  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.
