Torque and force on a magnetic dipole

Abstract

Recent controversies about torque and force on a magnetic dipole are discussed. Three essentially different current loop models are analyzed. Although all models yield the same expression for the torque, N=m×B, the detailed mechanisms that give rise to the torque in each case are very different. The expression for the force on a magnetic dipole is derived and analyzed for all models. The force expression is the same for all current loop models but it differs from the force on a magnetic charge dipole. The expression, obtained for the force on a current loop magnetic dipole, FCL=■(m⋅B)−(d/dt)(m×E/c), differs from what usually appears in the educational literature. The standard ‘‘naive’’ calculation of the force yields the correct expression for the rate of change of the total momentum, dP/dt=∇(m⋅B). However, the current loop in an externa l electric field has an internal ‘‘hidden momentum’’ m×E/c, which is not related to the motion of the center of mass of the dipole. Thus, for the force, defined as mass times acceleration, it is necessary to subtract the time derivative of this ‘‘hidden momentum.’’