Electric Vector Potential and the Biot-Savart like law in Electrostatics

A vector potential formulation is shown in this article to compute the electric field of planar surface electrodes. The electric field is derived from from the solution of the Laplace's equation in the free-charge space. Neumann-boundary conditions must be set on the region between planar metallic sheets as the separation goes to zero. It is shown that the electric field can be written via a Biot-Savart-like integral. The strategy enables to generalize the analytical result for its application in the gaped surface electrodes description.