Transverse wave cutoff in a cold unmagnetized plasma

With immobile ions, negligible collisions and electron density $n_e$, the plasma frequency is ${\omega }_p^2=n_e{e}^2/(m_e{ϵ}_0)$. Combining inertial electron current with vacuum displacement current gives ${\omega }^2={\omega }_p^2+c^2{k}^2$. Transverse fields are evanescent below ${\omega }_p$ and propagate above it.

A carrier response must appear once in an electromagnetic constitutive model. With harmonic convention $e^{-i\omega t}$, absolute permittivity $\epsilon$ describing the remaining polarization and an explicit conduction electric current density ${\mathbf{J}}_c=\sigma \mathbf{E}$, the Ampère-Maxwell equation containsThus an alternative description incorporates that carrier contribution into the complex permittivity ${\epsilon }_{\mathrm{eff}}=\epsilon +i\sigma /\omega$ and omits the explicit ${\mathbf{J}}_c$. Using this effective quantity and adding the same conduction current again doubles that response. For relative permittivity, the added term is $i\sigma /({\epsilon }_0\omega )$; changing the harmonic sign convention changes the displayed sign. Stating which charges are included in each term prevents a mathematical rearrangement from becoming an extra physical mechanism.