The document outlines the plan for Lecture 24 of a physics course. It will cover electromagnetic waves due to specific sources like dipoles. It will also discuss dipole radiation patterns. The lecture will start by reading Chapter 9 of the textbook. The document then provides mathematical formulations of Maxwell's equations in terms of vector and scalar potentials. It also discusses electromagnetic waves from time-harmonic sources and gives solutions to Maxwell's equations in the Lorentz gauge.

1. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 1
PHY 712 Electrodynamics
10-10:50 AM MWF Olin 107
Plan for Lecture 20:
Start reading Chap. 9
A. Electromagnetic waves due to
specific sources
B. Dipole radiation patterns

2. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 2

3. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 3

4. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 4

5. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 5
0
0
2
0
2
0
1
0
:
monopoles
magnetic
No
0
:
law
s
Faraday'
1
:
law
s
Maxwell'
-
Ampere
/
:
law
s
Coulomb'
:
0)
0;
(
form
or vacuum
c
Microscopi



























c
t
t
c
B
B
E
J
E
B
E
M
P

6. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 6
Formulation of Maxwell’s equations in terms of vector and
scalar potentials
t
t
t
t






































A
E
A
E
A
E
B
E
A
B
B
or
0
0
0

7. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 7
Formulation of Maxwell’s equations in terms of vector and
scalar potentials -- continued
4
1
:
form
equation
General
1
/
1
0
1
:
require
-
-
form
gauge
Lorentz
2
2
2
2
0
2
2
2
2
0
2
2
2
2
2
f
t
c
t
c
t
c
t
c
L
L
L
L
L
L








































J
A
A
A

8. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 8
Solution of Maxwell’s equations in the Lorentz gauge -- continued
   
 
c
t
t
t
t
G /
'
'
'
1
'
,
'
;
, r
r
r
r
r
r 



 
 
   
 
'
,
'
'
1
'
'
1
'
'
,
,
:
,
field
for
Solution
3
0
t
f
c
t
t
dt
r
d
t
t
t
f
r
r
r
r
r
r
r
r




























9. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 9
Electromagnetic waves from time harmonic sources
   
 
   
 
        0
,
~
,
~
0
,
,
:
condition
continuity
that the
Note
,
~
,
:
density
Current
,
~
,
:
density
Charge





























r
J
r
r
J
r
r
J
r
J
r
r
i
t
t
t
e
t
e
t
t
i
t
i
   
 
   
   








 
,
~
4
,
~
or
,
~
4
1
,
~
For
,
~
,
:
source
General
0
0
r
r
r
r
r
r
i
t
i
J
f
f
e
f
t
f



 

10. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 10
   
 
   
 
   
,
'
~
'
'
,
~
,
'
~
'
1
'
'
1
'
'
,
~
,
~
'
,
'
'
1
'
'
1
'
'
,
,
'
3
0
'
3
0
3
0
t
i
i
t
i
f
t
i
t
i
f
t
i
f
e
f
e
r
d
e
e
f
c
t
t
dt
r
d
e
e
t
f
c
t
t
dt
r
d
t
t
c











































































r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
Electromagnetic waves from time harmonic sources –
continued:

11. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 11
Electromagnetic waves from time harmonic sources –
continued:
     
,
'
~
'
'
4
1
,
~
,
~
)
gauge,
(Lorentz
potential
scalar
For
'
3
0
0 





r
r
r
r
r
r
r








ik
e
r
d
c
k
     
,
'
~
'
'
4
,
~
,
~
)
gauge,
(Lorentz
potential
For vector
'
3
0
0 





r
J
r
r
r
A
r
A
r
r






ik
e
r
d
c
k

12. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 12
Electromagnetic waves from time harmonic sources –
continued:
       
 
     
:
function
Hankel
Spherical
:
function
Bessel
Spherical
'
ˆ
ˆ
'
4
:
expansion
Useful
*
'
kr
in
kr
j
kr
h
kr
j
Y
Y
kr
h
kr
j
ik
e
l
l
l
l
lm
lm
l
lm
l
ik







 r
r
r
r
r
r

       
         
'
ˆ
,
'
~
'
,
~
ˆ
,
~
,
~
,
~
*
3
0
0









r
r
r
r
r
lm
l
l
lm
lm
lm
lm
Y
kr
h
kr
j
r
d
ik
r
Y
r










13. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 13
Electromagnetic waves from time harmonic sources –
continued:
       
 
     
:
function
Hankel
Spherical
:
function
Bessel
Spherical
'
ˆ
ˆ
'
4
:
expansion
Useful
*
'
kr
in
kr
j
kr
h
kr
j
Y
Y
kr
h
kr
j
ik
e
l
l
l
l
lm
lm
l
lm
l
ik







 r
r
r
r
r
r

       
         
'
ˆ
,
'
~
'
,
~
ˆ
,
~
,
~
,
~
*
3
0
0







r
r
J
a
r
a
r
A
r
A
lm
l
l
lm
lm
lm
lm
Y
kr
h
kr
j
r
d
ik
r
Y
r







14. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 14
Electromagnetic waves from time harmonic sources –
continued:
       
         









'
ˆ
,
'
~
'
,
~
ˆ
,
~
,
~
,
~
*
3
0
0
r
r
r
r
r
lm
l
l
lm
lm
lm
lm
Y
kr
h
kr
j
r
d
ik
r
Y
r









       
         
'
ˆ
,
'
~
'
,
~
ˆ
,
~
,
~
,
~
*
3
0
0







r
r
J
a
r
a
r
A
r
A
lm
l
l
lm
lm
lm
lm
Y
kr
h
kr
j
r
d
ik
r
Y
r






         
         





'
ˆ
'
,
'
~
'
,
~
'
ˆ
'
,
'
~
'
,
~
source)
of
(extent
For
*
3
0
*
3
0
r
r
J
a
r
r
lm
l
l
lm
lm
l
l
lm
Y
kr
j
r
d
kr
h
ik
r
Y
kr
j
r
d
kr
h
ik
r
r









15. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 15
Electromagnetic waves from time harmonic sources –
continued:
         
         





'
ˆ
'
,
'
~
'
,
~
'
ˆ
'
,
'
~
'
,
~
source)
of
(extent
For
*
3
0
*
3
0
r
r
J
a
r
r
lm
l
l
lm
lm
l
l
lm
Y
kr
j
r
d
kr
h
ik
r
Y
kr
j
r
d
kr
h
ik
r
r








   
   
         
       
 












'
ˆ
'
'
,
'
~
'
'
ˆ
'
,
'
~
'
,
~
0
,
~
,
~
:
condition
continuity
the
via
connected
are
,
'
~
and
,
'
~
that
Note
*
3
0
*
3
0
r
r
J
r
r
r
J
r
r
J
r
lm
l
l
lm
l
l
lm
Y
kr
j
r
d
kr
h
k
Y
kr
j
r
d
kr
h
ik
r
i















16. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 16
Electromagnetic waves from time harmonic sources –
continued:
   
   
 
     
       























'
ˆ
,
'
~
'
,
~
'
ˆ
3
'
,
'
~
'
,
~
:
expansions
in
ons
contributi
trivial)
-
(non
Lowest
!
!
1
2
'
'
1
'
1
:
ions
approximat
Various
00
*
3
0
00
1
*
3
0
1
1
r
r
J
a
r
r
Y
r
d
kr
e
i
ik
r
Y
kr
r
d
kr
e
ik
r
l
l
kr
kr
j
kr
kr
e
i
kr
h
kr
ikr
m
ikr
m
l
l
ikr
l
l









17. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 17
Electromagnetic waves from time harmonic sources –
continued:
     
   
   
r
e
kr
i
ω
i
r
e
ω
i
r
d
i
r
d
ω
ikr
ikr















 

1
ˆ
4
,
~
4
,
~
,
~
1
,
~
:
frequency
at
moment
dipole
Define
:
radiation
dipole
on;
contributi
order
Lowest
0
0
3
3
r
p
r
p
r
A
r
J
r
r
p











Note: in this case we have only assumed the throughout
the extent of the source kr’<<1.

18. 04/19/2014 PHY 712 Spring 2014 -- Lecture 24 18
Electromagnetic waves from time harmonic sources –
continued:
     
 
 
   
     
   
 
 
   
 
 
  2
0
0
2
4
2
*
0
2
2
2
2
0
2
2
0
ˆ
ˆ
32
,
~
,
~
ˆ
2
ˆ
ˆ
:
1
for
radiated
Power
1
1
ˆ
4
1
,
~
,
~
1
ˆ
ˆ
3
ˆ
ˆ
4
1
,
~
,
~
,
~
r
p
r
r
B
r
E
r
r
S
r
p
r
r
A
r
B
p
p
r
r
r
p
r
r
A
r
r
E





































 









ω
k
c
r
r
d
dP
kr
ikr
ω
k
r
e
c
ikr
r
ω
ω
ω
k
r
e
i
avg
ikr
ikr













