Effects of bulk and surface conductivity on the potential developed by dielectrics exposed to electron beams

The charging and discharging of a dielectric material which has bulk and surface conductivities is discussed. Two model problems are solved. In the first problem, a semi-infinite dielectric plane, attached to an infinite grounded conducting substrate and exposed to a monoenergetic electron beam, is analyzed. Bulk and surface conductivities and secondary emission characteristics are taken into account as parameters. In the second problem the dielectric is charged but the electron beam is shut off so only the bulk and surface conductivities enter the calculation. The principal result of the latter calculation is to show that steep tangential gradients develop in the presence of a surface conductivity during decay, and that for asymptotic times the temporal behavior, for a fixed position, is proportional to the square root of t rather than exponential, as expected in the presence of a bulk conductivity.