Poynting flux and local heating in a skin layer ($⟨S_z⟩=\sigma \delta E_0^2{e}^{-2z/\delta }/4$)

For a good conductor with surface electric-field peak amplitude $E_0$, the surface impedance of a good conductor gives the displayed time-averaged Poynting vector component into the conductor. The factor one half in $⟨\mathbf{S}⟩=\mathrm{Re}(\mathbf{E}\times {\mathbf{H}}^{\ast })/2$ is required for peak phasors. Its loss with depth obeys $-d⟨S_z⟩/dz=\sigma E_0^2{e}^{-2z/\delta }/2=⟨\mathbf{j}\cdot \mathbf{E}⟩$. Electromagnetic energy becomes Joule heating locally; averaged field-energy storage has zero time derivative.