Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/389

as a rotation of conduction-currents under the influence of a magnetic field; and if it be assumed that displacement-currents in dielectrics are rotated in the same way, the Faraday effect may evidently be explained. Considering the matter from the analytical point of view, the Hall effect may be represented by the addition of a term k [K.S] to the electromotive force, where K denotes the impressed magnetic force, and S denotes the current: so Rowland assumed that in dielectrics there is an additional term in the electric force, proportional to [K.Ḋ], i.e. proportional to the rate of increase of [K.D]. Now it is universally true that the total electric force round a circuit is proportional to the rate of decrease of the total magnetic induction through the circuit: so the total magnetic induction through the circuit must contain a term proportional to the integral of [K.D] taken round the circuit: and therefore the magnetic induction at any point must contain a term proportional to [curl K.D]. We may therefore write

$\mathbf {B} =\mathbf {H} +\sigma {\text{ curl }}[\mathbf {K.D} ]$,

where σ denotes a constant. But if this be combined with the customary electromagnetic equations

${\text{curl }}\mathbf {H} =4\pi \mathbf {\dot {D}} ,\qquad {\text{curl }}\mathbf {E} =-\mathbf {\dot {B}} ,\qquad \mathbf {D} =\epsilon \mathbf {E} /4\pi c^{2}$,

and all the vectors except B be eliminated (K being treated as a constant), we obtain the equation

$\mathbf {\ddot {B}} =(c^{2}/\epsilon )\nabla ^{2}\mathbf {B} +(\sigma /4\pi ){\text{ curl }}(\partial ^{2}\mathbf {B} /\partial t\partial \theta )$,

where ∂/∂θ stands for K_x∂/∂x + K_y∂/∂y + K_z∂/∂z (K20/0c + Ky0/0y + K20/02); and this is identical with the equation which Maxwell had given^[1] for the motion of the aether in magnetized media. It follows that the assumptions of Maxwell and of Rowland, different though they are physically, lead to the same analytical equations-at any rate so far as concerns propagation through a homogeneous medium.

The connexions of Hall's phenomenon with the magnetic rotation of light, and with the reflexion of light from magnetized

1. Cf. p. 308.

2 B