Skin effect

An alternating electric current and its electromagnetic field concentrate near a conductor's surface. For a local isotropic good conductor with real electrical conductivity $\sigma$, magnetic permeability $\mu$, and $\sigma \gg \omega ϵ$, the field penetrates over the skin depth $\delta =\sqrt{2/(\mu \sigma \omega )}$.

For a wave penetrating a nonmagnetic good conductor in the $+z$ direction with time convention $e^{-i\omega t}$, its wavenumber is $k=(1+i)/\delta$, with skin depth $\delta =\sqrt{2/({\mu }_0\sigma \omega )}$. If $\mathbf{E}\parallel \hat{\mathbf{x}}$, Faraday's law of induction gives $\mathbf{H}=[k/({\mu }_0\omega )]E_x\hat{\mathbf{y}}$. Thus the field ratio is $Z_s=E_x/H_y=(1-i)/(\sigma \delta )$, equal to the displayed expression. Magnetic and electric fields have equal exponential penetration lengths but a phasor phase difference $\pi /4$; $H_y$ has the positive phase relative to $E_x$ in this convention.

The skin depth is the distance over which a field amplitude falls by $e^{-1}$ inside a conductor. In the local good-conductor limit, $\delta =\sqrt{2/(\mu \sigma \omega )}$. Intensity then falls by $e^{-2}$ over one skin depth; amplitude and intensity penetration lengths differ by a factor two.