Page:Motion of Electrification through a Dielectric.djvu/2

and when this is constant everywhere, as we shall suppose now and later, the magnetic force is simply the circuital vector whose curl is 4π times the electric current; or the vector-potential of the curl of the current; or the curl of the vector-potential of the current, etc., etc. Thus, as found by J. J. Thomson,^[1] the magnetic field of a charge moving at a speed which is a small fraction of that of light is that which is commonly ascribed to a current-element itself. I think it, however, preferable to regard the magnetic field as the primary object of attention; or else to regard the complete system of closed current derived from it by taking its curl as the unit, forming what we may term a rational current-element, inasmuch as it is not a mere mathematical abstraction, but is a complete dynamical system involving definite forces and energy.

2. Let the axis of z be the line of motion of the charge q at the speed u; then the lines of magnetic force H are circles centred upon the axis, in planes perpendicular to it, and its tensor H at distance r from the charge, the line r making an angle θ with the axis, is given by

$\scriptstyle {H={\frac {q}{r^{2}}}u\sin \theta =cEuv,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(1)}}}$

where v=sin θ, E the intensify of the radial electric force, c the permittivity such that μ₀cv²=1, if μ₀ is the other specific quality of the medium, its inductivity, and v is the speed of propagation.

Since, under the circumstance supposed of u/v being very small, the alteration in the electric field is insensible, and the lines of E are radial, we may terminate the fields represented by (1) at any distance r=a from the origin. We then obtain the solution in the case of a charge q upon the surface of a conducting sphere of radius a, moving at speed u. This realization of the problem makes the electric and magnetic energies finite. Whilst, however, agreeing with J. J. Thomson in the fundamentals, I have been unable to corroborate some of his details; and since some of his results have been recently repeated by him in another place,^[2] it may be desirable to state the changes I propose, before proceeding to the case of a charge moving at any speed.

The Energy and Forces in the Case of Slow Motion.

3. First, as regards the magnetic energy, say T. This is the space-summation Σμ₀H²/8π; or, by (1),^[3],

$\scriptstyle {T={\frac {\mu _{0}q^{2}u^{2}}{8\pi }}\int \int \int {\frac {v^{2}}{r^{2}}}dr\ d\mu \ d\phi ={\frac {\mu _{0}q^{2}u^{2}}{3a}}.\quad \cdots \cdots \cdots \cdots \cdots \cdots {\text{(2)}}}$

The limits are such as include all space outside the sphere r=a. The coefficient $\scriptstyle {\frac {1}{3}}$ replaces $\scriptstyle {\frac {2}{15}}$.

4. Next, as regards the mutual magnetic energy M of the moving charge and any external magnetic field. This is the space-summation

1. Phil. Mag., April, 1881.
2. "Applications of Dynamics to Physics and Chemistry," chap. iv. pp. 31 to 37.
3. The Electrician, Jan. 24, 1885, p. 220. [vol. I., p. 446].