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The dynamics of hollow or solid beams in axial magnetic fields are analyzed, assuming electrons do not cross each other radially. The focusing properties are due to the angular velocity, which depends on magnetic flux linkages in the beam and at the cathode. The position of the cathode in the magnetic field is thus of vital importance. Radial oscillations about an equilibrium radius are always stable in the presence of a magnetic field, and can be made stable even without a magnetic field. Design formulas are presented for two cathode arrangements; in the first the cathode is in a uniform magnetic field, while in the second it is inside a magnetically shielded structure. Special cases of each of these arrangements are considered. Limited experimental results confirm most of the theory presented, and indicate the possibility of focusing without a magnetic field over the beam.