More than 80 years ago, Born-Infeld electrodynamics was proposed in order to remove the point charge singularity in Maxwell electrodynamics. In this work, after a brief introduction to Lagrangian formulation of Abelian Born-Infeld model in the presence of an external source, we obtain the explicit forms of Gauss's law and the energy density of an electrostatic field for Born-Infeld electrostatics. The electric field and the stored electrostatic energy per unit length for an infinite charged line and an infinitely long cylinder in Born-Infeld electrostatics are calculated. Numerical estimations in this paper show that the nonlinear corrections to Maxwell electrodynamics are considerable only for strong electric fields. We present an action functional for Abelian Born-Infeld model with an auxiliary scalar field in the presence of an external source. This action functional is a generalization of the action functional which was presented by Tseytlin in his studies on low energy dynamics of D-branes (Nucl. Phys. B469, 51 (1996); Int. J. Mod. Phys. A 19, 3427 (2004)). Finally, we derive the symmetric energy-momentum tensor for Abelian Born-Infeld model with an auxiliary scalar field.
