Skin effect

Definition

Let a homogeneous conductor fill the half-space $z > 0$. Let it have conductivity $\sigma > 0$, permeability $\mu$ and permittivity $\varepsilon$, and obey Ohm's law $\mathbf{J} = \sigma \mathbf{E}$. Consider time-harmonic fields with angular frequency $\omega > 0$ and time factor $e^{-i\omega t}$, in the good-conductor regime $\sigma \gg \omega \varepsilon$. The skin depth is $$\delta = \sqrt{\frac{2}{\omega \mu \sigma}}.$$ In this regime Maxwell's equations reduce to $\nabla^2 \mathbf{E} = -i \omega \mu \sigma \mathbf{E}$. The skin effect is the form of its bounded solutions that depend only on $z$: $$\mathbf{E}(z) = \mathbf{E}_0 \, e^{i z/\delta} \, e^{-z/\delta}.$$ So the field and the current density $\mathbf{J} = \sigma \mathbf{E}$ fall by the factor $e$ over each skin depth into the conductor.