$A$ copper rod of mass $m$ slides under gravity on two smooth parallel rails,with separation $l$ and set at an angle of $\theta$ with the horizontal. At the bottom,the rails are joined by a resistance $R$. There is a uniform magnetic field $B$ normal to the plane of the rails,as shown in the figure. The terminal speed of the copper rod is

$(a)$ 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.

The figure shows a metal rod $PQ$ resting on the smooth rails $AB$ and positioned between the poles of a permanent magnet. The rails,the rod,and the magnetic field are in three mutually perpendicular directions. $A$ galvanometer $G$ connects the rails through a switch $K$. Length of the rod $= 15 \; cm$,$B = 0.50 \; T$,resistance of the closed loop containing the rod $= 9.0 \; m\Omega$. Assume the field to be uniform.
$(a)$ Suppose $K$ is open and the rod is moved with a speed of $12 \; cm \; s^{-1}$ in the direction shown. Give the polarity and magnitude of the induced $emf$.
$(b)$ Is there an excess charge built up at the ends of the rod when $K$ is open? What if $K$ is closed?
$(c)$ With $K$ open and the rod moving uniformly,there is no net force on the electrons in the rod $PQ$ even though they do experience magnetic force due to the motion of the rod. Explain.
$(d)$ What is the retarding force on the rod when $K$ is closed?
$(e)$ How much power is required (by an external agent) to keep the rod moving at the same speed $(= 12 \; cm \; s^{-1})$ when $K$ is closed? How much power is required when $K$ is open?
$(f)$ How much power is dissipated as heat in the closed circuit? What is the source of this power?
$(g)$ What is the induced $emf$ in the moving rod if the magnetic field is parallel to the rails instead of being perpendicular?

The current-carrying wire and the rod $AB$ are in the same plane. The rod moves parallel to the wire with a velocity $v$. Which one of the following statements is true about the induced emf in the rod?

$A$ constant magnetic field of $1 \, T$ is applied in the $x > 0$ region. $A$ metallic circular ring of radius $1 \, m$ is moving with a constant velocity of $1 \, m/s$ along the $x$-axis. At $t = 0 \, s$,the center $O$ of the ring is at $x = -1 \, m$. What will be the value of the induced $emf$ in the ring at $t = 1 \, s$? (Assume the velocity of the ring does not change.) (In $V$)

$A$ conducting rod $PQ$ of length $L = 1.0\, m$ is moving with a uniform speed $v = 20\, m/s$ in a uniform magnetic field $B = 4.0\, T$ directed into the paper. $A$ capacitor of capacity $C = 10\, \mu F$ is connected as shown in the figure. Then: