In the circuit shown in the figure,a conducting wire $HE$ is moved with a constant speed $V$ towards the left. The complete circuit is placed in a uniform magnetic field $\vec B$ perpendicular to the plane of the circuit and directed inward. The current in $HKDE$ is

$A$ square loop of side $5 \ cm$ enters a magnetic field with a velocity of $1 \ cm/s$. The front edge enters the magnetic field at $t = 0$. Which graph best depicts the induced $emf$?

$A$ current $I = 10 \ A$ is passed through the part of a circuit shown in the figure. What will be the potential difference between $A$ and $B$ when $I$ is decreased at a constant rate of $10^2 \ A \ s^{-1}$ (in $V$)?

Two different coils have self-inductance $L_1 = 8 \, mH$ and $L_2 = 2 \, mH$. The current in both coils is increased at the same constant rate. At a certain instant of time,the power supplied to the two coils is the same. At that time,the current,induced voltage,and energy stored in the first coil are $i_1, V_1$,and $W_1$ respectively. Corresponding values for the second coil at the same instant are $i_2, V_2$,and $W_2$ respectively. Then:

$A$ 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
$(A) \frac{m}{r^5} \quad (B) \frac{m^2}{r^5} \quad (C) \frac{m^2}{r^6} \quad (D) \frac{m^2}{r^7}$

An electron is moving in a circular orbit of radius $R$ with an angular acceleration $\alpha$. At the centre of the orbit is kept a conducting loop of radius $r$ $(r \ll R)$. The $e.m.f.$ induced in the smaller loop due to the motion of the electron is