Nonlinear dependence of intensity effects on the number of particles in the ring current

The problem of the ring current self-action is considered. The exact Lienar-Vichert formulae are drawn for the fields of a particle moving along a ring with a constant velocity. With the help of the Plan summation method there is found a Lorentz force acting on a particle from all the other particles. A nonlinear dependence on the number of particles in the ring current has been obtained. Several specific cases of the formulae utilization are considered. The straight line path approximation is shown not to be applicable to the problem considered in the altrarelativistic range at any current density. The results obtained show that it is desirable to take account for the accurate formula for the particle field and beam current discretness when calculating intensity effects