An analytical treatment of self fields in a relativistic bunch of charged particles in a circular orbit

It is known that the electromagnetic field caused by a moving charge depends on its acceleration. Therefore, if a bunch of charged particles has a circular trajectory, the self fields in the bunch depend on the radius of curvature. We will treat these self fields analytically for a one-dimensional bunch, using the Lienard-Wiechert potentials. These depend on the retarded positions of the charges in the bunch. We will show that one only has to determine these positions explicitely for the endpoints of the bunch. The one-dimensional model predicts non-zero tangential and radial forces in the middle of the bunch which depend on its angular width and on its angular velocity. Expressions for these forces are presented. A comparison between the power loss due to coherent radiation and the tangential force exerted on the central electron of the bunch shows that there is a definite relation between these quantities.< >