Analytic solutions for some basic problems in electricity involving intersecting sphere and circular cylinder pairs

Abstract

The method of electrical inversion in classical electrostatics is employed to obtain exact solutions for basic electrostatic problems pertaining to overlapping spheres/cylinders. The problems considered here include (1) a pair of overlapping conducting spheres, intersecting at a vertex angle π/n, n an integer, placed in a constant potential field; (2) a pair of infinitely long conducting circular cylinders, intersecting at a vertex angle π/n, n an integer, placed in a uniform field; and (3) a composite hybrid geometry consisting of two orthogonally intersecting infinitely long circular cylinders whose boundary is a combination of conducting and dielectric surfaces (with mixed boundary conditions) submerged in a uniform field. Applying the basic idea of Kelvin’s inversion repeatedly, solutions for the electric potentials are derived in each case. An exact expression for the capacitance in terms of the two radii, center-to-center distance, and the vertex angle is found for the twin sphere geometry. The capacity is then used to find the steady-state rate coefficient of a perfectly absorbing body placed in a thermally conducting environment of lower temperature. The equipotentials are plotted using the exact solutions of the two-dimensional problems and their features are discussed as well. The simple method illustrated here can be useful both as a teaching tool and as a building block for further computations.