Perfect conductor

At the surface of a perfect conductor, the tangential electric field and normal magnetic field vanish. An incident plane electromagnetic wave is therefore accompanied by a reflected wave whose tangential electric field cancels the incident field at the surface.

The linear carrier acceleration equation $\dot{\mathbf{j}}={ϵ}_0{\omega }_p^2\mathbf{E}$ and Faraday's law imply this conserved local quantity. Setting its initial value to zero gives the second London equation and London screening. An undamped response alone does not select that initial value, so it must not be used to infer expulsion of a pre-existing magnetic field for every magnetic history.^[ref]

If a metal surface is $z=h(x)$ and the incident field is independent of $y$, Maxwell's equations separate into TE polarization and TM polarization. The metal condition that the tangential electric field vanish sets $E_y=0$ in the TE sector. In the TM sector $\mathbf{E}=i\nabla \times (H_y\hat{\mathbf{y}})/(\omega ϵ)$, so the same condition gives $({\partial }_z-h^{\prime }{\partial }_x)H_y=0$. This explains why the TE problem agrees with scalar Dirichlet acoustics while the TM problem has different rough-surface boundary forcing.

At normal incidence, the reflected electric amplitude is the negative of the incident amplitude at the surface, while the reflected magnetic amplitude has the same sign. Their superposition forms a standing wave with an electric node and a magnetic antinode at the conductor.