Question

A current-carrying coil is placed in an external uniform magnetic field. The coil is free to turn in the magnetic field. What is the net force acting on the coil? Obtain the orientation of the coil in stable equilibrium. Show that in this orientation the flux of the total field (field produced by the loop + external field) through the coil is maximum.

Hint:

The coil in a magnetic field experiences a torque that aligns its magnetic dipole moment with the field, and the flux through the coil is maximized in this orientation.

Answer 1

The net force on a current-carrying coil in a uniform magnetic field is zero because the magnetic field exerts equal and opposite forces on opposite sides of the coil. However, the coil experiences a torque \( \tau \), which tends to align the coil’s magnetic dipole moment \( \mathbf{M} \) with the external magnetic field \( \mathbf{B} \). The torque is given by: \[ \tau = \mathbf{M} \times \mathbf{B} \] The coil will be in stable equilibrium when \( \mathbf{M} \) is aligned with \( \mathbf{B} \). In this orientation, the potential energy of the coil is minimized, and the flux of the total magnetic field through the coil is maximum. The total flux \( \Phi_{\text{total}} \) through the coil is: \[ \Phi_{\text{total}} = B A \cos(\theta) \] Where \( \theta \) is the angle between the magnetic field and the normal to the coil’s surface. At stable equilibrium, \( \theta = 0 \), and the flux is maximized.