https://doi.org/10.1140/epjp/s13360-020-00864-0
Regular Article
Electric vector potential formulation in electrostatics: analytical treatment of the gaped surface electrode
1
Departamento de Física, Universidad de los Andes, Bogotá, Colombia
2
Universidad ECCI, Bogotá, Colombia
3
Facultad de Ingeniería, Universidad Nacional de Colombia, Bogotá, Colombia
4
Centro de Ingeniería Avanzada Investigación y Desarrollo, CIAID, Bogotá, Colombia
* e-mail: rp.salazar84@uniandes.edu.co
Received:
22
May
2020
Accepted:
13
October
2020
Published online:
3
November
2020
The electric vector potential 
is a legitimate—but rarely used—tool to calculate the steady electric field in charge-free regions. It is commonly preferred to employ the scalar electric potential 
rather than in most of the electrostatic problems. However, the electric vector potential formulation can be a viable approach to study certain systems. One of them is the gaped surface electrode (SE): a planar finite region 
kept at a fixed potential 
with a gap of thickness 
to the remaining grounded field. In this document, the Helmholtz decomposition theorem and the electric vector potential formulation are used to provide integral expressions for the surface charge density and the electric field of the gaped SE of arbitrary contour 
. It is shown that the electric field of the gaped circular SE in the 
space can be obtained from averaging the gapless SE solution over the gap. Even though the approach is illustrated with the circular SE, the strategy could be used in other geometries if the corresponding gapless solution is known. Analytic results are in agreement with numerical approximations of the electrostatic problem via Finite Element Method. Finally, the magnetic analogue of the gaped SE is provided.
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature, 2020
