Space-charge forces of a DC beam in a continuous bend

It has been shown that the net space-charge forces for a de beam with space-charge potential depression in a bend have the usual inverse energy square dependence to the first order in the beam radius a over the bend radius R. We extend the analysis to the second order in a/R and allow the beam to have a small transverse displacement. The net space-charge forces are no longer cancelled to inverse energy square factor. The non-cancelled part of self-induced magnetic forces are at the second order in a/R and independent of the beam energy. The nonlinear parts of these forces are much larger than that of the usual inverse energy square forces. Scaling laws for emittance growth caused by the curvature of the beam and a transverse beam displacement, respectively, are presented.