Biot-Savart-like law in electrostatics

Details Biot-Savart-like law in electrostatics Biot-Savart-like law in electrostatics

2000-11-07

arxiv.org/abs/physics/0011015v1

Physics

Classical Physics

14 pages, 6 figures, RevTex, accepted for publication in the European Journal of Physics

Scientific paper

10.1088/0143-0807/22/1/304

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.

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