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Computation of space-charge current-limiting effects across a vacuum cavity between parallel electrodes has previously been carried out only for thermionic emission spectra. In some applications, where the current arises from an injected electron beam or photo-Compton emission from electrode walls, the electron energy spectra may deviate significantly from Maxwellian. Considering the space charge as a collisionless plasma, we derive an implicit equation for the peak cavity potential assuming steady-state currents. For the examples of graphite, nickel, and gold electrodes exposed to x rays, we find that cavity photoemission currents are typically more severely space-charge limited than they would be with the assumption of a purely Maxwellian energy distribution.