Abstract
Particle trajectories in the presence of space charge are calculated for prescribed axial potential variations, including linear, parabolic, and sinusoidal variations, using the paraxial-ray equation including space charge for axial symmetry. Solutions are also presented for the potential as a function of distance corresponding to prescribed hyperbolic and parabolic trajectories. The off-axis equipotentials are calculated for some representative cases. These two-dimensional results are interpreted in the light of one-dimensional space-arge theory in which exact solutions can be obtained for the potential as a function of distance. Some of the limitations of the present theory in which either the potential or the trajectory is prescribed are pointed out.
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