Electric vector potential formulation in electrostatics: analytical treatment of the gaped surface electrode

Abstract

Electric vector potential Θ(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 Φ(r) rather than Θ(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 A− kept at a fixed electric potential with the rest grounded including a gap of thickness ν between electrodes. In this document we use the 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 ∂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 (ν = 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.

Expansion formula for the magnetic field of a periodically deformed circular current loop

“…Particularly, the three-dimensional vector field due to a circular-deformed loop with low symmetry. Analytic solutions of the BSL integrals are outstanding for solving steady electric, magnetic or fluid flow problems, as it has been demonstrated in [24,25] by the authors of the present study 3 .…”

“…3 For instance, the recent methodology in [25] demonstrates how the electric field of Gaped Surface Electrode systems can be obtained from the weighted average of their gapless counterparts. In that work, it was formally demonstrated that the electric field of the Gapped surface electrode with gap G is given by…”

“…Next, the analytic methodology using the expansion approach with the Gegenbauer Polynomials is presented in Section 3, which ends up giving the magnetic 2 Indeed, the gaped SE solution is essential to describe SE Radio Frequency ion traps: a collection of consecutive SE that have become promising candidates for manufacturing large-scale quantum processors. 3 For instance, the recent methodology in [25] demonstrates how the electric field of Gaped Surface Electrode systems can be obtained from the weighted average of their gapless counterparts. In that work, it was formally demonstrated that the electric field of the Gapped surface electrode with gap G is given by…”

Electric vector potential and the Biot-Savart like law in electrostatics

“…The present article demonstrated another plausible approach to studying the gaped SE advantageous regarding Ref. (Salazar et al, 2020) since it becomes mathematically simpler. That is constructing the electric vector potential from a superposition of vector potentials † As presented in the reference, the time-dependent version of Eq.…”

“…As shown in Ref. (Salazar et al, 2020), one can obtain the Biot-Savart law without solving the Laplace's equation. That alternative derivation is by employing the Helmholtz decomposition and Green's theorems.…”