Question

A magnetic dipole in a constant magnetic field has:
(A) Minimum potential energy when the torque is maximum.
(B) Zero potential energy when the torque is minimum.
(C) Zero potential energy when the torque is maximum.
(D) Maximum potential energy when the torque is maximum.

Answer 1

We are given the magnetic dipole in a constant magnetic field and are asked about the change in torque when there is a change in potential energy. Thus, we will take a formula of potential energy and then discuss the change in it. Then, we will take a formula for torque on a magnetic dipole. Then finally we will try to connect the change in both these parameters.
Formula Used
$U = - \vec \mu .\vec B = - \mu B\cos \theta$
Where, $U$ is the potential energy on a magnetic dipole, $\vec \mu$ is the magnetic dipole moment of the dipole and $\vec B$ is the uniform magnetic field in which the magnetic dipole is placed and $\theta$ is the angle between $\mu$ and $B$ .
$\vec \tau = \vec \mu \times \vec B = \mu B\sin \theta \hat n$
Where, $\vec \tau$ is the torque acting on the magnetic dipole, $\vec \mu$ is the magnetic dipole moment of the dipole and $\vec B$ is the uniform magnetic field in which the magnetic dipole is placed and $\theta$ is the angle between $\mu$ and $B$ . $\hat n$ is the direction of the torque which is perpendicular to the plane containing $\vec \mu$ and $\vec B$ .

Step By Step Solution
We know,
Potential energy of the magnetic dipole, $U = - \vec \mu .\vec B = - \mu B\cos \theta$
Now,
The magnetic dipole moment ($\vec \mu$) and magnetic field ($\vec B$) are constant parameters, only the $\cos \theta$ is the only varying parameter.
Now,
$\cos \theta$ is maximum when $\cos \theta = 1$ or $\theta = 2n\pi ;n = 0,1,2,...$ and then $\cos \theta$ is minimum when $\cos \theta = - 1$ or $\theta = (2n - 1)\pi ;n = 0,1,2,3,...$.
Also,
We know,
Torque acting on the magnetic dipole, $\vec \tau = \vec \mu \times \vec B = \mu B\sin \theta \hat n$
From this, we will only take the magnitude of the torque in order to compare the change with the potential energy change.
Thus,
$|\vec \tau | = \mu B\sin \theta$
Out of here also only the $\sin \theta$ is the varying parameter.
Now,
$\sin \theta$ is maximum when $\sin \theta = 1$ or $\theta = (2n + 1)\dfrac{\pi }{2};n = 1,2,3,...$ and $\sin \theta$ is minimum when $\sin \theta = 0$ or $\theta = n\pi ;n = 0,1,2,...$.
Thus, we can say that the points when the value of $\cos \theta$ is maximum when $\sin \theta$ is minimum and vice versa. Broadly speaking, when potential energy is zero, then the torque is maximum.

Hence, the answer is (C).

Note: We were asked to find the relation between potential energy and torque of the magnetic dipole. If we were asked for the relation between some other parameters, the calculations would be somewhat different but the workflow remains the same.