Question

A current-carrying circular coil of magnetic moment $ M $ is situated in a magnetic field $ B $ . The work done in deflecting it from an angle $ 0^\circ $ to $ \theta ^\circ $ will be :
A) $ MB $
B) $ \;MB\left( {1 - cos\theta } \right) $
C) $ - MB $
D) $ MB\left( {1 - sin\theta } \right) $

Answer 1

Hint: The current-carrying coil acts as a bar magnet placed in an external magnetic field. The potential energy of a current-carrying coil placed in an external magnetic field depends on the angle of orientation of the current-carrying coil with the external magnetic field. The work done in deflecting the current-carrying coil will be calculated as the difference of the potential energy of the current-carrying coil.

Formula used: In this solution, we will use the following formula:
 $ U = - M \cdot B $ where $ U $ is the potential energy stored in the apparent magnet formed by the current-carrying coil and $ B $ is the magnitude of the external magnetic field.

Complete step by step answer:
We’ve been given that a current-carrying coil is placed in an external magnetic field. We know that a current-carrying coil generates a magnetic field of its own which implies that the current-carrying coil can be treated as a bar magnet of magnetic moment $ M $ placed in an external magnetic field.
The potential energy associated with this configuration when the current-carrying coil and the external magnetic field have an angle $ \theta $ between them will be
 $ U = - M \cdot B = - MB\cos \theta $
Then the work done in deflecting the coil from an angle $ {\theta _1} $ to $ {\theta _2} $ will be due to the difference of the potential energies of the configuration in the two angles. So, we have
 $ W = {U_2} - {U_1} $
 $ \Rightarrow W = - MB\cos {\theta _2} - ( - MB\cos {\theta _1}) $
Substituting the value of $ {\theta _1} = 0^\circ $ and $ {\theta _2} = \theta ^\circ $ , we get
 $ W = - MB\cos \theta + MB $
$ \Rightarrow W = MB(1 - \cos \theta ) $ which corresponds to option (B).

Note:
The reason for the fact that the change in potential energy will be equal to the work done is that there is no external force acting on the coil and the only change in energy of the coil will due to a change in its alignment with the external magnetic field that is a change in its stored potential energy. The coil will not have kinetic energy since it will only rotate in the magnetic field and not translate in any direction.