Electromagnetic Fields Produced by Moving Sources in a Curved Beam Pipe

TL;DR

This paper introduces a new geometrical perturbation method to calculate electromagnetic fields generated by moving charged sources in curved, perfectly conducting beam pipes with complex shapes, using differential forms for a rigorous theoretical framework.

Contribution

A novel perturbation scheme for electromagnetic field calculation in non-straight beam pipes with variable curvature, employing differential forms and providing analytic solutions.

Findings

Derived explicit formulas for fields in pipes with piecewise constant curvature.

Established a formalism applicable to ultra-relativistic charged sources.

Demonstrated the method's effectiveness through pedagogical derivation from Maxwell's equations.

Abstract

A new geometrical perturbation scheme is developed in order to calculate the electromagnetic fields produced by charged sources in prescribed motion moving in a non-straight perfectly conducting beam pipe. The pipe is regarded as a perturbed infinitely long hollow right-circular cylinder. The perturbation maintains the pipe's circular cross-section while deforming its axis into a planar space-curve with, in general, non-constant curvature. Various charged source models are considered including a charged bunch and an off-axis point particle. In the ultra-relativistic limit this permits a calculation of the longitudinal wake potential in terms of powers of the product of the pipe radius and the arbitrarily varying curvature of the axial space-curve. Analytic expressions to leading order are presented for beam pipes with piecewise defined constant curvature modelling pipes with straight segments linked by circular arcs of finite length. The language of differential forms is used throughout and to illustrate the power of this formalism a pedagogical introduction is developed by deriving the theory ab-initio from Maxwell's equations expressed intrinsically as a differential system on (Minkowski) spacetime.