Green Function of the Point Source Inside/Outside Spherical Domain - Approximate Solution
A brief review of derivation of two groups of approximate closed form expressions for the electrical scalar potential (ESP) Green functions that originates from the current of the point ground electrode (PGE) in the presence of a spherical ground inhomoge
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Green Function of the Point Source Inside/Outside Spherical Domain Approximate Solution Nenad Cvetkovi´c, Miodrag Stojanovi´c, Dejan Jovanovi´c, Aleksa Risti´c, Dragan Vuˇckovi´c and Dejan Krsti´c Abstract A brief review of derivation of two groups of approximate closed form expressions for the electrical scalar potential (ESP) Green functions that originates from the current of the point ground electrode (PGE) in the presence of a spherical ground inhomogeneity are presented in this paper. The PGE is fed by a very low frequency periodic current through a thin isolated conductor. One of approximate solutions is proposed in this paper. Known exact solutions that have parts in a form of infinite series sums are also given in this paper. In this paper, the exact solution is solely reorganized in order to facilitate comparison to the closed form solutions, and to estimate the error introduced by the approximate solutions. Finally, error estimation is performed comparing the results for the electrical scalar potential obtained applying the approximate expressions and the accurate calculations. This is illustrated by a number of numerical experiments. Keywords Electrical scalar potential · Green’s function · Spherical inhomogeneity
N. Cvetkovi´c (B) · M. Stojanovi´c · D. Jovanovi´c · A. Risti´c · D. Vuˇckovi´c Faculty of Electronic Engineering, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia e-mail: [email protected] M. Stojanovi´c e-mail: [email protected] D. Jovanovi´c e-mail: [email protected] A. Risti´c e-mail: [email protected] D. Vuˇckovi´c e-mail: [email protected] D. Krsti´c ˇ Faculty of Occupational Safety, University of Niš, Carnojevi´ ca 10A, 18000 Niš, Serbia e-mail: [email protected] © Springer International Publishing Switzerland 2016 S. Silvestrov and M. Ranˇci´c (eds.), Engineering Mathematics I, Springer Proceedings in Mathematics & Statistics 178, DOI 10.1007/978-3-319-42082-0_6
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6.1 Introduction Problems of potential fields related to the influence of spherical material inhomogeneities, have a rather rich history of over 150 years in different fields of mathematical physics. In the fields of electrostatic field, stationary and quasi-stationary current field, and magnetic field of stationary currents, problems of a point source in the presence of a spherical material inhomogeneity are gathered in the book by Stratton [19] and all later authors that have treated this matter quote this reference as the basic one. The authors of this paper will also consider the results from [19] as the referent ones. The exact solution shown in [19, pp. 201–205] related to the point charge in the presence of the dielectric sphere, is obtained solving the Poisson, i.e. Laplace partial differential equation expressed in the spherical coordinate system, using the method of separated variables. Unknown integration constants are obtained satisfying the boundary conditions for the electrical scalar potentia
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