Properties of the electromagnetic fields generated by a circular-symmetric e-cloud pinch in the ultra-relativistic limit

The electromagnetic fields generated by an electron cloud in the beam pipe of a particle accelerator introduce additional forces in the dynamics of the particle beams, which can be responsible of beam instabilities, beam losses and transverse emittance blow-up. These effects are studied mainly employing numerical simulations, using macroparticle sets to model the circulating beam and the e-cloud. The fields are computed using the Particle in Cell method (PIC). In most simulation codes, the electrodynamics equations are solved in the quasi-electrostatic approximation. To understand the validity of this approximation, we investigate analytically the full electromagnetic problem for a circular-symmetric case, identifying the structure of the electromagnetic fields. Expressions for the error introduced by the quasi-electrostatic approximation are derived, showing that this has no effects when calculating the transverse forces exerted by the e-cloud on the particle beam. The longitudinal component of the field is also investigated and it is related to the power transferred from the beam to the e-cloud.