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If a charged particle is accelerated through a hole in a charged capacitor and makes a round trip back to the capacitor then the particle should come to a stop just before going through the capacitor again.

Question: If the particle goes through a set of capacitors arranged linearly and with the same polarity order (+,-) (+,-) (+,-) then does the particle continue to gain acceleration in the linear direction?

Or, is there a deceleration process limiting an increasing acceleration by going through more than one capacitor?

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Such an array of capacitors is not physically possible. The field travels from one capacitor to the next so that effectively you get an opposite polarized capacitor in between every capacitor of your array.

Even if you completely insulate the capacitor, for instance by placing it into a Faraday cage, you will still create two parasitic capacitors between the walls of the cage and the inner capacitor. Those make the whole contraption totally useless.

There is a small trick that is done in a class of electrostatic accelerators called "tandem". You have a structure like this:

|           |          |
^ ground    ^ V        ^ ground

You start with a negative ion and accelerate it through the first half up to the $e$V energy, then you strip away the electrons with a thin foil so that you get a positive ion which doubles its energy through the second half.

And, before you ask: yes, the cycle could be repeated endlessly, however, while stripping away electrons from a negative ion is quite straightforward, there is no (known) efficient way to realize the opposite process as the ion is accelerated to higher and higher energies.

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  • $\begingroup$ But, of course, if you manage to vary the voltage across the capacitor just right you get a linac or cyclotron derivative. $\endgroup$
    – Jon Custer
    Commented Feb 18, 2020 at 14:08
  • $\begingroup$ Thank you for your answer. I have a follow up question. Consider this arrangement of capacitors. A series connected array of capacitors. The separation between cells is a thin bipolar plate. The plate is a single sheet of conductive material. The series capacitors is equivalent to a single capacitor with the series voltage and bipolar plates removed. By adding the bipolar plates, is same limiting process occurring? $\endgroup$
    – MarkJanus1
    Commented Feb 19, 2020 at 14:43
  • $\begingroup$ @MarkJanus1 If I understand you properly, I think that between the two faces of the bipolar conductive plates, you get the same electric field as in the other capacitors. Try to make a draw, put the voltages where desired and then connect them with electric field lines. If you think to have a net accelerating field, then let me inspect the draw, maybe it is actually good ;) $\endgroup$
    – DarioP
    Commented Feb 20, 2020 at 10:11
  • $\begingroup$ @DarioP Bah, E=V/d and the thin plate has a proportional larger fridge field. Thank you on follow up. $\endgroup$
    – MarkJanus1
    Commented Feb 21, 2020 at 20:23

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