The force on a magnetic dipole

TLDR

The classical magnetic force on a dipole depends on the model: a current‑loop model yields \(F=\nabla(m\!\cdot\!B)\) while a separated‑charge model gives \(F=(m\!\cdot\!\nabla)B\), a distinction often overlooked in textbooks but crucial for interpreting experiments such as the Aharonov–Casher effect. The paper shows that these two force expressions can predict markedly different forces, as illustrated by elementary examples and by historical analyses of the neutron dipole moment in the 1930s–1940s.

Abstract

The classical magnetic force on a magnetic dipole depends upon the model for the dipole. The usual electric current loop model for a magnetic dipole leads to the force F=∇(m⋅B) on a magnetic dipole m in a magnetic field B. The separated magnetic charge model for a magnetic dipole leads to the force F=(m⋅∇)B on a magnetic dipole. The latter expression is analogous to the force experienced by an electric dipole in an electric field. Here, some elementary examples are given where the force expressions yield entirely different forces on a magnetic dipole. Electromagnetism textbooks usually do not emphasize the difference between these force expressions; however, occasionally the difference is important for understanding experimental results. In the 1930s and 1940s the difference in force expressions was involved in a determination of the nature of the neutron dipole moment. At present, in the 1980s, the difference in the force expressions is central to a controversy over an experiment to test the proposed Aharonov–Casher effect.