Self-Consistent Field of Single-Type Electrons in a Uniform Magnetic Field

Abstract

A problem which has been brought to the fore by Christofilos' proposed thermonuclear machine, "Astron," is that of electrons rotating in a uniform impressed magnetic field in such a way as to form a cylinder in space---a disembodied current-carrying helix. In the present paper the self-consistent magnetic field for such a "helix," infinitely long and made up of relativistic electrons of identical mass, axial velocity and canonical angular momentum, has been calculated for the interesting range of the two pertinent parameters. The first is the ratio, denoted by ${g}_{2}$, of apogee radius to the radius of gyration in the impressed field. The second is the quotient of the number of electrons per classical-electron diameter along the axis and the ratio of relativistic to rest mass. It is denoted by $(\frac{2{r}_{e}}{\ensuremath{\gamma}})N$ and is called the "strength" of the layer.When ${g}_{2}$ is less than 2, the trajectories of the electrons, when they are few in number, are approximately circles which include the axis. As the layer strength is increased, these trajectories stay closer and closer to the locus of apogees. The interior field never reverses and approaches $\frac{2}{{g}_{2}}\ensuremath{-}1$ as the strength reaches $2({g}_{2}\ensuremath{-}1)$, the largest value possible for a given ${g}_{2}$.When ${g}_{2}$ is greater than 2, the trajectories in a weak layer are approximately circles which do not include the axis, but with increasing strength the perigees approach, cross, and then recede from the axis. As they cross, the interior field decreases discontinuously even to the point of partial reversal. The approach to this discontinuity is attended by an enhancement of the interior field.When ${g}_{2}$ is greater than approximately 4, the layer strength peaks at a value in excess of $2({g}_{2}\ensuremath{-}1)$ at the axis crossing, and as the perigee locus then moves outward, the strength decreases toward $2({g}_{2}\ensuremath{-}1)$. Thus there can be two configurations of the layer which have the same strength, one consistent with enhancement of the interior field, the other consistent with reversal.