A small circular loop of area $A$ and resistance $R$ is fixed on a horizontal $x y$-plane with the center of the loop always on the axis $\hat{n}$ of a long solenoid. The solenoid has $m$ turns per unit length and carries current $I$ counterclockwise as shown in the figure. The magnetic field due to the solenoid is in $\hat{n}$ direction. List-I gives time dependences of $\hat{n}$ in terms of a constant angular frequency $\omega$. List-II gives the torques experienced by the circular loop at time $t=\frac{\pi}{6 \omega}$. Let $\alpha=\frac{A^{2} \mu_{0}^{2} m^{2} I^{2} \omega}{2 R}$.

List-I	List-II
(I) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\jmath}+\cos \omega t \hat{k})$	(P) 0
(II) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\imath}+\cos \omega t \hat{\jmath})$	(Q) $-\frac{\alpha}{4} \hat{\imath}$
(III) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\imath}+\cos \omega t \hat{k})$	(R) $\frac{3 \alpha}{4} \hat{\imath}$
(IV) $\frac{1}{\sqrt{2}}(\cos \omega t \hat{\jmath}+\sin \omega t \hat{k})$	(S) $\frac{\alpha}{4} \hat{\jmath}$
	(T) $-\frac{3 \alpha}{4} \hat{\imath}$

Which one of the following options is correct?

A special metal S conducts electricity without any resistance. A closed wire loop, made of S, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius a, with its center at the origin. A magnetic dipole of moment m is brought along the axis of this loop from infinity to a point at distance r (>> a) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole m, at a point on its axis at distance r, is $${{{\mu _0}} \over {2\pi }}{m \over {{r^3}}}$$, where $$\mu$$₀ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, m₁ and m₂, separated by a distance r on the common axis, with their north poles facing each other, is $${{k{m_1}{m_2}} \over {{r^4}}}$$, where k is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

When the dipole m is placed at a distance r from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to

A special metal S conducts electricity without any resistance. A closed wire loop, made of S, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius a, with its center at the origin. A magnetic dipole of moment m is brought along the axis of this loop from infinity to a point at distance r (>> a) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole m, at a point on its axis at distance r, is $${{{\mu _0}} \over {2\pi }}{m \over {{r^3}}}$$, where $$\mu$$₀ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, m₁ and m₂, separated by a distance r on the common axis, with their north poles facing each other, is $${{k{m_1}{m_2}} \over {{r^4}}}$$, where k is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

The work done in bringing the dipole from infinity to a distance r from the center of the loop by the given process is proportional to

A circular coil of radius R and N turns has negligible resistance. As shown in the schematic figure, its two ends are connected to two wires and it is hanging by those wires with its plane being vertical. The wires are connected to a capacitor with charge Q through a switch. The coil is in a horizontal uniform magnetic field Bo parallel to the plane of the coil. When the switch is closed, the capacitor gets discharged through the coil in a very short time. By the time the capacitor is discharged fully, magnitude of the angular momentum gained by the coil will be (assume that the discharge time is so short that the coil has hardly rotated during this time)