Manual Formal Structure of Electromagnetics: General Covariance and Electromagnetics

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Yan, M. Yan, and M. Keywords: Electrodynamics Relativity Constitutive law Anisotropic media 1. Introduction The development of the modern microscopic technology nanotechnology provides a possibility to manufacture materials of rather non-ordinary electromagnetic parameters. This situation calls for a theoretical investigation of electromagnetic wave propagation in media with a generic constitutive law.

Dispersion relation for electromagnetic waves in anisotropic media

Moreover it emerges in a broad class of theoretical and experimental subjects, particularly in the high energy physics, general relativity, astrophysics, materials science and the plasma physics. The theoretical issues connected to the wave propagation phenomena yield a class of intriguing mathematical physics problems. Two additional matrices describe relatively smaller electric—magnetic cross-term effects.

Maxwell's Equations

Even with this restriction, the corresponding dispersion relation was obtained in a rather complicated form. Such extensions of the standard electromagnetic materials properties are not only of a theoretical interest. In fact, the medium with negative permittivity and permeability parameters serves as a theoretical basis for recently manufactured metamaterials.

The medium with non-invertible matrices is recently discussed in form of a perfect electromagnetic conductor.

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Formal structure of electromagnetics ( edition) | Open Library

In the current Letter, we study the wave propagation in a generic medium in a framework of premetric electrodynamics approach [4—9]. In Section 3, the covariant metric-free form of the dispersion relation is represented. The main results are given in Section 4 where several compact forms of the generic dispersion relation and some straightforward consequences of them are derived.

In Section 5, Y. In the sequel, the Roman indices will be used for the spatial coordinates, i , j ,.


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  • In this notation, the system 2. Such a generic constitutive tensor can be represented by four three-dimensional matrices of 9 independent components.

    Dispersion relation for electromagnetic waves in anisotropic media

    In this Letter, they are taken to be equal to zero. We will consider, however, some type of a generalized anisotropic medium. Consequently we will use a constitutive tensor of 18 independent components 2. In three-dimensional form, the corresponding constitutive relation is given by These relations have a clear physical meaning. Thus we are looking for solutions of the system 3.

    ISBN 13: 9780486654270

    Consequently, our system always has a non-zero solution. However, due to the gauge invariance, we need more of that. In fact, we are looking for an additional linear independent solution.


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    • Only this one will be of a physical meaning. It is an algebraic fact, that a linear system has two independent solution only if the adjoint of the characteristic matrix is equal to zero. In fact, the situation is much different. Its explicit forms are given in [5,9]. Note some straightforward facts resulting from this expression: 1 The dispersion relation 4.

      For invertible matrices sion 4. The result is a scalar function multiplied by ki k j , 1 4. Isotropic case Substituting here 4. In correspondence with 3.