Electromagnetic Field Induced By an Infinitelysimally Thin Ring Moving in a Plane Condeser

Abstract

The main idea of a wake field transformer is to induce a high (wake) field by a beam of high intensity but low energy per particle (called the drive beam) accelerating a second beam with low intensity but high energy gain per particle. A geometry suitable for a wake field accelerator is a beam widened to a ring passing a thin pill-box cavity (that is a cylinder with a circular base) (Weiland, Voss 1982). In this report a problem with the same circular symmetry was investigated. The problem consists of a ring-shaped line charge in a plane parallel to the plates of a plane condensor and moving in a direction perpendicular to the plates. The ring is at rest in front of one condensor plate until it starts suddenly to move with constant velocity; then the ring traverses the plane condensor until it reaches the second plate, where it stops. The field is derived from the potentials. The only non-zero component, the longitudinal one, of the vector potential is computed with the method of Green's function. The scalar potential is computed by using the Lorentz gauge. The field can be expressed as an infinite sum over complete and incomplete elliptic integrals and simpler terms. But at any given finite time only a finite number of terms of the infinite sum contributes. The field is evaluated numerically with the computer and shown in several figures.