The time dependence of electrical disturbances in good conductors is governed by the frequency-dependent conductivity G.58). Consider longitudinal electric fields in a conductor, using Ohm's law, the continuity equation, and the differential form of Coulomb's law.

(a) Show that the time-Fourier-transformed charge density satisfies the equation

[σ (ω) – iωє₀]ρ(х, ω) = 0

(b) Using the representation σ(ω) = σ₀ / (l – iωτ), where σ₀ = є₀ω²_pτ and τ is a damping time, show that in the approximation ω_ρτ >> 1 any initial disturbance will oscillate with the plasma frequency and decay in amplitude with a decay constant Л = 1/2т. Note that if you use σ(ω) ≅ σ(0) = σ₀ in part a, you will find no oscillations and extremely rapid damping with the (wrong) decay constant λ_ω = σ₀/є₀.