Skin depth and conducting boundaries

Take a conductor occupying positive z, with no internal impressed source and a tangential electric field. Use the convention $e^{ikz-i\omega t}$. Let conductivity $\sigma$ be real and positive and permeability $\mu$ approximately real and positive at the driving frequency. Maxwell equations give a squared wavenumber with an imaginary conduction term. When $\sigma\gg\omega\epsilon$, where $\epsilon$ denotes the approximately real background permittivity, that term dominates and selects equal positive real and imaginary wavenumber components.

The decaying solution has a phase that advances into the conductor while its amplitude falls. Its imaginary wavenumber defines the amplitude skin depth. Increasing frequency, conductivity or permeability decreases that depth in this regime. The conclusion about frequency depends on treating the material parameters as approximately constant: a frequency-dependent Drude conductivity or magnetic resonance can invalidate simple inverse-square-root extrapolation. Neither a dc conductivity table nor an ideal-conductor argument proves this approximation remains valid at optical frequencies.

Faraday's law fixes the magnetic amplitude accompanying the penetrating electric field. Their ratio is a complex surface impedance. With the chosen convention its imaginary part is negative; changing the time convention changes that sign while leaving the positive resistive part and heating unchanged. Declare the field directions when forming this ratio: take electric field along x, magnetic field along y and energy flow into positive z. This removes an otherwise easy normal-vector sign ambiguity.

The real part of surface impedance gives the cycle-averaged power entering and dissipating per unit area for a prescribed peak tangential magnetic amplitude. A perfect conductor is the limiting boundary with vanishing surface impedance, not an ordinary material of literally infinite microscopic acceleration. For large but finite conductivity, a small tangential electric field remains and produces a finite surface loss. Replacing that field by exactly zero discards the very term needed to estimate heating or cavity damping.

The intensity of a single penetrating wave decreases as the square of its amplitude, so its intensity decay length is half the skin depth defined here. A sheet only a small fraction of a skin depth thick requires fields on both sides and cannot be treated as a semi-infinite absorber. Curved surfaces require the skin depth to be small compared with the relevant curvature scale before the planar local boundary approximation becomes reliable.

Local conductivity can also fail when the carrier mean free path is comparable to the penetration scale, producing an anomalous skin regime. Superconductors, strongly magnetic media and rapidly varying conductivity require their own constitutive treatment. Magnetic diffusion offers a useful connection within the ordinary quasistatic regime: neglecting displacement current gives a diffusion equation for magnetic field. The heat lab illustrates exact decay of periodic diffusion modes, but it does not solve the driven half-space boundary problem that defines skin depth.