The general motion of a spinning uniformly and rigidly electrifield sphere, III

Abstract

This investigation generalizes and shortens the method of determining the total mechanical reaction due to the electromagnetic field generated by the motion of a rigidly and uniformly electrified sphere, so as to make it apply to the most general case in which the centre of the sphere describes a tortuous curve in any manner with speed always less than that of light, whilst the sphere is spinning about any generally variable diameter with unrestricted angular velocity. It proved necessary at quite an early stage to consider this general case because it was found that a curvilinear motion of the centre of the sphere, a part from exceptional cases such as the rectilinear motion and the uniform circular motion, always generates a couple, which will produce a spin in the absence of some counteracting external couple, which such as might arise in a magnetic field. Conversely the spin was found to react on the orbital motion, generating in general both tangential and normal force components. The chief results obtained are as follows: (1) General expressions for the force and couple constituents are found in 5, superseding II (2·14) (Schott 1936 b , p. 492), but like it involving definite single integrals, whose limits depend only on the orbital motion of the centre and not at all on the spin of the sphere, so that the angular velocity of the spin is unrestricted both in magnitude and direction. Consequently, although the speed of the centre is still restricted to be less then that of light, that of any surface element can have any value greater or less than that of light, the ends of the spin axis of course excepted.