$A$ coil of wire having finite inductance and resistance has a conducting ring placed coaxially within it. The coil is connected to a battery at time $t = 0$,so that a time-dependent current $I_1(t)$ starts flowing through the coil. If $I_2(t)$ is the current induced in the ring and $B(t)$ is the magnetic field at the axis of the coil due to $I_1(t)$,then as a function of time $(t > 0)$,the product $I_2(t) B(t)$:

$A$ square loop of side $12 \; cm$ with its sides parallel to $X$ and $Y$ axes is moved with a velocity of $8 \; cm \, s^{-1}$ in the positive $x$-direction in an environment containing a magnetic field in the positive $z$-direction. The field is neither uniform in space nor constant in time. It has a gradient of $10^{-3} \; T \, cm^{-1}$ along the negative $x$-direction (that is, it increases by $10^{-3} \; T \, cm^{-1}$ as one moves in the negative $x$-direction), and it is decreasing in time at the rate of $10^{-3} \; T \, s^{-1}$. Determine the direction and magnitude of the induced current in the loop if its resistance is $4.50 \; m\Omega$.

$A$ metallic ring of mass $m$ and radius $l$ (ring being horizontal) is falling under gravity in a region having a magnetic field. If $z$ is the vertical direction,the $z$-component of the magnetic field is $B_z = B_0(1 + \lambda z)$. If $R$ is the resistance of the ring and the ring falls with a velocity $v$,find the energy lost in the resistance per unit time. If the ring has reached a constant velocity,use the conservation of energy to determine $v$ in terms of $m, B_0, l, \lambda, R$ and acceleration due to gravity $g$.

$A$ flexible wire bent in the form of a circle is placed in a uniform magnetic field perpendicular to the plane of the coil. The radius of the coil changes as shown in the figure. The graph of induced $emf$ in the coil is represented by

The figure shows a part of a complete circuit. The potential difference $V_B - V_A$ when the current $I$ is $5 \ A$ and is decreasing at a rate of $10^3 \ A \ s^{-1}$ is given by ........ $V$.

The physical quantity which is measured in the unit of $\text{wb A}^{-1}$ is