Torque and Force due to an external magnetic field

[wam_mi]

Torque N = μ ∧ B

where μ is the dipole moment of the loop,
and B is an external magnetic field.

Q1 Is it true that we can only write the Torque due to an external magnetic field in this form if and only iff B is a constant? What happens, say, if B = B * x (in the z-direction) where x is a variable, is it still possible to write Torque in this form, where μ is a constant.

Q2 Why is it that Force = 0 but Torque is not = 0 when the external magnetic field is constant?

Cheers guys

 

[Meir Achuz]

1. The torque is given by the cross product \mu\times B even if B is a function of position.
2. The force on a magnetic dipole is given by
{\vec F}=\nabla(\mu\cdot}{\vec B},so it vanishes if B is constant.
As a simple model, think of a bar magnet. The force on the N and S ends will be equal and opposite in a constant magnetic field.

 

[Bob S]

The torque on a magnetic dipole, like a compass needle, is non-zero whenever it is in a constant magnetic field B (and if the dipole is not aligned with B). There is no net translational force on the needle, however.

If B is non constant, specifically B = B * x, where B is along z, there can also be a net translational force on a magnetic dipole. A good example is the magnetic force on a beam of neutral silver atoms, which are now known to have a magnetic moment. Because the atoms did have a magnetic dipole moment, the beam was deflected (transverse force) in a magnetic field. This was the basis of the Stern Gerlach experiments in 1921-23, which later won the Nobel Prize in physics.