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
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
