# Evaluation of the Biot-Savart integral in electrostatic problems with non-uniform Dirichlet boundary conditions

## TL;DR

This paper introduces an analytical method to solve electrostatic problems with non-uniform Dirichlet boundary conditions, leveraging a Biot-Savart-like integral to compute electric fields for specific geometries.

## Contribution

It presents a novel analytical approach combining circulation and angular potential variations to solve Laplace's equation with non-uniform boundary conditions.

## Key findings

- Exact solutions for circular contours with periodic potentials
- Validation of analytical results with numerical and FEM methods
- Extension of Biot-Savart law to electrostatics with non-uniform boundaries

## Abstract

We present an analytical strategy to solve the electric field generated by a planar region $\mathcal{A}$ enclosed by a contour $c$ which is kept with a fixed but non-uniform electric potential. The approach can be used in certain situations where the electric potential on the space requires to solve the Laplace equation with non-uniform Dirichlet boundary conditions. We show that the electric field is due to a contribution depending on the circulation on the contour in a Biot-Savart way plus another one taking into account the angular variations of the potential in $\mathcal{A}$ valid for any closed loop $c$. The approach is used to find exact expansions solutions of the electric field for circular contours with fully periodic potentials. Analytical results are validated with numerical computations and the Finite Element Method. Keywords: Biot-Savart law, electrostatic problems, exactly solvable models.