Transcript 471/Lectures/notes/lecture 1 - Maxwell's eqns.pptx

What is optics?
Mathematical and physical background of:
 E/M waves in materials and at boundaries
 Control and description of polarization
 Waves: interference, diffraction, laser beams,
coherence
 photons and quantum optics
 rays, imaging
Divergence and curl
P1. Can field lines that stay parallel
(e.g. point only in the z direction)
have divergence?
a) yes
b) no
P2. Can field lines that stay parallel
(e.g. point only in the z direction)
have curl?
a) yes
b) no
Divergence and curl
Divergence
Curl
Know physical interpretations in terms of fluid flow!
Figures from Griffiths, 441/442 text
Light is an E/M wave, obeying…
Maxwell’s equations
Derivative form
Integral form
Narration of Maxwell’s equation

E 
0
 B  0
B
 E  
t
One source of E:
The part of E with div.
is due to charge right there
B never has div.
The other source of E:
The part of E that curls is due to
B change
E
  B  0 J   0 0
t
The sources of B are
physical currents and
change in E (displacement
current). It circles those
vectors, but has curl only E
right where there is J or t
Participation record with clicker:
a) I got it right (at most a sign or constant off) (10 pts)
b) I tried (9 pts)
Suppose you want to have the following E-field in this room
4
4
4 3
ˆ
E  z ( x  y  z )t
P1. Find the charge density
( x, y, z, t) that must be present
Could this charge density be consistent with a different E?
Participation record with clicker:
a) I got it right (at most a sign or constant off) (10 pts)
b) I tried (9 pts)
E  zˆ ( x  y  z )t
4
4
4
P3. Find
 E
P4. Find a
B ( x, y , z , t )
3
consistent with
 E
P5. Find the direction of B at the point (x,y,z) = (1,0,0)
B circles the room
a) CW viewed from above
b) CCW viewed from above
What can we say about the nature of the current density J?
Continuity equation
The part of J with divergence comes
from a changing charge density.

J  
t
The part of J with curl represents “divergence free
currents”, and can be supplied by neutral wires.
E  zˆ ( x 4  y 4  z 4 )t 3 (desired E-field)


  E  xˆ (4 y 3 )  yˆ (4 x 3 ) t 3
  E  4 z 3t 3 
 (t )
o
B  0
B
   E    xˆ (4 y 3 )  yˆ (4 x 3 )  t 3
t


B   xˆ ( y 3 )  yˆ ( x 3 ) t 4  f ( x, y, z )(optional )
J
 B
o
 o

zˆ3  x
E
t

t
  B  zˆ3 x 2  y 2 t 4
J
1
o
2
 y2
4
  o zˆ ( x 4  y 4  z 4 )3t 2
Of the 5 terms, only the last has divergence, and comes
from /t, so the first 4 terms have to be supplied by
wires to get the E field we want.