Dimensional Dependence of Electrostatic Equilibrium in Conductors: A Critical Analysis

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$W_{\Theta }/W_O=\lambda \left(n\right)=2n/\left(n+2\right)$ . 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.