# Modeling Magnetic Fields with Helical Solutions to Laplace's Equation

## TL;DR

This paper develops a series solution to Laplace's equation in helical coordinates, enabling precise modeling of solenoidal magnetic fields with minimal data and high accuracy, capturing complex helical features.

## Contribution

It introduces a novel series solution in helical coordinates and demonstrates its effectiveness for accurate magnetic field modeling with sparse data.

## Key findings

- Achieves better than one part per million accuracy in magnetic field modeling.
- Effectively captures helical features from solenoid winding.
- Uses minimal parameters and data for high-precision modeling.

## 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, 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.