There are two possible formulas for the magnetic force on a magnetic dipole moment due to the inhomogeneity of the magnetic field: $$ \vec{F}_{A} = \nabla (\vec{m}\cdot\vec{B}) \qquad\text{ and }\qquad \vec{F}_{B} = (\vec{m}\cdot\nabla)\vec{B} $$ (and there is an additional torque on the dipole: $\vec{\tau} = \vec{m}\times\vec{B}$ due to misalignment). The first formula comes from the "current loop model" and the second formula comes from considering two monopoles of opposite charges and taking the limit as they converge to one another. The two formulas are related by $$ \vec{F}_{A} = \vec{F}_{B} + \vec{m}\times(\nabla\times\vec{B}) - (\nabla\cdot\vec{B})\vec{m}. $$ Now $\nabla\cdot\vec{B} = 0$ by Gauss's law for magnetism (as of the time this post was created) and $\nabla\times\vec{B} = 0$ holds in magnetostatics where there are no currents. This means that in magnetostatics with no currents, the two formulas yield the same results, and in everyday life conditions we really can't distinguish between the two formulas.
My question is, have there been any experiments or attempts to find out which formula applies to a magnetic bar in electrodynamic regimes? Is it $F_{A}$, $F_{B}$, or something else? I can understand playing around with fridge magnets wouldn't get you anywhere, so I am willing to be liberal as to what counts as a valid experiment.
For example, I am wondering if anyone recorded the response of nanoscopic magnetic objects due to a curling magnetic field. Are there any precision tests of this sort? I would be very interested in finding out more about this type of research.
