On the magnetic dipole energy expression of an arbitrary current distribution

TL;DR

This paper investigates the correct sign of the magnetic dipole energy term in the potential energy expansion of a localized current distribution, revealing it differs from the traditionally accepted form and discussing its implications.

Contribution

It demonstrates that the magnetic dipole energy term has the opposite sign to the well-known expression, challenging conventional understanding.

Findings

The magnetic dipole energy term is actually positive, not negative.

Implications for the force and torque calculations on magnetic dipoles are discussed.

The result affects the interpretation of magnetic energy in classical electromagnetism.

Abstract

We show that the magnetic dipole energy term appearing in the expansion of the magnetic potential energy of a localized current distribution has the form $ U= + \bf{m} \cdot \bf{B}$ which is wrong by a sign from the well known $ - \bf{m} \cdot \bf{B}$ expression. Implication of this result in relation to the electric dipole energy $ - \bf{p} \cdot \bf{E}$, and the force and torque on the magnetic dipole based on this energy expression is also discussed.