Relativistically spinning charged sphere

When the equatorial spin velocity $v$ of a charged conducting sphere approaches $c$, the Lorentz force causes a remarkable rearrangement of the total charge $q$. Charge of that sign is confined to a narrow equatorial belt at latitudes $b\le \sqrt{3}(1-v^2/c^2{)}^{1/2}$ while charge of the opposite sign occupies most of the sphere’s surface. The change in field structure is shown to be a growing contribution of the “magic” electromagnetic field of the charged Kerr-Newman black hole with Newton’s $G$ set to zero. The total charge within the narrow equatorial belt grows as $(1-v^2/c^2{)}^{-(1/4)}$ and tends to infinity as $v$ approaches $c$. The electromagnetic field, Poynting vector, field angular momentum, and field energy are calculated for these configurations. Gyromagnetic ratio, $g$ factor, and electromagnetic mass are illustrated in terms of a 19th century electron model. Classical models with no spin had the small classical electron radius $e^2/m{c}^2\sim$ a hundredth of the Compton wavelength, but models with spin take that larger size but are so relativistically concentrated to the equator that most of their mass is electromagnetic. The method of images at inverse points of the sphere is shown to extend to charges at points with imaginary coordinates.

Figure 1

The electric lines of force for a relativistically rotating conducting spherical shell with $u=0.9$, i.e., $\gamma =19$, $v/c=0.9986$. Notice the charge concentration at the equator and the diminution of the field strength as the lines cross the sphere again, due to the opposing charge there. In contrast, the static shell has a radial field outside and none inside.