Question

A magnet of magnetic moment $M$ is rotated through ${{360}^{o}}$ in a magnetic field $H$. Work done will be
A.$0$
B.$2\,MH$
C.$MH$
D.$\pi MH$

Answer 1

Hint: When a magnet is placed in a magnetic field, it experiences a torque. If a magnet is rotated against the torque, work has to be done. Here the magnet is rotated through an angle ${{360}^{o}}$ , so by putting this angle value in the respective equation we can calculate the value of work done.

Formula used:
The total work done $W$through rotating this magnet from an angle ${{\theta }_{1}}\,\,to\,\,{{\theta }_{2}}$ is given by,
$W=MH(\cos {{\theta }_{1}}-\cos {{\theta }_{2}})$
Where,
$M=$Magnetic moment
$H=$Magnetic field

Complete answer:
When a magnet with the magnetic moment $\overrightarrow{M}$, is placed in a magnetic field $\overrightarrow{H}$, it experiences a torque,$\overrightarrow{\tau }=\overrightarrow{M}\times \overrightarrow{H}$
Or,$\tau =MH\sin \theta $ [Since $\theta =$angle] …….(i)

Now, this torque tends to align parallel to the direction of the magnetic field but if this magnet is rotated against the torque, work is to be done.
Thus, work done,$W=\tau d\theta $

Putting the value $\tau $from equation (i) we get,
$W=MH\sin \theta \,\,d\theta $

The work done in turning the magnetic dipole through a small angle is,
$dW=\tau d\theta $

If the magnet is rotated from an angle ${{\theta }_{1}}\,to\,\,{{\theta }_{2}}$,
Total work done,
$\int{dW}=\int\limits_{{{\theta }_{1}}}^{{{\theta }_{2}}}{MH\sin \theta \,d\theta }$
Or,
$W=MH(-\cos \theta )_{{{\theta }_{1}}}^{{{\theta }_{2}}}$
Or,
$W=MH(\cos {{\theta }_{1}}-\cos {{\theta }_{2}})$ ……..(ii)

In this problem ${{\theta }_{1}}={{0}^{o}}$and ${{\theta }_{2}}={{360}^{o}}$. Putting these values in equation (ii),
$\therefore W=MH(\cos {{0}^{o}}-\cos {{360}^{o}})$
Or,$W=MH(1-1)=0$

Therefore work done is zero when the magnet is rotated through an angle ${{360}^{o}}$ in a magnetic field.

Thus, option (A) is correct.

Note: Torque is an influence that causes a change in the rotational motion of any object. To make any object about an axis by imparting torque on it. Generally, torque is a vector quantity which means it has both magnitude and direction. The S.I unit of torque is $N.m$or $kg\,{{m}^{2}}{{\sec }^{-2}}$.