This document contains notes from a PHY 712 Electrodynamics lecture. It begins with the schedule for upcoming student presentations. It then reviews Maxwell's equations in SI units and various coordinate systems. It discusses the scalar and vector potential formulations and solution methods such as Green's functions. It also covers time-harmonic sources and the use of spherical Bessel and Hankel functions in expansions of the potentials and fields.

1. 1
PHY 712 Electrodynamics
10-10:50 AM MWF Olin 107
Plan for Lecture 35:
Comments and problem solving advice:
 Comment about PHY 712 final
 General review
04/25/2014 PHY 712 Spring 2014 -- Lecture 35

2. 2
04/25/2014 PHY 712 Spring 2014 -- Lecture 35

3. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 3

4. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 4
Time Presenter
Name
Presenter Title
10-10:20
AM
Sam Flynn “Group Theory and
Electromagnetism”
10:25-
10:40 AM
Ahmad ???????????????????????
??
Time Presenter
Name
Presenter Title
9:30-9:50
AM
Calvin Arter “Electrodynamics and the
interaction potential”
9:55-
10:20 AM
Ryan Melvin “Effects of electric fields on
small strands of human
RNA”
10:25-
10:40 AM
Drew Onken “The Electromagnetic Theory
Behind the Free Electron
Laser”
Monday
4/28/2014
Wednesday
4/30/2014

5. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 5
Final exam for PHY 712
 Available: Friday, May 2, 2014
 Due: Monday, May 12, 2014

6. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 6
0
0
2
SI units; Microscopic or vacuum form ( 0; 0):
Coulomb's law: /
1
Ampere-Maxwell's law:
Faraday's law: 0
No magnetic monopoles:
c t
t
 

 
 

  


  

P M
E
E
B J
B
E
2
0 0
0
1
c
 
 
 
B

7. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 7
0
0
0
1
SI units; Macroscopic form ( 0; = ):
Coulomb's law:
Ampere-Maxwell's law:
Faraday's law: 0
No magnetic monopol
free
free
t
t




   
 

  


  

D E P H B M
D
D
H J
B
E
es: 0
 
B

8. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 8
Gaussian units; Macroscopic form ( 4 0; = 4 ):
Coulomb's law: 4
1 4
Ampere-Maxwell's law:
1
Faraday's law: 0
No magn
free
free
c t c
c t
 



   
 

  


  

D E P H B M
D
D
H J
B
E
etic monopoles: 0
 
B

9. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 9
Energy and power (SI units)
 
1
Electromagnetic energy density:
2
Poynting vector:
u    
 
E D H B
S E H
   
t
i
t
i
t
i
)e
,
(
)e
,
(
)e
,
(
,t)
( 




 r
E
r
E
r
E
r
E *
~
~
2
1
~
:
fields
harmonic
for time
Equations



 

   
 
*
avg
1
2
t
,t ( , ) ( , )
 
  
S r E r H r
   
 
* *
avg
1
4
t
u ,t ( , ) ( , ) ( , ) ( , )
   

 


r E r D B r
H
r r

10. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 10
0 0
2
Solution of Maxwell's equations:
1
/
0 0
c t
t
  

    


    

E
E B J
B
E B
Introduction of vector and scalar potentials:
0
0 0
or
t t
t t
    
 
 
      
 
 
 
 
     
 
B B A
B A
E E
A A
E E

11. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 11
 
 
 
0
2
0
0
2
2
0
2 2
Scalar and vector potentials continued:
/ :
/
1
1
t
c t
c t t
 
 


 
 
   


  

 
 

    
 
 
 
E
A
E
B J
A
A J

12. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 12
 
   
J
A
A
A
J
A
A
A
0
2
2
2
2
0
2
2
2
2
2
0
2
2
2
0
2
1
/
1
0
1
require
-
-
form
gauge
Lorentz
1
/
:
equations
potential
vector
and
scalar
the
of
Analysis

























































t
c
t
c
t
c
t
t
c
t
L
L
L
L
L
L

13. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 13
2
2
0
2 2
2
2
0
2 2
Solution methods for scalar and vector potentials
and their electrostatic and magnetostatic analogs:
1
/
1
L
L
L
L
c t
c t
 

 
   


  

A
A J
In your “bag” of tricks:
 Direct (analytic or numerical) solution of
differential equations
 Solution by expanding in appropriate
orthogonal functions
 Green’s function techniques

14. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 14
How to choose most effective solution method --
 In general, Green’s functions methods work well when
source is contained in a finite region of space
2
2
3
0
0
3
2
( , ) 4 ( )
1
( )
Con
( ) ( , )
4
1
ˆ
( , ) (
sider the electrostatic problem:
/
Define:
) ( ) ( , ) .
4
' '
L V
S
L
G
d r G
d r G G








  
       

  
  
 
     
  

 



r r
r r r r
r r r r r r r
r r

15. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 15
   
 




 lm
*
lm
lm
l
l
,φ
θ
Y
θ,φ
Y
r
r
l
'
'
1
2
4
'
1
1

r
r
( ) is contained in a small
1
region of
For electrostat
space a
ic problems
nd , ( , )
'
where
S G


  

r r
r r
r

16. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 16
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
'
( )and ( ) are
contained in a small region of space and
For dynamic problems
,
wh
( , ', )
e , ,
'
re
i
c
S
e
G

  


 


r r
J
r
r r
r
r r

17. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 17
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

18. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 18
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










19. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 19
Model of dielectric properties of matter:
 
 


















i
i
i
t
i
t
i
t
i
e
i
m
q
q
i
m
q
e
m
m
e
q
m
r
r
p
P
P
E
E
D
E
r
p
E
r
r
r
r
r
E
r
3
0
2
2
0
0
2
2
2
0
0
0
0
2
0
0
:
field
nt
Displaceme
1
:
dipole
Induced
1
,
For























http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png
Drude model
Vibrations of charged particles near equilibrium:
r

20. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 20
r
r
E
r 

 2
0
0 


 
m
m
e
q
m t
i


 
Drude model:
Vibration of particle of charge q and mass m near
equilibrium:
r http://img.tfd.com/ggse/d6/gsed_0001_0012_0_img2972.png
 
dipoles
type
of
fraction
ume
dipole/vol
number
:
field
nt
Displaceme
3
0
i
f
N
f
N
i
i
i
i
i
i
i









 p
r
r
p
P
P
E
E
D




21. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 21
Drude model dielectric function:
 
   
 
 
 
 

















i i
i
i
i
i
i
I
i i
i
i
i
i
i
R
I
R
i i
i
i
i
i
m
q
f
N
m
q
f
N
i
i
m
q
f
N
2
2
2
2
2
0
2
0
2
2
2
2
2
2
2
0
2
0
0
0
2
2
0
2
0
1
1
1

































22. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 22
Kramers-Kronig transform – for use in dielectric analysis
 
z-α
f(z)
dz
-α
z
)
f(z
dz
πi
z-α
f(z)
dz
πi
f
rest
R
R
R
includes










 





2
1
2
1


Re(z)
Im(z)

=0
  f
-α
z
)
f(z
dz
P
πi
-α
z
)
f(z
dz
πi
f
R
R
R
R
R
R )
(
2
1
2
1
2
1

 

 








23. 04/25/2014 PHY 712 Spring 2014 -- Lecture 35 23
Kramers-Kronig transform – for dielectric function:
   
   
       




























I
I
R
R
R
I
I
R
-
d
P
-
d
P


























;
with
'
1
1
'
'
1
'
1
'
'
1
1
0
0
0
0
Further comments on analytic behavior of dielectric function
       
   















0
0
0
0
1
,
,
,
:
fields
and
between
ip
relationsh
Causal"
"










i
e
G
d
t
G
d
t
t r
E
r
E
r
D
D
E