Page 26 - Electrical Properties of Materials

Electromagnetic waves in solids 9

If the conductivity is large enough, the second term is the dominant one in
eqn (1.38) and we may write

±(i + 1) 1/2
~
1/2
k = (iωμσ) = √ (ωμσ) . (1.40)
2
So if we wish to know how rapidly an electromagnetic wave decays in a
good conductor, we may find out from this expression. Since
ωμσ
1/2
k imag = (1.41)
2
the amplitude of the electric field varies as

ωμσ
1/2
exp – z . (1.42)
|E x |= E x 0
2
The distance δ at which the amplitude decays to 1/e of its value at the
surface is called the skin depth and may be obtained from the equation
ωμσ
1/2
1= δ, (1.43)
2
yielding

1/2
2
δ = . (1.44)
ωμσ
You have seen this formula before. You need it often to work out the res-
istance of wires at high frequencies. I derived it solely to emphasize the major
steps that are common to all these calculations.
We can now go further, and instead of taking the constant σ, we shall look
a little more critically at the mechanism of conduction. We express the current
density in terms of velocity by the equation The symbol v still means the aver-
age velocity of electrons, but now
J = N e ev. (1.45) it may be a function of space and
time, whereas the notation v D is
This is really the same thing as eqn (1.7). The velocity of the electron is related generally restricted to d.c. phe-
to the electric and magnetic fields by the equation of motion nomena.

dv v
m + = e(E E E + v × B). (1.46) 1/τ is introduced again as a ‘vis-
dt τ cous’ or ‘damping’ term.
We are looking for linearized solutions leading to waves. In that approxim-
ation the quadratic term v × B can clearly be neglected and the total derivative
can be replaced by the partial derivative to yield

∂v v
m + = eE E E . (1.47)
∂t τ
Assuming again that the electric field is in the x-direction, eqn (1.47) tells
us that the electron velocity must be in the same direction. Using the rules set