The Conductivity of an Ionized Gas

The ionized gas has a conductivity σ such that
            Σ E = J_f                                     ... (1)

Now, Jf = ∑ NiQi (dx/dt);                                        ... (2)

Here, Ni = number of ions or electrons of a given type per m3,
    Qi = charge per ion or electron in coulombs,
(dx/dt)i = drift velocity along the x – axis = 1/ω Q/m E₀ sin ωt
J_f = convection current density.
Then
    σE₀ exp jωt = ∑ NiQ²i /ωm₁ E₀ exp j (ωt – π/2)                        ... (3)

and        σ = - j/ω ∑ NiQ²i                                     ... (4)

We retain only the term corresponding to the electrons.
    . :    σ = - jNeQ²e                                     ... (5)
Here Ne, is the number of free electrons per cubic meter.
So, the plasma conductivity σ_plasma becomes purely imaginary and negative.