The uniform circular motion with invariable normal spin of a rigidly and uniformly electrified sphere, IV

Abstract

In order to exhibit clearly and fully the possibilities inherent in classical electrodynamics when it is developed rigorously without approximations or unnecessary restrictions I have in this paper worked out completely the case in which the centre of the sphere describes a circle with any uniform speed less than that of light whilst it is spinning about a diameter normal to the plane of the circle with an invariable angular velocity unrestricted in magnitude or sense. It is to be noted that, although the speed of the centre is assumed for the sake of simplicity to be less than that of light, that of points on the surface of the sphere (other than the ends of the axis of spin) can be as large as we please. In §§ 2–5 general expressions for the tangential and normal force constituents and the couple constituent of the total reaction on the sphere of its own electromagnetic field are obtained from the general expressions given in paper III (§ 8), and the resulting equations of motion are written down (cf. (2·1)–(2·3) and (5·7)). There are two points to be noticed: (1) the tangential and normal force constituents are quadratic polynomials in the spin p with coefficients depending on the speed cβ of the centre and the radius R of its orbit, whilst the couple constituent is linear in p ; these results are true for any orbit with invariable spin. (2) The couple is found in § 5 to vanish identically, i. e. for all values of p, β and R , in the case of a circular orbit, owing to its symmetry with respect to a diameter; for this reason the result is probably peculiar to this class of orbit.