Page:Motion of Electrification through a Dielectric.djvu/5

It may be remarked that (if my calculations are correct) equation (7) or its equivalents expresses the mutual energy of any two rational current-elements (see § 1) in a medium of uniform inductivity, of moments q₁u₁, and q₂u₂, whether the currents be of displacement, or conduction, or convection, or all mixed, it being in fact the mutual energy of a pair of definite magnetic fields. But, since the hypothesis of instantaneous action is expressly involved in the above, the application of (7) is of a limited nature.

General Theory of Convection Currents.

8. Now leaving behind altogether the subject of current-elements, in the investigation of which one is liable to be led away from physical considerations and become involved in mere exercises in differential coefficients, and coming to the question of the electromagnetic effects of a charge moving in any way, I have been agreeably surprised to find that my solution in the case of steady rectilinear motion, originally an infinite series of corrections, easily reduces to a very simple and interesting finite form, provided u be not greater than v. Only when u>v is there any difficulty. We must first settle upon what basis to work. First the Faraday-law (p standing for d/dt),

$\scriptstyle {-\mathrm {curl} \mathbf {E} =\mu _{0}p\mathbf {H} ,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(11)}}}$

requires no change when there is moving electrification. But the analogous law of Maxwell, which I understand to be really a definition of electric current in terms of magnetic force, (or a doctrine), requires modification if the true current is to be

$\scriptstyle {\mathbf {C} +p\mathbf {D} +\rho \mathbf {u} ;\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(12)}}}$

viz. the sum of conduction-current, displacement-current, and convection-current ρu, where ρ is the volume-density of electrification. The addition of the term ρu was, I believe, proposed by G. F. Fitzgerald.^[1]

(This was not meant exactly for a new proposal, being in fact after Rowland's experiments; besides which, Maxwell was well acquainted with the idea of a convection-current. But what is very strange is that Maxwell, who insisted so strongly upon his doctrine of the quasi-incompressibility of electricity, never formulated the convection-current in his treatise. Now Prof. Fitzgerald pointed out that if Maxwell, in his equation of mechanical force,

$\scriptstyle {\mathbf {F} =\mathbf {VCB} -e\nabla \Psi -m\nabla \Omega ,}$

had written E for $\scriptstyle {-\nabla \Psi ,}$ as it is obvious he should have done, then the inclusion of convection-current in the true current would have followed naturally. (Here C is the true current, B the induction, e the density of electrification, m that of imaginary magnetic matter, Ψ the electrostatic and Ω the magnetic potential, and E the real electric force.)

Now to this remark I have to add that it is as unjustifiable to derive H from Ω as E from ψ; that is, in general, the magnetic force is not the slope of a scalar potential; so, for $\scriptstyle {-\nabla \Omega }$ we should write H, the real magnetic force.

1. Brit. Assoc., Southport, 1883.