3D Electron Path in Cylindrically Symmetric B-field

These equations for electron trajectories through a cylindrically symmetric azimuthal B-field were applied to the following gizmo. See page 12 (above) for a small schematic of it. A toroidal coil of fine wires was constructed, supported by a thin open plastic framework. A current pulse applied to the coil produced a uniform azimuthal B-field within the torus. This device was a magnetic lens. Its purpose was to focus electrons emitted at a point source, located on the axis of the torus, and some distance away (say, below). The relativistic electrons (generated from gamma rays at the source volume) are bent on transiting the torus, and focus to a point (say, above some distance along the axis), where a detector measures their arrival. Lead blocks with slits, between the source and the torus, select a specific angle of emitted electrons, and thus (by the Compton energy-angle formula) of a specific energy. The idea was for this magnetic lens to act as a monochromator. The detector measures the time history of the electron pulse, and thus of the gamma pulse. It worked great. I was fortunate to have a big team of excellent engineers to devise the hardware.