Penetration of the electric and magnetic velocity fields of a nonrelativistic point charge into a conducting plane

The electric and magnetic fields of a point charge moving with constant velocity outside and parallel to a conducting wall of finite conductivity are studied within classical electrodynamics for a nonrelativistic particle and a good conductor. Results are obtained through first order in the particle velocity for a wall with permeability $\mu =1\mathrm{}$, dielectric constant $\epsilon =1\mathrm{}$, and resistivity $\eta$. Calculations show the following: (i) There is no skin-depth behavior for the fields $\overrightarrow{E}$ or $\overrightarrow{B}$ or for the currents $\overrightarrow{\mathrm{J}}$ inside the conductor. (ii) The fields $\overrightarrow{B}$ and $\overrightarrow{E}$ fall off with distance as $r^{-2\mathrm{}}$ and $r^{-3\mathrm{}}$, respectively, inside the conductor. (iii) The magnetic field $\overrightarrow{B}$ and the current density $\overrightarrow{\mathrm{J}}$ inside the conductor are independent of the resistivity $\eta$, although the electric field $\overrightarrow{E}$ inside the conductor is suppressed by a factor of $\eta$. (iv) In the limit of many point charges moving so as to form a steady current outside the conductor, the magnetic field $\overrightarrow{B}$ penetrates the conductor as though it were not present while the electric field $\overrightarrow{E}$ and current density $\overrightarrow{\mathrm{J}}$ vanish inside the conductor. The results obtained here are quite different from those of the familiar calculations involving radiation fields where the penetration depth and the size of the fields inside the conductor are governed by the resistivity. The new results run contrary to the expectations of some physicists and contradict some earlier work in the literature. The calculations arose in connection with the Aharonov-Bohm effect where electrons moving with approximately constant velocity pass very close to a conducting solenoid.