Page 27 - Electrical Properties of Materials

10 The electron as a particle

out in eqn (1.32) we get the following algebraic equation

1
mv x –iω + = eE x . (1.48)
τ
The current density is then also in the x-direction:

J x = N e ev x
2
N e e τ 1
= E x
m 1–iωτ
σ
= E x , (1.49)
1–iωτ
where σ is defined as before. You may notice now that the only difference
from our previous (J – E ) relationship is a factor (1 – iωτ) in the denominator.
Accordingly, the whole derivation leading to the expression of k in eqn (1.38)
remains valid if σ is replaced by σ/(1 – iωτ). We get

σ 1/2
2
k = ω μ +iωμ
1–iωτ
1/2
iσ
1/2
= ω(μ ) 1+ . (1.50)
ω (1 – iωτ
If ωτ 1, we are back where we started from, but what happens when
ωτ 1? Could that happen at all? Yes, it can happen if the signal frequency
is high enough or the collision time is long enough. Then, unity is negligible
in comparison with iωτ in eqn (1.50), leading to
σ
1/2
1/2
k = ω(μ ) 1– . (1.51)
2
ω τ
Introducing the new notation
2
N e e 2 (N e e /m)τ σ
2
ω ≡ = = (1.52)
p
m τ τ
Equation (1.53) suggests a gen- we get
eralization of the concept of the
dielectric constant. We may intro- ω 2 1/2
duce an effective relative dielectric k = ω(μ ) 1/2 1– p 2 . (1.53)
constant by the relationship ω
ω 2 p Hence, as long as ω> ω p , the wavenumber is real. If it is real, it has (by the
ε eff =1 – . rules of the game) no imaginary component; so the wave is not attenuated. This
ω 2
is quite unexpected. By introducing a slight modification into our model, we
may come to radically different conclusions. Assuming previously J = σE ,we
It may now be seen that, depend-
ing on frequency, ε eff may be pos- worked out that if any electrons are present at all, the wave is bound to decay.
itive or negative. Now we are saying that for sufficiently large ωτ an electromagnetic wave may
travel across our conductor without attenuation. Is this possible? It seems to
contradict the empirical fact that radio waves cannot penetrate metals. True;