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# Electric and magnetic fields of two infinitely long parallel cylindrical conductors carrying a DC current

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## Abstract

This paper calculates the electric and magnetic fields and the Poynting vector around two infinitely long parallel cylindrical conductors, carrying a DC current. Also the charges on the surface of the wire are calculated, and their distribution is visualized. The wire is assumed to be perfectly electrically conducting. Furthermore, the Hall effect is ignored. In the literature [S.J. Orfanidis, Electromagnetic waves and antennas, 2008], the problem of determining the electric field is usually tackled using an equivalent model consisting of two line charge densities, producing the same electric field. In this work, the Laplace equation is rigorously solved. The authors found no work explaining the solution of the Laplace equation with boundary conditions for this problem and hence thought it was useful to dedicate a paper to this topic. The method of separation of variables is employed and a bipolar coordinate system is used. After solving the appropriate Sturm-Liouville problems, the scalar potential is obtained. Taking the gradient yields the electric field. Contribution to the Topical Issue "Numelec 2012", Edited by Adel Razek.

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... A similar solution for this problem has been found recently by Engelen et al [10]. They solve essentially the same problem using bi-cylinder coordinates (which they call bipolar coordinates). ...
... The similar solution presented by Engelen et al [10] lack the dependency on z for the potential, electric field and surface charges, rendering essentially an electrostatics problem. They were probably unaware of Russel's theorem [11]. ...
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... mm, much smaller compared to the cylinder diameter of 9.6 mm), this effect on the field on the axis of symmetry is small. This has been verified by comparing the electric field halfway between two metal cylinders (without dielectrics) calculated for the same cylinders diameter and effective gap, (a) from the superposition principle and (b) from exact solution of the Laplace equation [23]. The difference between the two results is approximately 2%. ...
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... Along similar lines, in case 6 of Fig. 3·9, voltage profile between phases A and B, which are maintained at zero potential, is affected by the presence of phase C electrodes of previous and current spatial period. Analytical models have been derived for the electric potential between two cylindrical conductors, like (Engelen et al., 2013) or two-wire transmission line (Orfanidis, 2014), that can be potentially used to provide better approximations for the voltage profile closer to what FEA method provides, rather than a linear one. ...
Thesis
Full-text available
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... On similar lines, in Fig. 5 case 3, when the first and second electrode are at 1 kV electric potential, the voltage profile provided by the FEA method shows a decrease in electric potential between the aforementioned electrodes down to 957.2 V while the voltage is assumed to be still at 1 kV along the x-axis in ANA method. Existing analytical models discussed in [32] and [33] have the potential to provide better approximations of the voltage profile than a linear onedcloser to what the FEA method yields. . Providing a detailed model of the voltage profile will improve the accuracy the analytical method, which can be a promising research topic for further studies. ...
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... Along similar lines, in case 6 of Fig. 5, the voltage profile between phases A and B, which are maintained at zero potential, is affected by the presence of phase C electrodes of the previous and current spatial period. Analytical models have been derived for the electric potential between two cylindrical conductors, like [16] or two-wire transmission line [17], that can be potentially used to provide better approximations for the voltage profile closer to what FEA method provides, rather than a linear one. Detailed analytical model of the voltage profile is out of the scope of this paper and is left for further studies. ...
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The electrodynamic screen, or EDS, has shown promising results in mitigation of dust accumulation losses in solar energy harvesting systems. In this paper, the electric field distributions in two EDS configurations have been thoroughly investigated. The analytical solutions for the electric potential and electric field distribution in the first EDS configuration have been provided and corroborated numerically using finite element analysis (FEA) software. The electrostatic model of second EDS configuration has been developed in the FEA software and its electric field distribution has been analyzed numerically. A comparison has been made between the two configurations regarding dust removal efficiency.
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Let us first state exactly what this book is and what it is not. It is a compendium of equations for the physicist and the engineer working with electrostatics, magne­ tostatics, electric currents, electromagnetic fields, heat flow, gravitation, diffusion, optics, or acoustics. It tabulates the properties of 40 coordinate systems, states the Laplace and Helmholtz equations in each coordinate system, and gives the separation equations and their solutions. But it is not a textbook and it does not cover relativistic and quantum phenomena. The history of classical physics may be regarded as an interplay between two ideas, the concept of action-at-a-distance and the concept of a field. Newton's equation of universal gravitation, for instance, implies action-at-a-distance. The same form of equation was employed by COULOMB to express the force between charged particles. AMPERE and GAUSS extended this idea to the phenomenological action between currents. In 1867, LUDVIG LORENZ formulated electrodynamics as retarded action-at-a-distance. At almost the same time, MAXWELL presented the alternative formulation in terms of fields. In most cases, the field approach has shown itself to be the more powerful.