Magnetic flux density B may vary in time 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, the electric displacement field D may vary in time because the field changes and/or because the field moves relative to the observation point. The Maxwell-Ampère law makes no distinction between these causes. But, by analogy with the above, the latter cause can be isolated by saying that a D field moving at velocity r induces the magnetizing field
H = r × D .
The above equations yield a simple theory of electromagnetic waves, including a derivation of Fresnel's equation for the ray-velocity surface of a 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.