Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/301

the constant μ be supposed to have the value unity, the equations may be written

${\begin{cases}\mathrm {div} \ \mathbf {H} &=&0,\\c_{1}^{2}\mathrm {curl} \ \mathbf {H} &=&{\overset {.}{\mathbf {E} }},\\-\mathrm {curl} \ \mathbf {H} &=&{\overset {.}{\mathbf {H} }}.\end{cases}}$

Eliminating E, we see^[1] that H satisfies the equations

${\begin{cases}\mathrm {div} \ \mathbf {H} &=&0,\mathrm {div} \ {\overset {.}{\mathbf {H} }}&=&c_{1}^{2}\nabla ^{2}\mathbf {H} .\end{cases}}$

But these are precisely the equations which the light-vector satisfies in a medium in which the velocity of propagation is c₁: it follows that disturbances are propagated through the model by waves which are similar to waves of light, the magnetic (and similarly the electric) vector being in the wave-front. For a plane-polarized wave propagated parallel to the axis of z, the equations reduce to

$-c_{1}^{2}{\frac {\partial H_{y}}{\partial z}}={\frac {\partial E_{x}}{\partial t}},c_{1}^{2}{\frac {\partial H_{x}}{\partial z}}={\frac {\partial E_{y}}{\partial t}},{\frac {\partial E_{y}}{\partial z}}={\frac {\partial H_{x}}{\partial t}},-{\frac {\partial E_{x}}{\partial z}}={\frac {\partial H_{y}}{\partial t}}$

whence we have

$c_{1}H_{y}=E_{x}$,    $-c_{1}H_{z}=E_{y}$

these equations show that the electric and magnetic vectors are at right angles to each other.

The question now arises as to the magnitude of the constant c₁.^[2] This may be determined by comparing different expressions for the energy of an electrostatic field. The work done by an electromotive force E in producing a displacement D is

$\int _{0}^{\mathbf {D} }\mathbf {E.} d\mathbf {D}$  or  ${\tfrac {1}{2}}\mathbf {ED}$.

per unit volume, since E is proportional to D. But if it be assumed that the energy of an electrostatic field is resident in the dielectric, the amount of energy per unit volume may be

1. For if a denote any vector, we have identically

   $\nabla ^{2}\mathbf {a} +\mathrm {grad\ div} \ \mathbf {a} +\mathrm {curl\ curl} \ \mathbf {a} =0$

2. For criticisms on the procedure by which Maxwell determined the velocity of propagation of disturbance, cf. P. Duhem, Les Théories Electriques de J. Clerk Maxwell, Paris, 1902.