The theorem on the magnetic field of rotating charged bodies

TL;DR

This paper derives simplified formulas for calculating the magnetic field of rotating charged bodies using retarded potentials, emphasizing axisymmetric cases and providing practical integral expressions.

Contribution

It introduces a theorem that expresses the magnetic field of rotating charged bodies through surface or volume integrals, simplifying calculations for axisymmetric and arbitrary charge distributions.

Findings

Magnetic field on the rotation axis has only one component along the axis.

The magnetic field can be expressed via surface or volume integrals without azimuthal angle integration.

Derived formulas enable quick calculation of external and central magnetic fields.

Abstract

The method of retarded potentials is used to derive the Biot-Savart law, taking into account the correction that describes the chaotic motion of charged particles in rectilinear currents. Then this method is used for circular currents and the following theorem is proved: The magnetic field on the rotation axis of an axisymmetric charged body or charge distribution has only one component directed along the rotation axis, and the magnetic field is expressed through the surface integral, which does not require integration over the azimuthal angle . In the general case, for arbitrary charge distribution and for any location of the rotation axis, the magnetic field is expressed through the volume integral, in which the integrand does not depend on the angle . The obtained simple formulas in cylindrical and spherical coordinates allow us to quickly find the external and central magnetic field of rotating bodies on the rotation axis.