Images in linearly conducting dielectrics

Abstract

Methods of images for electrostatic fields and steady conduction fields are fairly well known. Whenever applicable, they have made the solution of field problems much simpler. However, when dealing with lossy dielectrics both permittivity and nonzero conductivity are to be considered. A generalised method of images is developed which can deal with such linearly lossy (conducting) dielectrics. For linearly conducting dielectrics, a point charge is equivalent to a point current source and vice versa. At t = 0+, only the dielectric image will be seen. Subsequently, because of the finite nonzero conductivities of the associated media, a current will flow and surface charges will accumulate at the interface due to mismatch in the material properties. The equation governing this surface-charge accumulatioin is derived. Linearity of the media permits fields in either medium to be obtained by superposing the fields due to the dielectric images and that due to the interfacial charges. The field can be obtained in either medium by replacing these surface charges by an equivalent point charge kept ait the appropriate image point. This equivalent point charge satisfies a similar differential equation in time to that of the surface charge. The cases of time-varying chargeicurrent sources and the source-current requirement for keeping any point very close to the source at a specified potential are also discussed.