In the figure shown a conducting rod of length $l$, resistance $R$ and mass $m$ can move vertically downward due to gravity. Other parts are kept fixed. $B=\mathrm{constant}=B_0.$$MN \mathrm{and} PQ$are vertical smooth conducting rails. The capacitance of the capacitor is $C.$ The rod is released from rest. Find the maximum current in the circuit.

In the figure, a conducting rod of length $l=1 \mathrm{m}$ and the mass $m=1 \mathrm{kg}$ moves with initial velocity $u=5 \mathrm{m} {\mathrm{s}}^{-1}$ on a fixed horizontal frame containing inductor $L=2 \mathrm{H}$ and resistance $R=1 \mathrm{\Omega }.$$PQ \mathrm{and} MN$ are smooth conducting wires. There is a uniform magnetic field of strength $B=1 \mathrm{T}.$Initially there is no current in the inductor. Find the total charge in coulomb flown through the inductor by the time velocity of the rod becomes $v_{\mathrm{f}}=1 \mathrm{m} {\mathrm{s}}^{-1}$ and the rod has traveled a distance $x=3 \mathrm{m}.$

A conducting frame $ABCD$ is kept fixed in a vertical plane. A conducting rod $EF$ of the mass $m$ can slide smoothly on its remaining horizontal always. The resistance of the loop is negligible and inductance is constant having value $L$. The rod is left from rest and allowed to fall under gravity and the inductor has no initial current. A uniform magnetic field of magnitude $B$ is present throughout the loop pointing inwards. Determine, 

$\left(\mathrm{a}\right)$ The position of the rod as a function of time assuming the initial position of the rod to be $x=0$ and vertically downward as the positive $x$-axis,

$\left(\mathrm{b}\right)$ The maximum current in the circuit,

$\left(\mathrm{c}\right)$ The maximum velocity of the rod.

$L$ is s smooth conducting loop of radius $l=1.0 \mathrm{m}$ & fixed in a horizontal plane. A conducting rod of mass, $m=1.0 \mathrm{kg}$and length slightly greater than $l$ hinged at the center of the loop can rotate in the horizontal plane such that the free end slides on the rim of the loop. There is a uniform magnetic field of strength $B=1.0 \mathrm{T}$ directed vertically downward. The rod is rotated with angular velocity ${\omega }_0=1.0 \mathrm{rad} {\mathrm{s}}^{-1}$ and left. The fixed end of the rod and the rim of the loop are connected through a battery of e.m.f. $E$, a resistor of resistance $R=1.0 \mathrm{\Omega },$ and initially uncharged capacitor of capacitance $C=1.0 \mathrm{F}$ in series. Find:

$\left(\mathrm{i}\right)$ The time dependence of e.m.f. $E$ such that the current, $I_0=1.0 \mathrm{A}$ in the circuit is constant.

$\left(\mathrm{ii}\right)$ The energy is supplied by the battery by the time rod stops.

In the figure shown$PQRS$ is a fixed resistance less conducting frame in a uniform and constant magnetic field of strength $B$. A rod $EF$ of mass $m,$ length $l$ and resistance $R$ can smoothly move on this frame. A capacitor charged to a potential difference $V_0$ initially is connected as shown in the figure. Find the velocity of the rod as a function of time $t$if it is released at $t=0$ from rest.

In the figure shown a long conductor carries constant current $I$. A rod $PQ$ of length $l$ is in the plane of the rod. The rod is rotated about point $P$ with constant angular velocity $\mathrm{\omega }$ as shown in the figure. Find the emf induced in the rod in the position shown. Indicate which point is at high potential.

In the circuit shown, the switch S is shifted to the position $2$ from the position $1$ at, $t=0,$having been in position 1 for a long time. Find the current in the circuit as a function of time.