Ohmic-conductor complex wavenumber ($k^2=\mu ϵ{\omega }^2+i\mu \sigma \omega$)

With fields proportional to $e^{i(kz-\omega t)}$, a homogeneous Ohmic medium obeys the displayed relation by combining the two curl Maxwell equations. The physical passive branch has $\mathrm{Im}k\ge 0$. For real positive conductivity much larger than $\omega ϵ$, it reduces to $k=(1+i)/\delta$ with skin depth $\delta =\sqrt{2/(\mu \sigma \omega )}$. Field amplitude and intensity then decay with respective lengths $\delta$ and $\delta /2$. A frequency-dependent conductivity must be retained when charge-carrier inertia matters.