A study of arc discharges on various thicknesses of electron-beam-charged Mylar, FEP Teflon and Kapton shows that the peak substrate current and the energy dissipated in a load resistor both exhibit maxima at a particular thickness of the order of 50 mu m, for one set of experiment parameters. The experiments also show that, as thickness increases, this particular thickness is the transition from near-constant to decreasing released charge, and, for Mylar, from decreasing to near-constant arc duration. This transition is interpreted as being caused primarily for thin specimens by punchthrough formation and possibly influenced by the transition from conduction-dominated to emission-dominated charging. Additional low-energy ion exposure is shown to weaken and sometimes eliminate the arc discharges without radically altering the thickness scaling. At low fluxes, the incident ions are focused into a central spot.