Laminar-flow equilibria and limiting currents in magnetically focused relativistic beams

A self-consistent theory of laminar-flow equilibria in magnetically focused relativistic beams propagating inside a conducting tube that confines both the electric and the magnetic self-fields is presented. In the differential equations describing the equilibrium state, the canonical angular momentum, pϑ, defined by conditions at the source, uniquely determines the possible solutions. Three cases, pϑ=0 (magnetically shielded source), pϑ=const (hollow beam with immersed source), and pϑ∝r2 (solid beam with immersed source) are discussed. The case pϑ=0 is examined in some detail and an exact analytical expression for the limiting current is derived, which is compared with approximate formulae reported in the literature. It is shown that higher currents can be achieved by injecting the beam from a magnetically shielded source as compared with an immersed source.