An Equation For Charge Decay Valid in Both Conductors and Insulators

Gauss’ law and the equation of continuity must be satisfied in all materials be they solids, liquids or gases. Most materials are classified as simple materials; i.e., their electrical properties are linear, isotropic and homogeneous. Charge transport in these simple materials should be described by a constitutive equation known as Ohm’s law. When Ohm’s law is combined with Gauss’ law and the equation of continuity, a differential equation for volume charge density relaxation results. The usual solution to this equation shows that charge decays exponentially with a relaxation time given by the material’s permittivity divided by its electrical conductivity. Experiments show that good conductors follow this exponential decay but that poor conductors (insulators) tend to follow a decay that initially is more hyperbolic than exponential. This suggests that either Ohm’s law is not valid for insulator materials or that a deeper understanding of Ohm’s law is needed to explain charge decay in these less than good conductors. This paper examines the latter approach and shows that, when all the free charges within a simple material are taken into account, a new closed-form equation for unipolar charge decay can be derived which is valid for any simple material be it classified as a very good conductor, or a very poor conductor or anywhere in between. For good conductors the equation reduces to the standard exponential law of decay. For very poor conductors the equation reduces to the Vellenga-Klinkenberg modified hyperbolic law with the initial decay producing the characteristic Bustin hyperbolic law of decay. Explicit definitions for a good conductor and a good insulator are obtained and are used to define the range where explicit deviations from both of these hyperbolic laws occur.