In this section we consider wave propagation in an ionised plasma comprising ne electrons and rti ions per unit volume. In a dilute plasma we can ignore the (typically distant) ions as they provide little damping to the motion of the electrons, and because of their much higher mass, they contribute little to the current density.
We start by deriving the electrical conductivity from the motion of individual electrons which undergo acceleration in response to the electric field of the wave
me (dv/dt) = -e E(t)
The current density arises from the bulk flow of electrons in the plasma
J (t) = ne (-e) v (t)
And so, dJ/dt = ne (e2/me) E (t)
From this expression, we can work out the conductivity of an oscillating electric field E0e-iwt
We see that the conductivity is purely imaginary,
σ = i (ne e2/mew)
so that the current density and electric field are 90° out of phase. Consequently, the time-averaged power will be zero. This is analogous to the current and voltage in an LC circuit where energy is continually being exchanged between electric field energy of the capacitor and magnetic field energy of the inductor.