  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.
