Similarity transformations for decaying wall charge fields in a spherical shell

It is shown that ohmic dissipation of wall charges causes the electric field inside a spherical insulating shell to decay with a characteristic time constant, t0 =2aηε0 , where η is the surface resistance of the shell, a is its radius, and ε0 is the permittivity of a vacuum. It is demonstrated that the electric potential inside the shell at time t can be constructed by means of similarity transformations which are generated by multiplying the initial charge densities by a factor f and lengths by f 2 where f=exp(−t/2t0 ). Using this method the potential near the centers of oppositely charged disks of wall charge of radius b (b≪a) is calculated in terms of analytic expressions for the case where the externally applied field is absent. It is pointed out that the latter solution can be generalized to yield the potential distribution generated by small wall charge spots which have rotationally symmetric charge distributions. It is anticipated that the results will be useful for further studies of external electrode discharges in gases in insulating containers, Examples include low-frequency electric field sensors and gas-filled cavities in solid insulators.