Electrostatic force between rings and discs of charge inside a grounded metallic pipe using the Green’s function technique

We derive analytic formulae for the electrostatic force between ring and disc charge distributions inside a grounded metallic pipe using the Green’s function technique. These distribution models are useful in the modeling of electron beams commonly employed in microwave tubes. We analyze the electric force between two discs, between two rings, and between a disc and a ring and we compare the results for the electric potential, field, and force to numerical ones obtained from a 3D electrostatic solver. Present expressions were developed to avoid an oscillatory noise when the field diverges by axial proximity between source and observer.