Biot-Savart-like law in electrostatics

The Biot-Savart law is a well known and powerful theoretical tool used to c alculate magnetic fields due to currents in magnetostatics. We extend the r ange of applicability and the formal structure of the Biot-Savart law to el ectrostatics by deriving a Biat-Savart-like law suitable for calculating el ectric fields. We show that, under certain circumstances, the traditional D irichlet problem can be mapped onto a much simpler Biot-Savart-like problem . We find an integral expression for the electric field due to an arbitrari ly shaped, planar region kept at a fixed electric potential, in an otherwis e grounded plane. As a by-product we present a very simple formula to compu te the field produced in the plane defined by such a region. We illustrate the usefulness of our approach by calculating the electric field produced b y planar regions of a few non-trivial shapes.