Page:Motion of Electrification through a Dielectric.djvu/6

But this is not all. There is possibly a fourth term in F, expressed by 4πVDG, where D is the displacement and G the magnetic current; I have termed this force the "magneto-electric force," because it is the analogue of Maxwell's "electromagnetic force," VCB. Perhaps the simplest way of deriving it is from Maxwell's electric stress, which was the method I followed.^[1]

Thus, in a homogeneous nonconducting dielectric free from electrification and magnetization, the mechanical force is the sum of the "electromagnetic" and the "magnetoelectric," and is given by

$\scriptstyle {\mathbf {F} ={\frac {1}{v^{2}}}{\frac {d\mathbf {W} }{dt}},}$

where W=VEH/4π is the transfer-of-energy vector.

It must, however, be confessed that the real distribution of the stresses, and therefore of the forces, is open to question. And when ether is the medium, the mechanical force in it, as for instance in a light-wave, or in a wave sent along a telegraph-circuit, is not easily to be interpreted.)

The companion to (11) in a nonconducting dielectric is now

$\scriptstyle {\mathrm {curl} ~\mathbf {H} =cp\mathbf {E} +4\pi \rho u.\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(13)}}}$

Eliminate E between (11) and (13), remembering that H is circuital, because μ₀ is constant, and we get

$\scriptstyle {\left(p^{2}/v^{2}-\nabla ^{2}\right)\mathbf {H} =\mathrm {curl} ~4\pi \rho u,\quad \cdots \cdots \cdots \cdots \cdots \cdots {\text{(14)}}}$

the characteristic of H. Here $\scriptstyle {\nabla ^{2}=d^{2}/dx^{2}+\cdots ,}$ as usual.

Comparing (14) with the characteristic of H when there is impressed force e instead of electrification ρ, which is

$\scriptstyle {\left(p^{2}/v^{2}-\nabla ^{2}\right)\mathbf {H} =\mathrm {curl} ~cp\mathbf {e} ,}$

we see that ρu becomes cpe/4π. We may therefore regard convection-current as impressed electric current. From this comparison also, we may see that an infinite plane sheet of electrification of uniform density cannot produce magnetic force by motion perpendicular to its plane. Also we see that the sources of disturbances when ρ is moved are the places where ρu has curl; for example, a dielectric sphere uniformly filled with electrification (which is imaginable), when moved, starts the magnetic force solely upon its boundary.

The presence of "curl" on the right side tells us, as a matter of mathematical simplicity, to make H/curl the variable. Let

$\scriptstyle {\mathbf {H} =\mathrm {curl} ~\mathbf {A} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(15)}}}$

and calculate A, which may be any vector satisfying (15). Its characteristic is

$\scriptstyle {\left(p^{2}/v^{2}-\nabla ^{2}\right)\mathbf {A} =4\pi \rho \mathbf {u} .\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(16)}}}$

The divergence of A is of no moment, and it is only vexatious complication to introduce ψ. The time-rate of decrease of A is not the real

1. "El. Mag. Ind. and its Prop." xxii. The Electrician, Jan. 15, 1886, p. 187. [vol. I., p. 545].