Spreading of surface charge on a thick insulator

We show that a patch of charge on the surface of a thick insulator will spread in a self-similar fashion. We have calculated the corresponding charge density by a numerical approximation. Guided by these results, we deduce the exact analytic solution. The radius increases as the cube root of the time. Substituting appropriate values for the carrier mobility and charge density, we show that the time taken for the patch to double in size may be measured in weeks, in agreement with the stability observed in practice.