= Carrier-current double counting in effective permittivity
{title2=$\varepsilon_{\rm eff}=\varepsilon+i\sigma/\omega$}
A carrier response must appear once in an electromagnetic constitutive model. With harmonic convention $e^{-i\omega t}$, absolute <permittivity> $\varepsilon$ describing the remaining polarization and an explicit conduction <electric current density> $\mathbf J_c=\sigma\mathbf E$, the <Ampère-Maxwell equation> contains
$$
\mathbf J_c+\partial_t\mathbf D=(\sigma-i\omega\varepsilon)\mathbf E=-i\omega\left(\varepsilon+\frac{i\sigma}{\omega}\right)\mathbf E.
$$
Thus an alternative description incorporates that carrier contribution into the <complex permittivity> $\varepsilon_{\rm eff}=\varepsilon+i\sigma/\omega$ 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 $i\sigma/(\varepsilon_0\omega)$; 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.
Back to article page
