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

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