Magnetic fields of spherical, cylindrical, and elipsoidal electric charge superficial distributions at rotation

Abstract

The vector potentials $ A( r)$ produced by spherical, cylindrical, and elipsoidal uniform superficial distributions of electrical charge rotating at a constant angular velocity $ \omega$, are found. This is done by modeling such a distributions as if they were simple bobbins made of $N$ loops of a very thin coil carrying a current $I$ and calculating simply the dipolar potential $ A_{\text{dip}}( r)$ produced by them. Due that in the case of the spherical geometry the potential $ A( r)$ has already been calculated its value is used as a consistence test of the present approach, for the two other geometries the analytical calculation of the potentials is not so trivial by this reason the equalness between $ A_{\text{dip}}( r)$ and $ A( r)$ is proved trough a numerical evaluation of the complex integrals appearing in the Biot-Savart expression for $ A( r)$. The respective magnetic fields generated by these three rotating distributions have an identical structure: they are constant inside the surfaces while outside them they are dipolar-like (nearby to radiation zone). An application of the above results to quark confinement inside hadrons is proposed.