P1. Which equation can be used to
get the continuity equation from
Maxwell’s eqns?
a) 3 b) 4 c) 5 d) 6 e)7

J  
t
Maxwell’s equations in vacuum
E  0
 B  0
B
 E 
0
t
E
  B   0 0
0
t
To derive a wave equation, we have to differentiate
equations and then substitute into each other.
P2. What is the starting Maxwell’s Eqn (1-4) to get:
 2E
 E    
0
2
t
2
Wave equation in free space (vacuum)
“To Maxwell, mystery was revealed—
how light could move through space.
A change in E makes changing B,
but B makes E, and off they race!”
2

E
2
 E   
0
2
t
 2E
 E    
0
2
t
2
means:
 

 
x  E y ˆy  Ez z    
 2  2  2  Ex ˆ
y
z 
 x
or
2
2
2


 2  Ex ˆx  E y ˆy  Ez z 
t
2
2
2

Ex  x, y, z 
 2



0
 2  2  2  Ex ( x, y, z)    
2
y
z 
t
 x
2
2
2

E y  x, y, z 
 2



0
 2  2  2  E y ( x, y, z)    
2
y
z 
t
 x
2
2
2

Ez  x, y, z 
 2



0
 2  2  2  Ez ( x, y, z)    
2
y
z 
t
 x
a 3-D wave equation for each component!
2
0
Nature of wave equation
2
1  2 f ( x, t)
f ( x, t)  2
0
2
2
x
c
t
P2. Sketch f(t) at
points x1 and x2.
Get the signs of first
and second
derivative right
a) did it
b) tried
f(x, t) = F(x - c t) + G(x + c t)
What does this mean mathematically?
Wave equation
P3.
2
2

1

f ( x, t)
f ( x, t)  f ( x  ct) is a solution to
f
(
x
,
t
)

0
2
2
2
x
c
t
It is also a solution to ____:
1)
3
1  3 f ( x, t)
x
2)
a)
b)
c)
d)
3
f ( x, t) 
c
3
t
3
0
4
1  4 f ( x, t)
f ( x, t)  4
0
4
4
x
c
t
eqn 1
eqn 2
both
neither
With light in a material, we add “source terms” on right.

E 
0
 B  0
B
 E 
0
t
E
  B   0 0
 0 J
t
Electrons bound in atoms form dipoles in a “dielectric”
(insulator)
Dipole from two equal charges +q, -q.
Dipole of a polarized atom
patom (t)  e  r  (r , t)d r  ermicro
2
3
Electrons bound in atoms form dipoles in a “dielectric”
(insulator)
Need to define a useful volume for calculating how atoms
influence light fields: it must be large compared to
___________, and small compared to ______________
All these atoms
get replaced by
So E, B, J , , P in our theory will be continuous, smooth functions in
space…no atomic-scale features
Polarization P of a material with many atoms or molecules:
1
P
V
N
N
pi 
p  qrmicro if all identical

V
V
dipoles in small V
N
J  qvmicro
V
P will oscillate when light goes through a material!
P

t
So
E
  B  0 J P   0 0
t
P
E
  B  0
  0 0
t
t
Oscillating P is a
current, and a direct
source of B
Can P provide a charge density, and be a direct source of E?

E 
0
Yes, if
E 
 P
  P
  JP 

t
t
  P  P
P  0
  P
0
If the polarization grows to
the right, it acts like a
steadily increasing net
charge density
e.g. P(r )  Po x2 xˆ
Which is hardest to model (most likely to have) charge   P (r )?
Glass
For light (l>>d_atom), amorphous materials are more
uniform than crystals, because of directional
differences in crystals
So in the most common optical materials (glass, plastics,
water, air…all amorphous):
E  0
 B  0
B
 E 
0
t
E
P
  B   0 0
 0
t
t
Review
What does
b.
c.
d.
e.
 mean?
E 
0
E field at a point spreads out due to charge nearby
E fields near a point spread out due to charge there
The E field’s change at a point is due to charge there
The E field’s change at a point is due to charge nearby
Review
What does “displacement current” at some point represent?
a.
b.
c.
d.
charges building up at that point
charges passing that point
electric field changing at that point
magnetic field changing at that point