Fields of a relativistic beam of charged particles between parallel planes: Exact two-dimensional solutions obtained by the method of images

We calculate exact two-dimensional analytic expressions for the electric and magnetic fields and their potentials generated by a linear beam of relativistic charged particles between infinite ideally conducting planes and ferromagnetic poles. We use the method of summing an infinite sequence of fields of linear image charges and image currents to obtain solutions and also calculate the density distribution of the electric charge on the conductor surface induced by the charged particle beam. We obtain new formulas for the linear approximation of the fields near the beam and discuss the mathematical features of the exact solutions and the applicability constraints for the linear approximations.