Modeling magnetic fields with helical solutions to Laplace’s equation

Abstract

The series solution to Laplace’s equation in a helical coordinate system is derived and refined using symmetry and chirality arguments. These functions and their more commonplace counterparts are used to model solenoidal magnetic fields via linear, multidimensional curve-fitting. A judicious choice of functional forms motivated by geometry, a small number of free parameters, and sparse input data can lead to highly accurate, fine-grained modeling of solenoidal magnetic fields. These models capture the helical features arising from the winding of the solenoid, with overall field accuracy at better than one part per million.