Modeling the Electrostatic Potential of Disks with Arbitrary Radial Charge Profiles

TL;DR

This paper derives an exact analytical expression for the electrostatic potential of a charged disk with arbitrary radial charge distributions, enabling efficient computation and analysis of various charge configurations.

Contribution

It introduces a general analytical framework for calculating the potential of radially symmetric charged disks using Bessel functions, covering many special cases.

Findings

Potential behavior changes qualitatively with parameter variations

Analytical expressions are computationally efficient

Framework includes special cases like uniform charge distribution

Abstract

In this work, we present an analytical study of the electrostatic potential generated by a charged disk with a surface charge distribution that possesses radial axial symmetry. We express the potential in cylindrical coordinates and apply a Bessel function representation for the Coulomb kernel to simplify its calculation. We consider a broad family of surface charge densities and derive an exact expression for the potential based on a classical integral identity involving Bessel functions. This general formulation includes several important special cases such as the uniformly charged disk and edge-concentrated charge density distributions. We explore how variations in various parameters affect the resulting potential, thereby illustrating the influence of radial charge modulation. It is found that the potential undergoes a qualitative change of behavior when the chosen set of parameters is varied. The results are analytically elegant and computationally efficient as all functions involved are implemented in standard scientific computing libraries.