Calculation of potential gradients for a dielectric slab placed between a sphere and a plane

Successive overrelaxation methods have been used to calculate potential gradients around a spherical high-voltage electrode separated from an earthed plate by a plane or recessed dielectric slab. The distance from the sphere to the earthed plate has been made one half of the sphere radius for most of the calculations, and the ratio of the permittivities of the dielectric slab and the surrounding medium has been varied. For a limited range of slab-thickness/sphere-radius ratios, it is shown that the highest potential gradient in a plane slab may be calculated by assuming that a uniform permittivity exists throughout and by using a fictitious sphere-radius/electrode-separation ratio. For a plane slab having a permittivity 4 times that of the surrounding medium, and for slab a thickness equal to the sphere radius, the highest potential gradient in the medium is found to be nearly 14 times the average gradient in the shortest gap. When the slab thickness is one half of the sphere radius, the highest potential gradient in the medium is still about 9 times the gap average, but this is considerably reduced when a recessed slab is used. The potential-gradient transitions at the point where the surface of the recessed dielectric slab meets the sphere surface have been determined and analysed.