= Solution
In a homogeneous Ohmic conductor, $\mathbf J=\sigma\mathbf E$. Combining this with <Maxwell's equations> gives
$$
\nabla^2\mathbf E=\mu\sigma\,\partial_t\mathbf E+\mu\epsilon\,\partial_t^2\mathbf E.
$$
For a good conductor, $\sigma\gg\omega\epsilon$, so conduction dominates <displacement current>. With time dependence $e^{-i\omega t}$ and depth $z>0$, take $E\propto e^{ikz}$. Then
$$
k^2=i\mu\sigma\omega,\qquad k=\frac{1+i}{\delta},
\qquad
\boxed{\delta=\sqrt{\frac{2}{\mu\sigma\omega}}.}
$$
The field is consequently proportional to $e^{-z/\delta}e^{iz/\delta}$: it both attenuates and changes phase with depth. The <skin depth> is the distance for amplitude to fall by $e^{-1}$; <power> density falls as $e^{-2z/\delta}$, so its attenuation length is $\delta/2$.
Physically, changing fields induce currents whose <magnetic field> opposes penetration. Equivalently, after neglecting <displacement current>, <magnetic diffusion> obeys $\partial_t\mathbf B=(\mu\sigma)^{-1}\nabla^2\mathbf B$. Faster changes have less time to diffuse into the metal. Increasing <frequency>, conductivity or <magnetic permeability> reduces $\delta$, provided the relevant material parameters remain applicable.
For a thick conductor the tangential surface <electrical impedance> is $Z_s=(1-i)/(\sigma\delta)$ with the stated time convention and consistent field orientation. Its real part $1/(\sigma\delta)$ represents dissipative resistance per square. Current crowding near the surface raises the AC resistance of a wire when $\delta$ is small compared with its radius; thin strands and laminated magnetic cores reduce suitable eddy-current losses. When $\sigma\not\gg\omega\epsilon$, the full complex propagation constant must replace the good-conductor approximation. In a magnetic metal the <magnetic permeability> can itself be nonlinear and <frequency> dependent.
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