Electric potential of insulated conducting objects in presence of electric charges -- some exact and approximate results

TL;DR

This paper introduces a new $J$ formalism for calculating the electric potential of insulated conducting objects in the presence of external charges, providing exact solutions for spheres and efficient approximations for other shapes.

Contribution

The paper presents a novel $J$ formalism that simplifies potential calculations without needing surface charge distributions, applicable to various geometries.

Findings

Exact potential calculation for spherical objects.

Efficient approximation method for complex geometries.

Numerical validation using the Robin Hood method.

Abstract

Determination of the electric potential of insulated conducting objects is an important problem both theoretically and practically. For an insulated conducting object in the presence of external charges or charges distributed on the object surface, the problem of potential determination is reformulated using a newly introduced $J$ formalism. Using the $J$ formalism, it is shown how the electric potential can be calculated exactly for spherical objects and efficiently approximated for other object geometries using geometrical properties of the insulated conducting object. This approach does not require calculation of the surface charge distribution at the object surface of the calculation of the electric potential in the surrounding space. Properties and the performance of the approach are investigated numerically using the Robin Hood method. Possible applications of the approach based on the $J$ formalism are outlined for calculation of capacitance of conducting objects.