The solution of Laplace’s equation in cylindrical and toroidal configurations with rectangular sectional shapes and rotation‐symmetrical boundary conditions

摘要

Basic formulas have been obtained for the contribution to the potential in any point of a cylindrical or toroidal volume with rectangular sectional shapes and rotation‐symmetrical boundary conditions. This is done by solving Laplace’s equation in cylindrical coordinates using the method of separation of variables for the cases in which simple but characteristic rotation‐symmetrical potential overlays on the boundaries exist. Knowing these solutions and limiting the widths of the areas where the potential overlays are defined, differential contributions to the potential in any point of the observed volumes in analytical form are derived. With these solutions also numerical formulas for the contribution in a point of the volume can be formulated so that the relaxation method can be avoided.