Electric field of a 2 D elliptical charge distribution inside a cylindrical conductor ∗

Many applications in beam physics, particularly those concerning transverse beam dynamics, call for an approximate computation of the transverse electric field produced by the charge distribution of the beam, namely the field components in the plane perpendicular to the beam motion. Mathematically, this approximation becomes exact in the limit of an infinitely long charge distribution that is also constant in the longitudinal direction. In practice, the approximation is a reasonable starting point in many cases. Examples include proton beams used in spallation neutron sources, or heavy-ion fusion ion beams, in which the characteristic length of variation of the charge density along the longitudinal direction is much larger than the transverse beam size. In this case the longitudinal component of the electric field is much smaller than the transverse, hence the field is effectively contained in the 2D transverse plane. Another example arises in the case of ultra-relativistic beams, for which the electric field is effectively squeezed into a 2D transverse “pancake” owing to the Lorentz contraction of the longitudinal component of the field. Furthermore, the approximation of an elliptical charge distribution, as defined in Sec. II below, is also a reasonable starting point for numerous beam dynamics problems both for lepton and hadron beams. A few recent examples can be found in Refs. [1–3]. In Ref. 4 we developed a formalism to compute the 2D electric field for elliptical charge distributions in free space. The formalism makes essential use of Cauchy’s theorem, and yields a simple and quite general formula for the field in complex form. In this article we extend the formalism to elliptical distributions contained inside a perfectly conducting circular cylinder by applying the method of images. Our formalism naturally yields the electric field itself