Figure shows three conducting rails connected with resistances $R, R$ and $2R.$ (here $R=5 \mathrm{\Omega }$). A wire in form of $\sin$ curve $x=2 \sin \pi y$ is being moved with constant velocity $V=6 \mathrm{m} {\mathrm{s}}^{–1}$ towards $x$-axis. A uniform magnetic field $B=5.0 \mathrm{T}$ is applied perpendicular to the plane of rails and wire. Find out the current(in $\mathrm{A}$) induced in $2R$ resistance. (Here $x \& y$ both are in metres).

A long circular tube of length $10 \mathrm{m}$ and radius $0.3 \mathrm{m}$ carries a current $I$ along its curved surface as shown. A wire-loop of resistance $0.005 \mathrm{ohm}$ and of radius $0.1 \mathrm{m}$ is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as $I=I_0\cos (300 t)$ where $I_0$ is constant. If the magnetic moment of the loop is $N{\mu }_0{I}_0\sin (300 t)$, then $\text{'}N\text{'}$ is

A conducting rod of length $5 \mathrm{m}$ slides in a vertical plane against two insulated perpendicular surfaces as shown. Outward uniform magnetic field $B_0=3$ Tesla exists in the region. If end $A$ of the rod is being pulled with constant speed $V_0=2 \mathrm{m} {\mathrm{s}}^{–1} \hat{i}$, then find the magnitude of induced Emf (in volt) across end $A$ and $B$ of the rod at the instant when the rod makes an angle of $\theta =37°$ with horizontal.

In the system shown, distance between two parallel rails are $10 \mathrm{cm}.$ Two metallic rods having resistance of $10 \mathrm{\Omega }$ and $15 \mathrm{\Omega }$ are pulled with constant speed $4 \mathrm{m} {\mathrm{s}}^{–1}$ and $2 \mathrm{m} {\mathrm{s}}^{–1}$ respectively. A uniform magnetic field of magnitude $0.01 \mathrm{T}$ is applied perpendicular to the plane of the rails. The current in the $5 \mathrm{\Omega }$ resistor is $\frac{k}{55} \mathrm{mA}.$ Value of $k$, is

A rod of length $l$ Starts sliding down the smooth parabolic curve $y=\frac{x^2}{a}$ from $y=a$ as shown in figure. Uniform magnetic field $B_0\hat{j}$ is present in region. EMF developed across the ends of rod when it reaches the point $x=\frac{a}{\sqrt{2}}$ is $B_{0}l\sqrt{\frac{ga}{n}}.$ Value of $n$, is

Consider the circuit shown in figure, initially capacitor is uncharged and switch is closed at $t=0.$ Value of $\frac{I_1}{I_2}$ at $t=\ln 2 \mathrm{S}$, will be

A parallel $L-R$ circuit consists of two ideal inductances $L$ and $2L$ and a resistor $R.$ At an instant the same current $I_0$ is flowing in both the inductor in the same direction. How much heat (in $\mu \mathrm{J}$) will be dissipated in resistance after a long time? $\left(L{I}_0{}^2=24 \mu \mathrm{J}\right)$

In a square loop of side length $\text{'}a\text{'},$ current $I_0$ is flowing as shown. If magnetic flux associated with region $x>0$ in $xy$ plane is $\frac{{\mu }_0{I}_{0}a}{2\pi }\ln N$, then value of $N$ is

A conducting wire of length $L$ fixed at both ends is vibrating with displacement equation given by $y=2A \sin \left(kx\right) \cos \omega t.$ A constant magnetic field $B$ is present in the region as shown. If maximum value of emf induced across the wire is $\frac{4 AB\omega L}{5\pi }$, then value of $\frac{kL}{\pi }$ is