Torque acting on the magnetic dipole kept in non uniform magnetic field
Answers
Answer:
This depends on exactly what you mean by non-uniform, or (equivalently) on how big the loop is. In particular, the important criterion is whether the field changes appreciably over distances that are about the same size as the loop.
If the field changes throughout space, but the loop is small enough that the field doesn't change much from point to point on the loop, then the uniform-field formula τ=μ×B(r) still applies. In essence, the field is locally uniform, though the direction and magnitude it's uniform on can change from place to place.
If the loop is big enough that the field changes appreciably over its span then there's nothing for it but to integrate the local torque on each bit of circuit and add them up, which gives you
τ=∮Cr×F(r)dl=∮Cr×(
ˆ
t
I×B(r))dl=∮Cr×(Idl×B(r)).
There really isn't much you can do to simplify it beyond that without special assumptions. The integral is a line integral, of exactly the same sort you use to calculate the magnetic dipole moment μ itself.