Modeling Magnetic Fields with Helical Solutions to Laplace’s Equation

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, a small number of free parameters and sparse input data can lead to
highly accurate, fine-grained modeling of solenoidal magnetic fields, including helical features arising
from the winding of the solenoid, with overall field accuracy at better than one part per million.