On sector magnets or transverse electromagnetic fields in cylindrical coordinates

TL;DR

This paper introduces special functions based on radial harmonics for solving Laplace's equations in cylindrical coordinates, aiding the design and modeling of sector magnets in accelerator physics.

Contribution

It presents a new set of special functions derived from radial harmonics, including a symmetric description for electric and magnetic fields in cylindrical coordinates.

Findings

Derived special functions based on radial harmonics.

Provided formulas relating transverse fields to these functions.

Discussed applications in accelerator magnet design.

Abstract

The Laplace's equations for the scalar and vector potentials describing electric or magnetic fields in cylindrical coordinates with translational invariance along azimuthal coordinate are considered. The series of special functions which, when expanded in power series in radial and vertical coordinates, in lowest order replicate the harmonic homogeneous polynomials of two variables are found. These functions are based on radial harmonics found by Edwin~M.~McMillan in his more-than-40-years "forgotten" article, which will be discussed. In addition to McMillan's harmonics, second family of adjoint radial harmonics is introduced, in order to provide symmetric description between electric and magnetic fields and to describe fields and potentials in terms of same special functions. Formulas to relate any transverse fields specified by the coefficients in the power series expansion in radial or vertical planes in cylindrical coordinates with the set of new functions are provided. This result is no doubt important for potential theory while also critical for theoretical studies, design and proper modeling of sector dipoles, combined function dipoles and any general sector element for accelerator physics. All results are presented in connection with these problems.