Jackson 7.4 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
UMASS LOWELL
C.S.BAIRD
PROBLEM: A plane-polarized electromagnetic wave of frequency w in free space is incident normally on the flat surface of a non-permeable medium of conductivity o and dielectric constant e.
(a) Calculate the amplitude and phase of the reflected wave relative to the incident wave for arbitrary o and e.
(b) Discuss the limiting case of a very poor and a very good conductor, and show that for a good conductor the reflection coefficient (ratio of reflected to incident intensity) is approximately
R~1-2"g c
where is the skin depth.
SOLUTION: (a) If we set up the problem in the usual way with the incident wave E traveling in the positive z direction, the transmitted wave E' traveling in the positive z direction, and the reflected wave E" traveling in the negative z direction, we get from the boundary conditions:
E,'=E.+E."
k'E.'=kE.-kE."
We can use both of these to solve for the reflected wave in terms of the incident wave:
k
k-k' k+k'
c
The wave number in the other material obeys: k '=Ve(w) o w
For a conducting material we can use e(w)=e+i - so that we have 3
toa
Plugging these in above leads to:
E
E.
Eo
m3
The problem asks us to explicitly expand this in terms of amplitude and phase. The amplitude of a complex number is defined as |z|= V z * z so that we have
E
E VEO
Eo
E
m3
m3
E E
m3
m3
g
E
Eo
m 3
Ec
E
The phase of a complex number is Arg(z)=tan-1
(b) Discuss the limiting case of a very poor and a very good conductor, and show that for a good conductor the reflection coefficient (ratio of reflected to incident intensity) is approximately
R~1-2"s
where is the skin depth.
For a very poor conductor o << &o which means the same as o/2o << 1. This leads to:
E. E.
Eo E +2 VE
E
E.
Now recognize:
2
=ly/x so that
(
Jvx+iy= 2
We must be careful because the arctan leads to some ambiguity of what quadrant the phase should be in. We know from physical arguments that the phase difference should be for a perfect dielectric, so that our answer should reduce to 1 for o->0. The answer must be in the third quadrant:
9
Arg
Eo
For a very good conductor o >> &oW
V2 m3
m 03
1+ m3
E." which leads to E.
after dropping small terms m 3
E.
0