3D Diagonalization and Supplementation of Electrostatic Field Equations in Fully Anisotropic and Inhomogeneous Media Proof of Existence and Consistency

Authors

  • A. R. Baghai-Wadji 1Electrical Engineering Department University of Cape Town, Cape Town, 7701, South Africa , 2College of Science, Xi’an University of Science and Technology, Xi’an, Shaanxi, 710054, China

Keywords:

Anisotropic and inhomogeneous dielectric media, diagonalization, electrostatic field, supplementation

Abstract

Consider Maxwell’s homogeneous curl equation divergence × E = 0 for the electric field vector E and the inhomogeneous divergence equation divergence · D = rho for the dielectric displacement vector D and the charge density function rho in the static limit. Assume an (x, y, z)–Cartesian coordinate system. Consider the constitutive equation D = epsilon E, with the 3 × 3 position-dependent positive-definite permittivity matrix epsilon (x, y, z) modeling fully anisotropic and inhomogeneous dielectric media. This paper proves that divergence × E = 0 and divergence · D = rho along with D = epsilon E are diagonalizable with respect to the arbitrarily chosen z–axis leading to the Dc–form. The existence of an associated supplementary equation, the Sc–form, has also been demonstrated. Finally, it is shown that the constructed (Dc, Sc)–forms are sharply equivalent with the originating set of equations divergence × E = 0, divergence · D = rho, and D = epsilon E, and, thus, internally consistent. The proof scheme is relative in the sense that its validity hinges on the consistency of Maxwell’s equations in the static limit and the material realizability conditions.

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References

A. R. Baghai-Wadji, ‘‘3D diagonalization and supplementation of Maxwell’s equations in fully bi-anisotropic and inhomogeneous media, Part I: Proof of existence by construction,’’ ACES Journal, this issue.

A. R. Baghai-Wadji, ‘‘3D diagonalization and supplementation of Maxwell’s equations in fully bi-anisotropic and inhomogeneous media, Part II: Relative proof of consistency,’’ ACES Journal, this issue.

A. R. Baghai-Wadji, ‘‘3-D electrostatic charge distribution on finitely thick busbars in micro acoustic devices: Combined regularization in the near- and far-field,’’ IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control (UFFC), vol. 62, no. 6, June 2015, pp. 1132-1144.

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Published

2019-02-01

How to Cite

[1]
A. R. Baghai-Wadji, “3D Diagonalization and Supplementation of Electrostatic Field Equations in Fully Anisotropic and Inhomogeneous Media Proof of Existence and Consistency”, ACES Journal, vol. 34, no. 02, pp. 246–251, Feb. 2019.

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