Two infinitely long conducting parallel rails are connected through a capacitor $C$ as shown in the figure. $A$ conductor of length $l$ is moved with constant speed $v_0$. Which of the following graphs truly depicts the variation of current through the conductor with time?

$(a)$ Obtain an expression for the mutual inductance between a long straight wire and a square loop of side $a$ as shown in Figure.
$(b)$ Now assume that the straight wire carries a current of $50\; A$ and the loop is moved to the right with a constant velocity, $v=10\; m / s$. Calculate the induced $emf$ in the loop at the instant when $x=0.2\; m$. Take $a=0.1\; m$ and assume that the loop has a large resistance.

Consider the situation shown in the figure. The wire $PQ$ has a negligible resistance and is made to slide on the three rails with a constant speed of $5 \, cm/s$. Find the current in the $10 \, \Omega$ resistor when the switch $S$ is connected to the middle rail. (Given: $B = 1.0 \, T$) (in $mA$)

$A$ rectangular metallic loop is moving out of a uniform magnetic field region to a field-free region with a constant speed. When the loop is partially inside the magnetic field,the plot of the magnitude of induced $\text{emf} \ (\varepsilon)$ with time $(t)$ is given by:

$A$ coil of area $10 \ m^2$ is placed in a uniform magnetic field of $0.3 \ Wb \cdot m^{-2}$,with its plane perpendicular to the field. The coil rotates at a uniform rate to complete one revolution in $8 \ s$. Find the average emf (in $V$) in the coil during intervals when the coil rotates from:
$i. 0^{\circ}$ to $90^{\circ}$ position
$ii. 90^{\circ}$ to $180^{\circ}$ position
$iii. 180^{\circ}$ to $270^{\circ}$ position
$iv. 270^{\circ}$ to $360^{\circ}$ position

$A$ rod of length $l$ is rotating in the plane of the paper with an angular speed $\omega$ about one of its ends,which is at a distance $a$ from an infinitely long conducting wire carrying current $i$ inwards the plane of the paper. Find the induced $emf$ in the rod at the instant as shown in the figure.