Sector magnets or transverse electromagnetic fields in cylindrical coordinates

Abstract

Laplace’s equation is considered for scalar and vector potentials describing electric or magnetic fields in cylindrical coordinates, with invariance along the azimuthal coordinate. A series of special functions are found which, when expanded to lowest order in power series in radial and vertical coordinates, replicate harmonic polynomials in two variables. These functions are based on radial harmonics found by Edwin M. McMillan forty years ago. In addition to McMillan’s harmonics, a second family of radial harmonics is introduced to provide a symmetric description between electric and magnetic fields and to describe fields and potentials in terms of the same functions. Formulas are provided which 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. This result is important for potential theory and for theoretical study, 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.