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 $\delta =\sqrt{2/(\mu \sigma \omega )}$.

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.

A coating substantially thinner than the skin depth does not screen the substrate as a bulk conductor would. Thickness of a few skin depths suppresses the field reaching the substrate exponentially. A unique required thickness additionally needs an allowed residual field or loss criterion.

In the normal skin-effect regime of a good conductor, $\delta =\sqrt{2/(\mu \sigma \omega )}$ and the surface resistance is $R_s=1/(\sigma \delta )=\sqrt{\mu \omega /(2\sigma )}$. For identical geometries, resistance ratios depend on the square root of magnetic permeability divided by conductivity.

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.