Classical Physics


Simple Theory of Electromagnetic Waves and Biaxial Birefringence: from Lorentz Force to Conical Refraction in 37 Pages

Authors: Gavin R. Putland

A time-variation in magnetic flux density B may occur because the field changes and/or because the field moves relative to the observation point. Faraday's law for a fixed circuit makes no distinction between these causes. But the latter cause is isolated by the magnetic term in the Lorentz force law, which, in a reference frame fixed with respect to the particle, implies that a field B moving at velocity r induces an electric field E = −r × B. In the case of a traveling electromagnetic wave, r is the ray velocity (hence the symbol).

Similarly, a time-variation in the electric displacement field D may occur because the field changes and/or because the field moves. The Maxwell-Ampère law makes no distinction between these causes. But, by analogy with the Lorentz force law, the latter cause can be isolated by saying that a D field moving at velocity r induces a magnetizing field H = r × D.

The two "moving field" laws, combined with the relations between D and E and between B and H, yield an unusually simple theory of electromagnetic waves, including a derivation of Fresnel's equation for the ray-velocity surface of a non-chiral birefringent crystal. Taking cross-products of the "moving field" laws with the wave-slowness vector, we obtain two more "moving field" equations in terms of wave slowness (generalizing the conventional formulation in terms of the wave vector). The last two equations, by analogy with the first two, yield Hamilton's wave-slowness surface. Comparing the results, we can conclude that the ray-velocity and wave-slowness surfaces of a biaxial crystal have curves of contact with tangent planes, and deduce the associated polarizations. Eigenvectors are introduced to show that, in general, the permitted polarizations for a given propagation direction are orthogonal. A coordinate transformation (simpler than Hamilton's) shows that the curves of contact are circles and yields their linear and angular diameters.

Among the footnotes are interpretations of the Poynting vector and the Minkowski momentum density. The text includes introductory material intended to make it comprehensible to high-school graduates.

(P.S.: In this abstract, vectors are shown in italics because boldface is not permitted.)

Comments: 41 pages (main text: 37 pages).

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Submission history

[v1] 2017-05-19 01:41:04

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