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:

Two long parallel wires whose centres are at a distance $d$ apart carry equal currents in opposite directions. If the flux within the wires is neglected,the inductance of such an arrangement of wire of length $l$ and radius $a$ will be

Discuss the types of $AC$ generator. How much power they deliver? What is the frequency of $AC$ generator?

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

$(a)$ $A$ closed loop is held stationary in the magnetic field between the north and south poles of two permanent magnets held fixed. Can we hope to generate current in the loop by using very strong magnets?
$(b)$ $A$ closed loop moves normal to the constant electric field between the plates of a large capacitor. Is a current induced in the loop
$\quad (i)$ when it is wholly inside the region between the capacitor plates
$\quad (ii)$ when it is partially outside the plates of the capacitor? The electric field is normal to the plane of the loop.
$(c)$ $A$ rectangular loop and a circular loop are moving out of a uniform magnetic field region to a field-free region with a constant velocity $v$. In which loop do you expect the induced emf to be constant during the passage out of the field region? The field is normal to the loops.
$(d)$ Predict the polarity of the capacitor in the situation described by the figure.

The plane of a circular coil of resistance $7.5 \ \Omega$ is placed perpendicular to a uniform magnetic field. The flux $\phi$ (in weber) through the coil varies with time $t$ (in second) as $\phi = 2t^2 + 3t - 2$. The induced power in the coil at time $t = 3 \ s$ is (in $W$)