# Parallel-plate and spherical capacitors in Born-Infeld electrostatics: An analytical study

## TL;DR

This paper analytically investigates how Born-Infeld nonlinear electrodynamics modifies the classical electrostatic properties of parallel-plate and spherical capacitors, revealing significant nonlinear effects at high potential differences.

## Contribution

It provides explicit formulas for capacitance and energy in Born-Infeld electrostatics, highlighting deviations from classical relations and analyzing nonlinear corrections.

## Key findings

- Nonlinear corrections to capacitance are significant at high voltages.
- Classical energy relations do not hold in Born-Infeld electrostatics.
- Numerical estimates show considerable deviations from Maxwell theory at large potentials.

## Abstract

In 1934, Max Born and Leopold Infeld suggested and developed a nonlinear modification of Maxwell electrodynamics, in which the electrostatic self-energy of an electron was a finite value. In this paper, after a brief introduction to Lagrangian formulation of Born-Infeld electrodynamics with an external source, the explicit forms of Gauss's law and the electrostatic energy density in Born-Infeld theory are obtained. The capacitance and the stored electrostatic energy for a parallel-plate and spherical capacitors are computed in the framework of Born-Infeld electrostatics. We show that the usual relations $U=\frac{1}{2}C_{_{\textrm{Maxwell}}}(\triangle \phi)^{2}$ and $U=\frac{q^{2}}{2C_{_{\textrm{Maxwell}}}}$ are not valid for a capacitor in Born-Infeld electrostatics. Numerical estimations in this research show that the nonlinear corrections to the capacitance and the stored electrostatic energy for a capacitor in Born-Infeld electrostatics are considerable when the potential difference between the plates of a capacitor is very large.