Accurate determination of the capacitance of a thin rectangular plate

THE PURPOSE of this paper is two fold: to determine by the method of subareas the charge distribution and capacitance of a rectangular area; and to advance therefrom a universal curve which furnishes accurate values of the capacitance of a rectangular area of any specified dimensions. From well-known theory, rigorous determination of these electrical quantities hinges on determination of the corresponding electrostatic potential function V. However, determination of the function for a specified plane area is of such mathematical difficulty that solution to date has been achieved for only one plane area, the elliptical disk of which the circular disk is a special case. In consequence, explicit equations for the charge distribution and electrical capacitance are known only for these two plane areas. However, need of accurate knowledge of the capacitance of a rectangular plane area, or of a certain parameter1 equal in magnitude to the capacitance, occurs in numerous physical problems. (Thus, the normal force R on a thin rectangular plate moving in the direction of its normal with a velocity V through a liquid of viscosity μ is R = 8πμCV, where C is a parameter equal to the capacitance of a rectangular planar conductor of the same dimensions as the thin plate.) To cite an example of much current interest in electrical engineering, a knowledge of the capacitance of rectangular plates is required in the rational design of grounding systems for high-voltage power systems.2–4 Accordingly, it is obviously most desirable to have in hand curves which readily afford values of the capacitance with accuracy sufficient for design purposes. Precisely such a set of curves is advanced in this paper, the essential content of which may be summarized as follows: The method of subareas, earlier formulated and used by the authors to obtain excellent approximate solutions of the difficult, long-standing problems of determination of the capacitance and charge distribution of a plane annulus5 and a cube,6,7 is recast to enable solution of the present problem. Therewith equations for the capacitance and charge distribution of an arbitrarily dimensional rectangular area are formulated; and the solutions of these equations, as effected through use of International Business Machines equipment, are then utilized to plot a universal curve of capacitance versus width/height from which can be read the capacitance of any specified rectangular area with sufficient accuracy for most design purposes. Corroborative of the correctness of the solution in general, this curve falls within the narrow upper and lower bounds to be calculated from the known expressions for the capacitances of the circumscribed and inscribed ellipses. Finally, comparison of this universal curve with the corresponding curve plotted from Howe'8 wellknown and much used expression for the approximate capacitance of a rectangular area reveals that this expression yields values in considerable error over much of the range of w/h (width/height).