The Motion of a Charged Sphere. The Condition at a Surface of Equilibrium (Footnote).
14. If, in the solutions (29), we terminate the fields internally at r=a, the perpendicularity of E and the tangentiality of H to the surface show that (29) represents the solutions in the case of a perfectly conducting sphere of radius a, moving steadily along the z-axis at the speed u, and possessing a total charge q. The energy is now finite. Let U be the total electric and T the total magnetic energy. By space-integration of the squares of E and H we find that they are given by
![{\displaystyle {\begin{aligned}&\scriptstyle {U={\frac {q^{2}}{2ca}}\cdot {\frac {1-u^{2}/v^{2}}{4}}\left[1+{\frac {\frac {3}{2}}{1-u^{2}/v^{2}}}+{\frac {{\frac {3}{2}}\tan ^{-1}{\frac {u/v}{\left(1-u^{2}/v^{2}\right)^{\frac {1}{2}}}}}{(u/v)(1-u^{2}/v^{2})^{\frac {3}{2}}}}\right],\quad \cdots \cdots \cdots \cdots \cdots \cdots {\text{(36)}}}\\&\scriptstyle {T={\frac {q^{2}}{2ca}}\cdot {\frac {1-u^{2}/v^{2}}{4}}\left[1+{\frac {2u^{2}/v^{2}-{\frac {1}{2}}}{1-u^{2}/v^{2}}}+{\frac {\left(2u^{2}/v^{2}-{\frac {1}{2}}\right)\tan ^{-1}{\frac {u/v}{\left(1-u^{2}/v^{2}\right)^{\frac {1}{2}}}}}{(u/v)(1-u^{2}/v^{2})^{\frac {3}{2}}}}\right],\quad {\text{(37)}}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/873caa4c2c2d0ecdcc544ac2a11582a078d40abb)
in which u<v. When u=v, with accumulation of the charge at the equator of the sphere, we have infinite values, and it appears to be only possible to have finite values by making a zone at the equator cylindrical instead of spherical. The expression for T in (37) looks quite wrong; but it correctly reduces to that of equation (2) when u/v is infinitely small.[1]
- ↑ [I am indebted to Mr. G. F. C. Searle, of Cambridge, for the opportunity of making a somewhat important correction before going to press. In a private communication (August 19, 1892) he informed me that he had verified the accuracy of the solution for a point-charge, which he had also obtained in another way, from equations equivalent to (33), without the use of the function A of §§ 8 to 10; but he cast doubt upon the validity of the extension made in § 14, from a point-charge to a charged conducting sphere, and asked the plain question (in effect), What justification is there for terminating the displacement perpendicularly, to make a surface of equilibrium?
On examination, I find that there is no justification whatever, exceptions excepted. The true boundary condition may, however, be found without a fresh investigation. On p. 499 the problem of uniform motion of electrification through a dielectric medium, or conversely, of the uniform motion of the whole medium past stationary electrification, is reduced to a case of eolotropy in electrostatics. The effect of the motion of the isotropic medium on the displacement emanating from stationary electrification is there shown to be identical with the effect of keeping the medium stationary and reducing its permittivity in lines parallel to the (abolished) motion from c to c(1-u²/v²), whilst keeping the transverse permittivity the same. The transverse concentration of the displacement is obvious. Now the function P (equation (14), p. 499) is the electrostatic potential in the stationary eolotropic problem, so that its slope , which call F, is the electric force, and the displacement D is a linear function thereof, say D=λF, where λ is the permittivity operator. The condition of equilibrium is that F is perpendicular to the surface where it terminates, this being required to make curl F=0, or the voltage zero in every circuit. Now, in the corresponding problem of the same electrification in a moving isotropic medium, we have the same function P (no longer the electrostatic potential) and the same derived vector F, whilst the displacement D is also derived from F in the same way. But whilst the meaning of D is the same in both cases, that of F is not. In the eolotropic case, F is the
