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

A current-carrying coil within a uniform magnetic field experiences zero net force. This is due to the magnetic field applying equal and opposing forces on opposite sides of the coil. However, a torque \( \tau \) is generated, aiming to align the coil’s magnetic dipole moment \( \mathbf{M} \) with the external magnetic field \( \mathbf{B} \). The torque is mathematically expressed as:\[\tau = \mathbf{M} \times \mathbf{B}\]Stable equilibrium is achieved when \( \mathbf{M} \) is parallel to \( \mathbf{B} \). In this configuration, the coil's potential energy is at its minimum, and the total magnetic field flux through the coil is maximized.The total flux \( \Phi_{\text{total}} \) through the coil is calculated as:\[\Phi_{\text{total}} = B A \cos(\theta)\]Here, \( \theta \) represents the angle between the magnetic field and the normal vector of the coil’s surface.When \( \theta = 0 \), the coil is in stable equilibrium, and the flux is at its maximum value.