Abstract
We analyze the propagation of electromagnetic fronts in unbounded electric conductors. Our analysis is based on the Maxwell model of electromagnetism that includes the displacement current and Ohm’s law in its simplest forms. A weak electromagnetic front is a propagating interface at which the electromagnetic field remains continuous while its first- and higher-order derivatives experience finite jump discontinuities. Remarkably, analysis of such fronts can be performed autonomously, i.e. strictly in terms of the quantities defined on the front. This property opens the possibility of establishing exact analytical solutions of the exact Maxwell system along with the evolution of the front.
Highlights
Maxwell’s theory of electromagnetism was initially met with skepticism
A weak electromagnetic front is a propagating interface at which the electromagnetic field remains continuous while its first- and higher-order derivatives experience finite jump discontinuities
Where ∆iE, ∆iH are unit vectors known as the directors, and AE, AH are the magnitudes of the jump discontinuity vectors
Summary
Maxwell’s theory of electromagnetism was initially met with skepticism. upon Hertz’s experimental verification of some of its key predictions, it began gaining traction and eventually became the dominant model for electricity and magnetism. Of particular value are solutions to the nonlinear version of the equations In this regard, of considerable interest are exact wavefronts, i.e. moving surfaces along which the electromagnetic field experiences discontinuities. Exact solutions to Maxwell’s equations, i.e. solutions found without making approximations of any kind, are of great value since they exhibit the essential features of the system that include various symmetries and conservation laws These critical features are typically not preserved by approximation procedures, such as linearization. The key tool in the analysis of wavefronts is compatibility conditions, originally developed by Hadamard [1], Levi-Civita [2], and Thomas [3]. The method of compatibility conditions, though applicable to nonlinear problems, faces its own difficulties and limitations One of those difficulties stems from the technical complexity of the conditions themselves. The tensorial aspects of the theory of compatibility conditions have been discussed in detail in [8] where the technique received further analytical development
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