Variations on the magnetic torque acting on a wire

TL;DR

This paper provides a comprehensive and accessible derivation of the magnetic torque on a wire, extending the basic relation to more complex wire shapes and magnetic field configurations.

Contribution

It offers a general, elementary proof of the magnetic torque relation and extends it to non-closed wires and non-uniform magnetic fields.

Findings

Derived a general proof of M = mu x B relation

Extended the relation to non-closed wires

Extended the relation to non-uniform magnetic fields

Abstract

The relation M = mu x B is presented in all elementary courses on electromagnetism but it is usually given just for the simple case of a rectangular wire. We will present a completely general but elementary proof of this relation together with two more advanced proof methods. We will then provide some extensions: non-closed wires and non-uniform magnetic field.