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I haven't been able to find the answer to this anywhere.

When calculating the energy gained by a particle that is accelerated across the gap of two cavities (e.g. Dees in a cyclotron, or charged cylindrical cavities of a linear accelerator), does one need to take into account the size of the conductor cavity--or more accurately the distance of the charged particle from the surfaces of these cavities?

If an electron, or proton were simply traveling perpendicular from the face of one charged plate to the face of another plate with a potential difference of, let's say say +10V and -10V (for simplicity sake) in a vacuum, the energy gained, as I understand it, is an increase of 20 electron volts. However, when the particle is inside of (not touching) a charged cylinder or a charged cyclotron Dee at one potential traveling across a gap to the inside of another charged cylinder or cyclotron Dee it is always at some distance from the conductor surface, so it seems like the particle would "feel" less of a potential than if it were going from being in direct contact with one conductor to being in direct contact the other conductor because the potential should drop more the further away it is located from the surface of the conductor. There is a visual representation of a linear accelerator on Wikipedia here: https://en.wikipedia.org/wiki/Linear_particle_accelerator, which may be helpful where you can see the particle traveling down the center of the cylindrical cavities some distance away from the conductor surfaces.

I'm not necessarily looking for a mathematical formula to solve this right now, I just want to know if my thinking is correct that the potential energy gain would actually be somewhat less than E =

Thanks for any help.

Todd