$A$ thin conducting rod $MN$ of mass $20 \text{ g}$,length $25 \text{ cm}$ and resistance $10 \text{ }\Omega$ is held on frictionless,long,perfectly conducting vertical rails as shown in the figure. There is a uniform magnetic field $B_0 = 4 \text{ T}$ directed perpendicular to the plane of the rod-rail arrangement. The rod is released from rest at time $t = 0$ and it moves down along the rails. Assume air drag is negligible. Match each quantity in List-$I$ with an appropriate value from List-$II$,and choose the correct option. [Given: The acceleration due to gravity $g = 10 \text{ m s}^{-2}$ and $e^{-1} = 0.4$]

List-$I$	List-$II$
$(P)$ At $t = 0.2 \text{ s}$,the magnitude of the induced emf in Volt	$(1)$ $0.07$
$(Q)$ At $t = 0.2 \text{ s}$,the magnitude of the magnetic force in Newton	$(2)$ $0.144$
$(R)$ At $t = 0.2 \text{ s}$,the power dissipated as heat in Watt	$(3)$ $1.20$
$(S)$ The magnitude of terminal velocity of the rod in $\text{m s}^{-1}$	$(4)$ $0.12$
	$(5)$ $2.00$

Two similar circular loops carry equal currents in the same direction. On moving the coils further apart,the electric current will

$A$ square loop of side $1 \, m$ is placed in a perpendicular magnetic field. Half of the area of the loop is inside the magnetic field. $A$ battery of $emf$ $10 \, V$ and negligible internal resistance is connected in the loop. The magnetic field changes with time according to the relation $B = (0.01 - 2t) \, Tesla$. The resultant $emf$ in the loop will be.....$V$

$A$ short magnet is allowed to fall along the axis of a horizontal metallic ring. Starting from rest,the distance fallen by the magnet in one second may be.....$m$

$A$ 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 $r \gg a$ from the center of the loop with its north pole always facing the loop,as shown in the figure.
The magnitude of the magnetic field of a dipole $m$,at a point on its axis at distance $r$,is $\frac{\mu_0}{2 \pi} \frac{m}{r^3}$,where $\mu_0$ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments $m_1$ and $m_2$,separated by a distance $r$ on the common axis,with their north poles facing each other,is $\frac{k m_1 m_2}{r^4}$,where $k$ is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.
$(1)$ 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
$(A) \frac{m}{r^3} \quad (B) \frac{m^2}{r^2} \quad (C) \frac{m}{r^2} \quad (D) \frac{m^2}{r}$
$(2)$ 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
$(A) \frac{m}{r^5} \quad (B) \frac{m^2}{r^5} \quad (C) \frac{m^2}{r^6} \quad (D) \frac{m^2}{r^7}$

Two circular coils $P$ and $Q$ are fixed coaxially and carry currents $I_1$ and $I_2$ respectively.