Solution of a set of Maxwell-Lorentz equations for a ring relativistic electron beam

Nonlinear solutions to a set of Maxwell’s equations and the relativistic equations of electron motion are obtained that describe the equilibrium of a high-power ring relativistic electron beam against the background of immobile ions. By transforming the basic equations, a set of equations for a three-component vortex vector field is derived that describes ring beam configurations for plasma confinement. An example of a numerical calculation of the steady state of a compact beam torus of immobile ions and relativistic electrons is presented.