We derive analytical expressions for external fields of a charged relativistic bunch with a circular and an elliptical cross section. The particle density in the bunch is assumed to be uniform as well as non-uniform. At distances far apart from the bunch, the field reduces to the relativistic modified Coulomb form for a point-like charge and at small distances the expressions reproduce the external fields of a continuous beam. In an ultra relativistic limit the longitudinal components of the internal and external electric fields of the bunch are strongly suppressed by the Lorentz factor. If the bunch is surrounded by conducting surfaces, the bunch self-fields are modified. Image fields generated by a bunch between two parallel conducting plates are studied in detail. Exact summation of electric and magnetic image fields allows the infinite series to be represented in terms of elementary trigonometric functions. The new expressions for modified fields are applied to study image forces acting on the bunch constituents and the bunch as a whole and calculated in the framework of the linear theory the coherent and incoherent tune shifts for an arbitrary bunch displacement from the midplane. Moreover, the developed method allows to generalize the Laslett image coefficients
The image of a point charge in an infinite conducting cylinder^ https://arxiv.org/abs/2001.10651
Matt Majic
The electrostatics problem of a point charge next to a conducting plane is best solved by placing an image charge placed on the opposite side. For a charge between two parallel planes this can be solved with image charges outside the planes at evenly spaced intervals moving out to infinity. What is the corresponding image of a point charge is when placed on the axis of a cylinder?. The potential of a point charge in a cylinder is well known and may expressed in many forms involving integrals or series of Bessel functions, but none of which elude to an image. In fact the image is complex (in both senses), consisting of infinitely many rings on a disk with some surface charge distribution. This manuscript attempts to describe the image as accurately as possible, and in doing so finds simple accurate approximations for the potential.
Comments: 7 pages, 8 figures