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Electrostatic field of angular-dependent surface electrodes

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Abstract

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 \({\mathscr {A}}\) 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 \({\mathscr {A}}\). 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.

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Notes

  1. Being only a an analogy between two different physical systems: the electrostatic where charges are static and the magnetostatic where current is time-independent.

  2. Generally speaking, an electric current along a wire is considered as constant. However, there are devices that emulate non-uniform distributed currents along wires (see for instance the setup in [24] which can be used in the experimental evaluation of pipe-lines’ cathodic protection, the DC version of this setup would also emulate a steady non-uniform current along a wire). Another case concerns the electric current of a point-like charge moving in a loop: the electric current associated with this charge is not uniform.

  3. For instance, a smooth linear function \({\mathscr {V}}(\phi ) = \phi \) can be approximated by selecting \(V(\phi )\) as an staircase function satisfying periodicity conditions Eq. (6). However, in that situation, there would exist a discontinuity in which \(\partial _\phi {\mathscr {V}}\) is a Dirac delta function, and the Taylor series expansion would diverge, no matter how large becomes N.

  4. At this point, it is necessary emphasizing that we deal with the electrostatic problem which involves no currents. However, in the SE with variable potential \(V(\phi )\), the lines of the electric field can be matched with the field lines of two superposed magnetic fields of its magnetostatic analogue.

  5. The angle in the Gegenbauer Polynomial is computed as usual, \(\theta ({\varvec{r}}) = \arctan (\rho (x,z)/y)\) if \(y>0\), \(\pi - \arctan (-\rho (x,z)/y)\) if \(y<0\), otherwise \(\pi /2\), with \(\rho (x,z)^2 = x^2 + z^2\) accounting for the coordinate transformation. In other words, \(\theta ({\varvec{r}}) =(1-{\mathscr {U}}(\Upsilon _n({\varvec{r}}))\pi + \Theta _n({\varvec{r}})\) if \(\Lambda _n({\varvec{r}}) \ne 0\), otherwise \(\pi /2\), where \(\Theta _n({\varvec{r}})=\arctan (\Upsilon ({\varvec{r}})/\Lambda _n({\varvec{r}}))\) and \(\Upsilon _n({\varvec{r}})=\sqrt{\lambda _n({\varvec{r}})^2-\Lambda _n({\varvec{r}})^2}\).

  6. We chose this potential, because the coefficients \({\mathscr {A}}^{(n)}_{m}(r)\) and \({\mathscr {B}}^{(n)}_{m}(r)\) are found easily. However, other type of potentials portray as good options, e.g., gaussian-like potentials, power-law potentials, etc.

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Acknowledgements

This work was supported by Vicerrectoría de investigación, Universidad ECCI. Robert Salazar also thanks Fundación Colfuturo and Departamento de Ciencias Básicas, Universidad ECCI. Camilo Bayona acknowledges the postdoctoral fellowship received from Departamento Administrativo de Ciencia, Tecnología e Innovación - Colciencias.

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Authors and Affiliations

  1. Universidad ECCI, Cra. 19 No. 49-20, Bogotá, Código Postal 111311, Colombia

    Robert Salazar, Camilo Bayona-Roa & J. S. Solís-Chaves

  2. Departamento de Física, Universidad de los Andes, Bogotá, Colombia

    Robert Salazar

  3. Centro de Ingeniería Avanzada Investigación y Desarrollo-CIAID, Bogotá, 111111, Colombia

    Camilo Bayona-Roa

  4. Universidad Nacional de Colombia, Bogotá, 111321, Colombia

    Camilo Bayona-Roa

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  1. Robert Salazar
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Correspondence to Robert Salazar.

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Salazar, R., Bayona-Roa, C. & Solís-Chaves, J.S. Electrostatic field of angular-dependent surface electrodes. Eur. Phys. J. Plus 135, 93 (2020). https://doi.org/10.1140/epjp/s13360-019-00090-3

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