Electrostatic potential of a uniformly charged triangle in barycentric coordinates

Abstract

We compute the electrostatic potential of a uniformly charged triangle. Barycentric coordinates are employed to express the field point, the parametrization of the surface integral, and the gradient operator. The resultant analytic expression for the electrostatic potential is expressed in terms of the side lengths of the triangle, the altitude of the field point from the plane in which the triangle is placed, and the barycentric coordinates of the field point relative to the triangle. Our results are in good agreement with available numerical results. The asymptotic behavior of the analytic expression is investigated in special limits that satisfy known values. The resultant analytic expressions for the asymptotic regions are useful in improving the numerical convergence at boundaries. As an application, we provide a strategy to compute the electrostatic potential of a uniformly charged polygon. The electrostatic potential of a uniformly charged rectangle is considered as a simple example that agrees with a previous result. Appendices provide a complete set of integral tables that are necessary to evaluate the double integral over the barycentric coordinates, an explicit parametrization of the gradient operator in the barycentric coordinates, and useful coordinate-transformation rules between the barycentric and Cartesian coordinates.