Potential in the “Clearance” of torus: Development on spherical functions

Abstract

The method of solution of the Dirichlet problem for potentials of volume bodies with torus topology when the boundary conditions are set in the form of a series on spherical harmonics on pieces of two spherical surfaces is developed. The problem representation of an exterior potential homogeneous gravitating (or charged by static electric charge) circular torus out of a substance in a special (“intermediate”) spherical zone is presented and solved. The solution is received in the form of a combination of the Laplace series on even positive and odd negative degrees of the radius-vector of a test point. Coefficients of this series are received in a final analytical form. The general member of the series at the limit of big n tends to zero so the series converges fast and the radius of convergence are defined by torus geometry. The specified solution meets a gap in the theory, connecting together earlier discovered by us, two expansions in the Laplace series of a torus potential in “interior” and “exterior” spherical space zones. Thus, it is proved that it is possible to present the torus potential by power rows in all free space from the substance. For control of results by means of the obtained series, equipotential surfaces of the torus were calculated.