Electrostatic Field of Angular-Dependent Surface Electrodes

TL;DR

This paper introduces an analytic method to compute the electric field of surface electrodes with angular-dependent potentials, combining circulation-based and angular variation contributions, validated by numerical methods.

Contribution

It provides an exact analytic approach to determine electric fields for planar regions with angular-dependent potentials, including circular and polygonal shapes.

Findings

Derived explicit formulas for electric fields with angular-dependent potentials.

Validated analytic solutions with numerical and finite element methods.

Applicable to various planar electrode geometries and potential distributions.

Abstract

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 $\mathcal{A}$ kept at a fixed but non-uniform electric potential $V(\phi)$ 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 $\mathcal{A}$. 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. Keywords: Surface-electrode, Biot-Savart law, electrostatic problems, exactly solvable models.