Alternative separation of Laplace's equation in toroidal coordinates and its application to electrostatics

The usual method of separation of variables to find a basis of solutions of Laplace's equation in toroidal coordinates is particularly appropriate for axially symmetric applications; for example, to find the potential outside a charged conducting torus. An alternative procedure is presented here that is more appropriate where the boundary conditions are independent of the spherical coordinate θ (rather than the toroidal coordinate η or the azimuthal coordinate ψ). Applying these solutions to electrostatics leads to solutions, given as infinite sums over Legendre functions of the second kind, for (i) an arbitrary charge distribution on a circle, (ii) a point charge between two intersecting conducting planes, (iii) a point charge outside a conducting half plane. In the latter case, a closed expression is obtained for the potential. Also the potentials for some configurations involving charges inside a conducting torus are found in terms of Legendre functions. For each solution in the basis found by this separation, reconstructing the potential from the charge distribution (corresponding to singularities in the solutions) gives rise to integral relations involving Legendre functions.