Electrostatic field of angular-dependent surface electrodes

We present an analytic strategy to find the electric field generated by surface electrode SE with angular-dependent potential. This system is a planar region A kept at a fixed but non-uniform electric potential V(ϕ) with an arbitrary angular dependence. We show that the generated electric field is due to the contribution of two fields: one that depends on the circulation on the contour of the planar region—in a Biot–Savart-Like (BSL) term—and another one that accounts for the angular variations of the potential in A. This approach can be used to find exact solutions of the BSL electric field for circular or polygonal contours of the planar region with periodic distributions of the electric potential. Analytic results are validated with numerical computations and the Finite-Element Method.