A unified Fourier transform approach to electrostatics: Potentials and self-energies

This paper develops a unified Fourier transform (FT) framework for solving Poisson’s equation in classical electrostatics. Rather than treating each geometry separately, the method provides a systematic and algebraic procedure that applies uniformly to a wide class of charge distributions. Known results for standard configurations – including point charges, dipoles, infinite geometries, and spherical and cylindrical systems – are recovered in a compact and consistent manner, demonstrating the generality of the approach. The main results are the derivation of closed-form expressions for the electric potential of uniformly charged rings and disks at arbitrary spatial points, expressed in terms of complete elliptic integrals, together with a unified calculation of the electrostatic self-energy of finite-sized objects within the same formalism. These results illustrate how the FT method yields analytically tractable solutions for geometries where direct integration is cumbersome. • A unified Fourier transform framework solves Poisson’s equation. • Standard charge systems are recovered within one formalism. • Closed-form potentials for ring and disk using elliptic integrals. • On/off-axis and near/far-field behaviors analyzed explicitly. • Electrostatic self-energies are evaluated via Fourier space charge density.