Wake and Impedance

We begin with interactions of particles that moving with constant velocity in free space. If the material walls are far from the particles, their effect in the first approximation can be neglected. Let us consider a leading particle of charge q moving with velocity v, and a trailing particle of unit charge moving behind the leading one on a parallel path at a distance s with an offset x, as shown in Fig. 1. We want to find the force which the leading particle exerts on the trailing one. We will use the following expressions for the electric and magnetic fields of a particle moving with a constant velocity (see, e.g., [4]):