  The Biot-Savart law is a well-known and powerful theoretical tool used to
calculate magnetic fields due to currents in magnetostatics. We extend the
range of applicability and the formal structure of the Biot-Savart law to
electrostatics by deriving a Biot-Savart-like law suitable for calculating
electric fields. We show that, under certain circumstances, the traditional
Dirichlet problem can be mapped onto a much simpler Biot-Savart-like problem.
We find an integral expression for the electric field due to an arbitrarily
shaped, planar region kept at a fixed electric potential, in an otherwise
grounded plane. As a by-product we present a very simple formula to compute the
field produced in the plane defined by such a region. We illustrate the
usefulness of our approach by calculating the electric field produced by planar
regions of a few nontrivial shapes.
