For high-voltage devices such as particle accelerators, novel geometries for a triple-junction at which an insulator, an anode and a vacuum meet are disclosed. A singularity in the electric field at the triple-junction is eliminated, reducing dielectric flashover and allowing the devices to operate at higher voltages without breakdown. In one aspect, such a device includes a cathode, an anode having an anode surface exposed to a vacuum, and a dielectric body disposed between the cathode and anode, the dielectric body having a dielectric surface that is exposed to the vacuum, wherein the dielectric surface and the anode surface approach each other such that an angle measured across the vacuum between the dielectric surface and the anode surface decreases with decreasing distance between the dielectric surface and the anode surface until the dielectric surface and the anode surface meet and are parallel.