Electrostatic excitation of a conducting toroid: Exact solution and thin-wire approximation

Laplace's equation is solved via separation of variables in toroidal coordinates for the electrostatic potential external to a conducting torus placed in a uniform electric field and excited by an arbitrarily located point charge. The accuracy of the static thin-wire kernel approximation in an integral equation applied to the circular loop is verified using the exact results in the limit as the toroid shrinks to a ring. An equivalent lineal charge density from the exact solution agrees remarkably well with the integral equation solution for the conducting ring. Since the singularity in the Helmholtz Green's function for the electrodynamic problem is the static singularity considered herein, the results confirm the applicability of the thin-wire kernel to the scattering and radiation problems of the circular loop.