Dimensional Dependence of Electrostatic Equilibrium in Conductors: A Critical Analysis

Abstract

The study examines the electrostatic charge distribution in n -dimensional conductors. By generalizing Gauss’s law and using hyperspherical coordinates, the ratio of electrostatic potential energies between a uniform distribution in the volume ( W Θ ) and a uniform distribution on the surface ( W O ) is calculated. For n ≤ 2, the two distributions are energetically equivalent, whereas for n > 2, the surface distribution is more favorable, with a ratio W Θ / W O = λ ( n ) = 2 n / ( n + 2 ) . The main conclusion is that the concentration of charge on the surface constitutes an energetically favorable state only in spaces with more than two dimensions, revealing a fundamental difference in the behavior of electrostatic systems depending on spatial dimensionality. This study, providing new theoretical insight into the dimensional dependence of electrostatic equilibrium, could have applications in higher-dimensional theories.