Solutions to Chen's Plasma Physics

We also have Gauss’s law:
∇ · E = − n ee
ɛ 0
⇒ k · E = i n ee
ɛ 0
= i m e ω2 p (86)
We will dot equation (84) with k:
(iωm − mν)
k · E = k · v
e
(87)
Plugging in equations (86) and (87), we obtain
i m e ω2 p = (iω2 m − mων)
e
⇒ ω 2 p = ω 2 + ων ✷ (88)
So we see that if we include collisions, the oscillation frequency is different from the plasma frequency.
b) Lets let ω = ω R + iω I . Then expression (88) becomes
ω 2 p = ω 2 R − ω 2 I + 2iω R ω I + iω I ν + ω R ν (89)
This means that
2ω R ω I + ω I ν = 0 ⇒ ω I = − ν 2 ✷ (90)
Now we suppose a plane wave solution for the field quantities, i.e.
we obtain
E ∝ e −iω Rt+ω I t
Thus, the wave is exponentially attenuated in time.
E ∝ e −iωt (91)
⇒
E ∝ e −iω Rt e − νt
2 ✷ (92)