Comparative study between toroidal coordinates and the magnetic dipole field

TL;DR

This paper compares toroidal coordinates and magnetic dipole fields from circular loops, assessing how well the former can approximate the latter, especially at large aspect ratios, with quantitative criteria for accuracy.

Contribution

It provides a detailed evaluation of the similarities and differences between toroidal coordinates and dipole magnetic fields, establishing criteria for their approximation.

Findings

Toroidal coordinates and dipole fields are similar for large aspect ratios.

Centers of magnetic dipole tori displace faster from the axis than toroidal coordinates.

Quantitative criteria for approximation accuracy are established.

Abstract

There is a similar behaviour between the toroidal coordinates and the dipole magnetic field produced by a circular loop. In this work we evaluate up to what extent the former can be used as a representation of the latter. While the tori in the toroidal coordinates have circular cross sections, those of the circular loop magnetic field are nearly elliptical ovoids, but they are very similar for large aspect ratios.The centres of the latter displace from the axis faster than the former. By making a comparison between tori of similar aspect ratios, we find quantitative criteria to evaluate the accuracy of the approximation.