A Linear Structure from Magnetic-Dipole Systems and Its Geometry

TL;DR

This paper explores the algebraic structures derived from magnetic dipole systems, revealing how certain sub-algebras enable the identification of dipole moments that maximize magnetic forces, with implications for understanding magnetic field geometries.

Contribution

It introduces a new algebraic framework for magnetic dipole systems and characterizes how sub-algebra structures influence the maximum magnetic forces achievable.

Findings

Existence of specific algebraic decompositions in magnetic dipole systems.

Identification of dipole moments that produce maximal translational forces.

Establishment of upper bounds for magnetic force strengths.

Abstract

We investigate a class of algebras on $\mathbb{R}^3$ arising and generalized from the algebraic structure of magnetic gradient fields induced by systems of synchronous magnets with identical dipole moments (i.e., $\mathbf{M}_i=\mathbf{M},\,\forall i$). We show that when there is a $2$ dimensional sub-algebra, the linear structure associated to such an algebra admits a certain type of decompositions, which allows the locating of the dipole moment $\bar{\mathbf{M}}$ that yields the strongest translational force(s) on a test magnet $\mathfrak{m}$. Upper bounds to the strength of this magnetic force are then established.