Force on a current carrying loop in a non-uniform magnetic field

Question

I was taught that a current carrying loop in a non-uniform magnetic field will always experience both a torque and a net force.

Is this always true? I can't think of any examples where the force would be zero; are there any?

Answer 1

It is, of course, not true that there is always a force and a torque. There are usually equilibrium points where they vanish for a certain orientation of the magnetic moment (aka current loop). The torque on a magnetic moment in an external magnetic field (no matter if homogeneous or inhomogeneous) is $$\vec T=\vec \mu \times \vec B$$ so if the magnetic moment is locally parallel (tangent) to the magnetic field, the torque vanishes. Similarly, the force on a magnetic moment in an inhomogeneous magnetic field is $$\vec F=\vec \nabla (\vec \mu \cdot \vec B)$$ so if there is a point where the inner product $(\vec \mu \cdot \vec B)$ is locally constant, the force vanishes.

This situtation does not change in principle if the current loop has finite size (can't just be represented by a singular magnetic moment), the derivation, that there could be equilibrium configurations, only gets more complicated. And it is not necessary to consider this case, because the situation for the point-like magnetic moment already constitutes a counter example to the attribute "always".

However, these equilibrium configurations usually form only a finite set of points. So it is correct to say that almost always a magnetic moment (and hence, a current loop) experiences a force and a torque in an inhomogeneous magnetic field.