Differential energy loss for a particle in a square pulse of charge traveling between infinite conducting plates

In a recent paper by Papiernik et al., the problem of including the beam-induced fields in a cavity to determine the steady-state bunch shape was studied. In order to understand how the fields felt by a particle in the beam are generated, the authors have rederived these fields from a slightly different approach and have concentrated on obtaining the physical picture of the fields. In a previous note, we calculated the electromagnetic fields in a cavity produced by an ultra-relativistic point charge. The cavity was approximated by two infinite conducting plates and the particle was assumed to travel with the speed of light c. It was found that a particle q2 following at a finite distance behind another particle q1 will not see any fields produced by q1 until q1 has left the cavity, and, when doing so, has sent (accelerating) δ-function fields backwards toward the oncoming q2. In this case, q2 always gains energy from the fields generated by q1, and as the distance between q1 and q2 decreases to zero, the energy gain goes to positive infinity. At first glance, this appears to contradict the results of Ref. 1; however, if q1 is exactly on top of q2, then q2 will travel across the cavity together with the (decelerating) δ-function field of q1. Due to this charge, q2 loses an infinite amount of energy. In the following treatment, we attempt to clarify this confusion. 2 refs., 3 figs