Abstract
Maxwell’s equations in materials are studied jointly with Euler equations using new knowledge recently appeared in the literature for polyatomic gases. For this purpose, a supplementary conservation law is imposed; one of the results is a restriction on the law linking the magnetic field in empty space and the electric field in materials to the densities of the 4-Lorentz force να and its dual μα: These are the derivatives of a scalar function with respect to να and μα, respectively. Obviously, two of Maxwell’s equations are not evolutive (Gauss’s magnetic and electric laws); to simplify this mathematical problem, a new symmetric hyperbolic set of equations is here presented which uses unconstrained variables and the solutions of the new set of equations, with initial conditions satisfying the constraints, restore the previous constrained set. This is also useful because it allows to maintain an overt covariance that would be lost if the constraints were considered from the beginning. This is also useful because in this way the whole set of equations becomes a symmetric hyperbolic system as usually in Extended Thermodynamics.
Highlights
Up to now it has been shown that Maxwell’s Equations are compatible with a supplementary conservation law [1]; but this property was demonstrated only in the case of the empty space
Maxwell’s equations in materials must necessarily be coupled with the balance equations of this material and we begin to couple them with the Euler equations for polyatomic gases; the whole set of equations is:
As usual in Extended Thermodynamics, restrictions on these functions can be found by imposing the Entropy Principle which requires the existence of the entropy-entropy flux 4-vector hα and of the entropy production Σ such that the following supplementary equation holds for each solution of the system (1)1,2:
Summary
Up to now it has been shown that Maxwell’s Equations are compatible with a supplementary conservation law [1]; but this property was demonstrated only in the case of the empty space. Gαγ is the Lorentz 4-force, Jβ is the free current density and, in any fixed reference frame, the tensors Fαβ and Gαβ can be decomposed as follows:. For references on this subject, see for example [2–7] which contain only marginally the results of the present article (for example, Maxwell equations are not coupled with the equations for the material), or belong to another context such as general relativity, quantistic mechanics or the use of a Lagrangian function. It follows that it is necessary to express a part of these components as functions of the rest; they are called constitutive functions and “the closure problem” deals with how to find them. We adopt well-known procedures which we describe
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