THE THEOREM ON THE MAGNETIC FIELD OF ROTATING CHARGED BODIES

Sergey G Fedosin

doi:10.2528/pierm21041203

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.

Highlights

In the general case, stationary motion of a charged particle consists of rectilinear motion at a constant velocity and rotational motion at a constant angular velocity
If we consider the stationary motion of a set of particles or motion of a charged body, the total magnetic field of the system can be found based on the superposition principle as the sum of the magnetic field vectors of individual particles
We have proved the following theorem: 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 φ

Summary

INTRODUCTION

Stationary motion of a charged particle consists of rectilinear motion at a constant velocity and rotational motion at a constant angular velocity. Let us suppose that a certain volume V is uniformly filled with a set of moving charge distributions, so that V = dV , dV = dV , and we need to find the total magnetic field outside the volume V In this case, we can use the superposition principle for the magnetic fields. If the motion of charged particles is rectilinear, the magnetic field is expressed only in terms of E in full accordance with the Lorentz transformations for the components of the electromagnetic field tensor in inertial reference frames. This once again underlines the difference between rectilinear motion and rotational motion and the need for different formulas for the magnetic field, depending on the type of motion. We will briefly present the relativistic expression for the Biot-Savart law and estimate its accuracy, and in Section 3 we will pass on to the analysis of rotational motion of charges and currents and to the proof of the theorem on the magnetic field for this case

RECTILINEAR MOTION OF CHARGES

ROTATIONAL MOTION OF CHARGES

MAGNETIC FIELD ON THE CYLINDER’S AXIS

MAGNETIC FIELD ON THE BALL’S AXIS

CONCLUSION