# Analytic models of magnetically enclosed spherical and solenoidal coils

## TL;DR

This paper derives analytic solutions for the magnetic fields of spherical and solenoidal coils within high-permeability magnetic shields, aiding design and analysis through theoretical models and comparisons.

## Contribution

It provides new analytic models for magnetic fields of enclosed coils, including generalizations for winding misplacements, useful for design and benchmarking.

## Key findings

- Solenoidal coils produce more homogeneous fields than spherical coils.
- Models enable theoretical analysis and comparison of coil field homogeneity.
- Analytic solutions assist in coil design and finite-element benchmarking.

## Abstract

We provide analytic solutions of the net magnetic field generated by spherical and solenoidal coils enclosed in highly-permeable, coaxial magnetic shields. We consider both spherical and cylindrical shields in the case of the spherical coil and only cylindrical shields for the solenoidal coil. Comparisons of field homogeneity are made and we find that the solenoidal coil produces the more homogeneous field for a given number of windings. The models are useful as theoretical and conceptual guides for coil design, as well as for benchmarking finite-element analysis. We also demonstrate how the models can be generalized to explore field inhomogeneities related to winding misplacement.