Gavin R. Putland,  BE PhD

Friday, March 31, 2017 (Comment)

EM waves with a flux-preserving ray velocity

In a time-varying electromagnetic (EM) field, let us say that a certain velocity field preserves magnetic flux if and only if the magnetic flux through every closed curve, every part of which moves at that velocity, is constant. In the case of a directed beam of EM radiation, it is obvious that any flux-preserving velocity field is to be identified with the ray velocity. Let us therefore call it r.

If the velocity field r preserves magnetic flux, Faraday's law reduces to

E = −r × B .

Similarly, if r preserves electric displacement flux, the Maxwell-Ampère law (with no conduction current) reduces to

H = r × D .

These equations yield a simple theory of EM waves, including a derivation of Fresnel's equation for the ray-velocity surface of a biaxial birefringent crystal.

Taking cross-products of the above equations with the wave slowness s gives, respectively,

B = s × E ;
D = −s × H .

The last two equations, by analogy with the first two, yield Hamilton's wave-slowness surface.

For sinusoidal waves, we can write k/ω for s , so that the last two equations become

ωB = k × E ,
ωD = −k × H ,
which may be more familiar.

[Revised 9 June 2018; last modified 11 June 2018.]

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