 Carrier-current double counting in effective permittivity

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.