Potential Distributions in Large Cylindrical Disks with Partially Penetrating Electrodes

Abstract

Because of the equivalence of the pressure in a liquid bearing sand to a velocity potential, an analysis has been carried through of the potential theory of the electrical analogue of an artesian well partially penetrating a water bearing sand. Part I. In Part I the method of analysis is illustrated by the solution of the problem of a hemispherical electrode imbedded in one of the faces of the disk. Two types of formulae are developed suitable for study of the potential distribution at small and large distances from the electrode respectively. The resistance of this system as a function of the disk thickness is computed and plotted. Part II. In Part II is treated the more general problem of the potential distribution for a partially penetrating electrode, with a uniform flux density along the electrode surface. Separate formulae for the potential function are again derived that are convenient for discussion at small and large separations from the electrode. It is shown that they reduce in the limit of small penetrations to those of Part I and in the limit of complete penetration to well-known radial flow formulae. Expressions are also given for the stream functions corresponding to the derived potential functions. Finally it is shown that when the flux density along the electrode is assumed to be constant the potential over the electrode is far from uniform, falling sharply near the electrode extremity to a value slightly more than half of that at the top of the electrode. Part III. The problem of physical interest in which the electrode is at uniform potential is attacked in Part III. After discussing the case of a flux density over the electrode surface increasing with distance from the top the direct method of superposing discontinuous flux density elements is finally used to obtain electrode surface potentials uniform to within 2 percent of the average surface value. The equivalence of this method to the direct solution of the integral equation for the problem is pointed out. The resistances of the disk for various electrode penetrations and disc thicknesses are then computed and plotted. From these a semi-empirical approximation is derived permitting the resistances for other cases to be derived from a single expression without the necessity of first adjusting the electrode flux distribution so that its potential be uniform.