An electromagnetic wave of frequency `omega` propagates in dilute plasma. The free electron concentration in plasma is equal to `n_(0)`. Neglecting the interaction of the wave and plasma ions, find:
(a) the frequency dependence of plasma permittivity,
(b) how the phase velocity of the electromagnetic wave depends on its wavelength `lambda in plasma.

Explanation

It turns out that one can neglect the spatial dependence of the electric field as well as the magnetic field. Thus for a typical electron
`m overset rarr(r ) = eoversetrarr(E_(0)) sin omegat`
so `oversetrarr(r ) =- (eoversetrarr(E_(0)))/(m omega^(2)) sin omega t` (neglecting any nonsinusoidal part)
The ions will be practially unaffected. Then
`oversetrarr(P) = n_(0) eoversetrarr(r) =- (n_(0)e^(2))/(m omega^(2)) oversetrarr(E)`
and `oversetrarr(D) = epsilon_(0) oversetrarr(E) + oversetrarr(P) = epsilon_(0) (1- (n_(0)e^(2))/(epsilon_(0)m omega^(2)))oversetrarr(E)`
Hence the permittivity `epsilon = 1- (n_(0)e^(2))/(epsilon_(0)m omega^(2))`.
(b) The phase velocity is given by
`v = omega//K = (c)/(sqrt(epsilon))`
So `ck = omega sqrt(1- (omega_(P)^(2))/(omega^(2)), omega_(P)^(2) = (n_(0)e^(2))/(epsilon_(0)m)`
`omega^(2) = c^(2)k^(2) + omega_(P)^(2)`
Thus `v = c sqrt(1- (omega_(P)^(2))/(c^(2)k^(2))) = c sqrt(1+((n_(0)e^(2))/(4pi^(2)mc^(2)epsilon_(0))))lambda^(2)`