Page:Electromagnetic effects of a moving charge.djvu/8

vary inversely as its distance from the apex, and the forces, electric and magnetic, would therefore vary inversely as the square root of the distance from the apex, instead of inversely as the distance, which is obviously necessary in order that the displacement may be confined to the surface. This conflict of conditions constitutes instability. In the Phil. Mag. for April, 1889, I suggested that whilst there must be a solution of some kind, one representing a steady state was impossible. This conclusion is confirmed by the failure of Prof. Thomson's proposed surface-wave to keep itself going.

Prof. Thomson, who otherwise confirms my results, has also extended the matter by supposing that the medium itself is set in motion, as well as the electrification. This is somewhat beyond me. I do not yet know certainly that the ether can move, or its laws of motion if it can. Fresnel thought the earth could move through the ether without disturbing it; Stokes, that it carried the ether along with it, by giving irrotational motion to it. Perhaps the truth is between the two. Then there is the possibility of holes in the ether, as suggested by a German philosopher. When we get into one of these holes, we go out of existence. It is a splendid idea, but experimental evidence is much wanting.

But if we consider that the medium supporting the electric and magnetic fluxes is really set moving when a body moves, and assume a particular kind of motion, it is certainly an interesting scientific question to ask what influence the motion exerts on the electromagnetic phenomena. I do not, however, think that any new principles are involved.

The general connections of E and H, referred to fixed space without conductivity, being

${\begin{aligned}\scriptstyle {\mathrm {curl} (\mathbf {e} -\mathbf {E} )=\mu p\mathbf {H} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots }&\scriptstyle {\text{(1)}}\\\scriptstyle {\mathrm {curl} (\mathbf {H} -\mathbf {h} )=cp\mathbf {E} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots }&\scriptstyle {\text{(2)}}\\\end{aligned}}$

where p stands for d/dt and e and h are the impressed parts of E and H; if there is also motion of electrification, we have to consider it to constitute a convection-current, a part of the true current, and so make (2) become

$\scriptstyle {\mathrm {curl} (\mathbf {H} -\mathbf {h} )=cp\mathbf {E} +4\pi \rho \mathbf {u} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdot \cdot {\text{(3)}}}$

where ρ is the density of electrification, whose velocity is u. [See Part II.] It now remains to specify e and h. They are zero when the medium supporting the fluxes is at rest. But if it moves, and its velocity is w, there is, first, the electric force due to motion in a magnetic field,

$\scriptstyle {\mathbf {e} =\mu \mathbf {VwH} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(4)}}}$

which is well known; and next the magnetic force due to motion in an electric field,

$\scriptstyle {\mathbf {h} =c\mathbf {VEw} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(5)}}}$