  Electric vector potential $\Theta(\boldsymbol{r})$ is a legitimate but rarely
used tool to calculate the steady electric field in free-charge regions.
Commonly, it is preferred to employ the scalar electric potential
$\Phi(\boldsymbol{r})$ rather than $\Theta(\boldsymbol{r})$ in most of the
electrostatic problems. However, the electric vector potential formulation can
be a viable representation to study certain systems. One of them is the surface
electrode SE, a planar finite region $\mathcal{A}_{-}$ kept at a fixed electric
potential with the rest grounded including a gap of thickness $\nu$ between
electrodes. In this document we use the \textit{Helmholtz Decomposition
Theorem} and the electric vector potential formulation to provide integral
expressions for the surface charge density and the electric field of the SE of
arbitrary contour $\partial\mathcal{A}$. We also present an alternative
derivation of the result found in [M. Oliveira and J. A. Miranda 2001 Eur. J.
Phys. 22 31] for the gapless ($\nu=0$) surface electrode GSE without invoking
any analogy between the GSE and magnetostatics. It is shown that electric
vector potential and the electric field of the gapped circular SE at any point
can be obtained from an average of the gapless solution on the gap. Keywords:
Electric vector potential, surface-electrode, Helmholtz Decomposition, Green's
theorem.
