Electrostatics of a Conducting Toroidal Surface and a Point Charge

Electrostatics of a conducting toroidal surface and a point charge
J. A. Hernandes
Departamento de Fisica, ICE, Universidade Federal de Juiz de Fora, 36036-900 Juiz de Fora, MG, Brazil

Abstract We find the exact expression for the electrostatic potential due to a grounded toroidal conducting surface and a point charge outside the toroidal surface. We use the approach provided by Green’s method. We also find the induced surface charge on the toroidal conductor and the net force between the toroid and the point charge. We show how the potential and induced charges behave on some special cases. Keywords: Electric potential, induced charges, electrostatics, toroidal conductor

1. Introduction Recently, there’s been a renewed interest in the toroidal geometry and toroidal functions, particularly on the fields of astrophysics [CT99, CTRS00], nanoparticles [MKH+ 07] and toroidal function calculations [And06, GS97]. The toroidal coordinates system is well known, and Laplace’s equation is separable in this system. This is suitable for solving problems involving the gravitational or electric potential in the toroidal geometry, like the electrostatics of a charged toroid, [Smy89, p. 239], [MF53, p. 1304], or even the magnetostatics of steady currents along the toroid, [HA03, HMLA08]. The aim of this work is to calculate the electrostatic potential that arises due to the proximity of a point charge outside a conducting toroid. We present the solution of this problem for the electric potential, along with the force between them and the surface charges induced on the toroid surface. To our knowledge this has never been done before. To this end we consider the Green’s function method, [Jac99, Chap. 1-3].
Email address: jahernandes@fisica.ufjf.br (J. A. Hernandes)

Preprint submitted to Elsevier

September 14, 2010

2. Toroidal coordinates, scale factors and differential operators We use toroidal coordinates, given by (η, χ, ϕ) [MS88, p. 112], defined by: x=a sinh η cos...