
tfc
ttdtrd
tt
c
rrr
r
rrrrr
rr
rrrrr
rr
rr
Electromagnetic waves from time harmonic sources – continued:
PHY 712 Spring 2015 -- Lecture 22 903/20/2015
Electromagnetic waves from time harmonic sources – continued:
,'~'
'4
1,~,~
) gauge, (Lorentz potentialscalar For
'3
00
rrr
rrrr
ikerd
ck
,'~'
'4
,~,~
) gauge, (Lorentz potential For vector
'30
0
rJrr
rArArr
ikerd
ck
PHY 712 Spring 2015 -- Lecture 22 1003/20/2015
Electromagnetic waves from time harmonic sources – continued:
:function Hankel Spherical
:function Bessel Spherical
'ˆˆ'4
:expansion Useful
*'
krinkrjkrhkrj
YYkrhkrjike
lll
l
lmlmllm
l
ik
rrrr
rr
'ˆ,'~',~
ˆ,~,~,~
*3
0
0
rr
rrr
lmlllm
lmlm
lm
Ykrhkrjrdikr
Yr
PHY 712 Spring 2015 -- Lecture 22 1103/20/2015
Electromagnetic waves from time harmonic sources – continued:
:function Hankel Spherical
:function Bessel Spherical
'ˆˆ'4
:expansion Useful
*'
krinkrjkrhkrj
YYkrhkrjike
lll
l
lmlmllm
l
ik
rrrr
rr
'ˆ,'~',~
ˆ,~,~,~
*30
0
rrJa
rarArA
lmlllm
lmlm
lm
Ykrhkrjrdikr
Yr
PHY 712 Spring 2015 -- Lecture 22 1203/20/2015
Forms of spherical Bessel and Hankel functions:
x
exx
iixhx
xxxx
xj
xe
xixh
xx
xxxj
ixexh
xxxj
ix
ix
ix
22232
121
00
331 cos3sin13
1 cossin
sin
x
eixhx
lxxjx
ixl
l
l
l
1 1
!!12 1
:behavior cAssymptoti
PHY 712 Spring 2015 -- Lecture 22 1303/20/2015
Electromagnetic waves from time harmonic sources – continued:
'ˆ,'~',~
ˆ,~,~,~
*3
0
0
rr
rrr
lmlllm
lmlm
lm
Ykrhkrjrdikr
Yr
'ˆ,'~',~
ˆ,~,~,~
*30
0
rrJa
rarArA
lmlllm
lmlm
lm
Ykrhkrjrdikr
Yr
'ˆ','~',~
'ˆ','~',~source) of(extent For
*30
*3
0
rrJa
rr
lmlllm
lmlllm
Ykrjrdkrhikr
Ykrjrdkrhikr
r
PHY 712 Spring 2015 -- Lecture 22 1403/20/2015
Electromagnetic waves from time harmonic sources – continued:
'ˆ','~',~
'ˆ','~',~source) of(extent For
*30
*3
0
rrJa
rr
lmlllm
lmlllm
Ykrjrdkrhikr
Ykrjrdkrhikr
r
'ˆ'','~'
'ˆ','~',~0,~,~ :condition continuity
the viaconnected are ,'~ and ,'~ that Note
*3
0
*3
0
rrJ
rr
rJr
rJr
lmll
lmlllm
Ykrjrdkrhk
Ykrjrdkrhikr
i
PHY 712 Spring 2015 -- Lecture 22 1503/20/2015
Electromagnetic waves from time harmonic sources – continued:
'ˆ,'~',~
'ˆ3
','~',~
:expansions in onscontributi trivial)-(nonLowest !!12
'' 1'
1
:ionsapproximat Various
00*3
000
1*3
01
1
rrJa
rr
Yrdkreiikr
Ykrrdkreikr
llkrkrjkr
kreikrhkr
ikr
m
ikr
m
l
l
ikrl
l
PHY 712 Spring 2015 -- Lecture 22 1603/20/2015
Electromagnetic waves from time harmonic sources – continued:
r
ekriωi
reωi
rdi
rdω
ikr
ikr
1ˆ4
,~4
,~
,~ 1,~
:frequency at moment dipole Define :radiation dipole on;contributiorder Lowest
0
0
33
rpr
prA
rJrrp
Note: in this case we have assumed a restricted extent of the source such that kr’<<1.
PHY 712 Spring 2015 -- Lecture 22 1703/20/2015
Electromagnetic waves from time harmonic sources – continued:
2
0
02
42
*
0
22
22
0
22
0
ˆˆ32
,~,~ˆ2
ˆˆ
:1for radiatedPower
11ˆ4
1
,~,~
1ˆˆ3ˆˆ
41
,~,~,~
rpr
rBrErrSr
pr
rArB
pprrrpr
rArrE
ωkc
rrddP
krikr
ωkr
ec
ikrr
ωωωkr
e
i
avg
ikr
ikr
PHY 712 Spring 2015 -- Lecture 22 1803/20/2015
Example of dipole radiation source
RrRr eRi
JeJ /0/0 cos,~ ˆ,~
rzrJ
''cos ,~
'' ˆ,~
10
1/'2
0
0
00
0/'2
00
krjkrhedrrRkJ
krjkrhedrrikJ
Rr
Rr
r
zrA
RkR
kri
rekJ
RkR
reJ
Rr
ikr
ikr
222
3
0
0
222
3
00
121 cos ,~
12ˆ,~
:for Evaluation
r
zrA
PHY 712 Spring 2015 -- Lecture 22 1903/20/2015
Example of dipole radiation source -- continued
RkR
kri
rekJ
RkR
reJ
Rr
ikr
ikr
222
3
0
0
222
3
00
121 cos ,~
12ˆ,~
:for Evaluation
r
zrA
Relationship to pure dipole approximation (exact when kR0)
r
ekriωi
reωi
iJRrd
irdω
ikr
ikr
1ˆ4
,~4
,~ :fields dipole ingCorrespond
ˆ8,~ 1,~
0
0
03
33
rpr
prA
zrJrrp