Write the expression for the magnetic moment m due to a planar square loop of side ℓ carrying a steady current I in vector form. This loop is placed in a horizontal plane near a long straight conductor carrying a steady current I1 at a distance ℓ as shown. Give reason to explain that the loop will experience a net force but no torque. Write the expression for this force acting on the loop.
Solution
The magnetic moment is m=IA=Iℓ2n^, directed normal to the loop. Near the straight wire the field is perpendicular to the plane of the horizontal loop, so m∥B and the torque τ=m×B=0. However the field is non-uniform (stronger on the near side), so the forces on the two sides parallel to the wire do not cancel and there is a net force. The near side (distance ℓ) and far side (distance 2ℓ) give
