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Efficient operation of gyro-devices depends critically on the quality of the beam. Single-particle motion of relativistic electrons in a cusped magnetic field has been studied analytically. The Lagrangian formulation is used to solve for the electron trajectories in a general magnetic cusp configurations. The exact configurations of the solenoid coils and pole pieces were designed to generate the magnetic reversal field. Results from computer aided design of a novel electron gun generating axis-encircling beams are presented and discussed. This gun has small azimuthal velocity spread of less than 1% and relatively small beam ripple.