Electromagnetic Induction

Consider a circuit consisting of a capacitor of capacitance $C$ and a coil with $N$ turns per unit length, cross sectional area $S$ and length $d$, where $d^2\gg S$. There is another coil of length $d/2$, cross sectional area $S/2$ and $2N$ turns per unit length completely inside the larger coil, as shown in the figure. The ends of this smaller coil are connected with each other by an insulated conducting wire. The self-inductance of the larger coil is $L$. Neglecting edge effects and all the Ohmic resistances, the resonant frequency of the circuit is:

List-I contains four conducting loops lying in the $XY$ plane, as shown in the figures. The loops are rotating about $Z$ axis passing through the point $O$ with time period $T$ in clockwise direction. The region $x>0$ contains a uniform magnetic field $B$ in the $+z$ direction. List-II contains the qualitative variation of the induced current $i(t)$ for each of these loops. Choose the option which describes the correct match between the entries in List-I to those in List-II.

When a coil is placed in a time dependent magnetic field the power dissipated in it is $P$. The number of turns, area of the coil and radius of the coil wire are $N$, $A$ and $r$ respectively. For a second coils number of turns, area of the coil and radius of the coil wire are $2N$, $2A$ and $3r$ respectively. When the first coil is replaced with second coil the power dissipated in it is $\sqrt2\,\alpha P$. The value of $\alpha$ is ________.

A circular current loop of radius $R$ is placed inside square loop of side length $L$ ($L\gg R$) such that they are co-planar and their centers coincide. The permeability of free space is $\mu_0$. The mutual inductance between circular loop and square loop is ________.

In the given circuit below inductance values of $L_1$, $L_2$ and $L_3$ are same. The magnetic energy stored in the entire circuit is $(U_t)$ and that stored in the $L_2$ inductor is $(U_l)$. $U_t/U_l$ is ________.
(Ignore the mutual inductance if any)

A metal rod of length $L$ rotates about one end at origin with a uniform angular velocity $\omega$. The magnetic field radially falls off as $B(r)=B_oe^{-\lambda r}$; $\lambda$ being a positive constant. The emf induced (neglecting the centripetal force on electrons in the rod) is :

A circular loop of radius 20 cm and resistance 2 Ω is placed in a time varying magnetic field $\vec B=\left(2t^2+2t+3\right)$ T. At $t=0$, for the plane of the loop being perpendicular to the magnetic field and, the induced current in the loop at $t=3$ s is $\dfrac\alpha{50}$ A. The value of $\alpha$ is ________.

(Take $\pi=22/7$)

A small cube of side 1 mm is placed at the centre of a circular loop of radius 10 cm carrying a current of 2 A. The magnetic energy stored inside the cube is $\alpha\times10^{-14}$ J. The value of $\alpha$ is ________.

($\mu_o=4\pi\times10^{-7}$ Tm/A, $\pi=3.14$)

An inductor of inductance 10 mH having resistance of 100 Ω is connected to battery of E.M.F. 1.0 V through a switch as shown in the figure below. After switch is closed, the ratio of instantaneous voltages across the inductor when the current passing through it is 2 mA and 4 mA is ________.

A square loop of side 2 cm is placed in a time varying magnetic field with magnitude as $B=0.4\sin(300t)$ Tesla. The normal to the plane of loop makes an angle of 60° with the field. The maximum induced emf produced in the loop is ________ mV.

A 30 cm long solenoid has 10 turns per cm and area of 5 cm². The current through the solenoid coil varies from 2 A to 4 A in 3.14 s. The e.m.f. induced in the coil is $\alpha\times10^{-5}$ V. The value $\alpha$ is ________.

A conducting square loop initially lies in the $XZ$ plane with its lower edge hinged along the $X$-axis. Only in the region $y \ge 0$, there is a time dependent magnetic field pointing along the $Z$-direction, $\vec B(t) = B_0(\cos\omega t)\hat k$, where $B_0$ is a constant. The magnetic field is zero everywhere else. At time $t = 0$, the loop starts rotating with constant angular speed $\omega$ about the $X$ axis in the clockwise direction as viewed from the $+X$ axis (as shown in the figure). Ignoring self-inductance of the loop and gravity, which of the following plots correctly represents the induced e.m.f. ($V$) in the loop as a function of time:

A conducting square loop of side $L$, mass $M$ and resistance $R$ is moving in the $XY$ plane with its edges parallel to the $X$ and $Y$ axes. The region $y \ge 0$ has a uniform magnetic field, $\vec B = B_0\hat k$. The magnetic field is zero everywhere else. At time $t = 0$, the loop starts to enter the magnetic field with an initial velocity $v_0\hat j$ m/s, as shown in the figure. Considering the quantity $K = \dfrac{B_0^2L^2}{RM}$ in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

The circuit shown in the figure contains an inductor $L$, a capacitor $C_0$, a resistor $R_0$ and an ideal battery. The circuit also contains two keys K₁ and K₂. Initially, both the keys are open and there is no charge on the capacitor. At an instant, key K₁ is closed and immediately after this the current in $R_0$ is found to be $I_1$. After a long time, the current attains a steady state value $I_2$. Thereafter, K₂ is closed and simultaneously K₁ is opened and the voltage across $C_0$ oscillates with amplitude $V_0$ and angular frequency $\omega_0$.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

A region in the form of an equilateral triangle (in $x$-$y$ plane) of height $L$ has a uniform magnetic field $\vec B$ pointing in the $+z$-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height $L$, is placed in the $x$-$y$ plane with its vertex P at $x = 0$ in the orientation shown in the figure. At $t = 0$, the loop starts entering the region of the magnetic field with a uniform velocity $\vec v$ along the $+x$-direction. The plane of the loop and its orientation remain unchanged throughout its motion.

Which of the following graph best depicts the variation of the induced emf ($E$) in the loop as a function of the distance ($x$) starting from $x = 0$?

A thin conducting rod MN of mass 20 gm, length 25 cm and resistance $10\,\Omega$ is held on frictionless, long, perfectly conducting vertical rails as shown in the figure. There is a uniform magnetic field $B_0 = 4$ T directed perpendicular to the plane of the rod-rail arrangement. The rod is released from rest at time $t = 0$ and it moves down along the rails. Assume air drag is negligible. Match each quantity in List-I with an appropriate value from List-II, and choose the correct option.

[Given: The acceleration due to gravity $g = 10\ \mathrm{m\,s^{-2}}$ and $e^{-1} = 0.4$]

A rectangular conducting loop of length 4 cm and width 2 cm is in the $xy$-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction $\dfrac{\sqrt3}{2}\hat x + \dfrac12\hat y$ with a constant speed v. The wire is carrying a steady current $I = 10$ A in the positive $x$-direction. A current of $10\ \mu$A flows through the loop when it is at a distance $d = 4$ cm from the wire. If the resistance of the loop is $0.1\ \Omega$, then the value of v is _____ $\mathrm{m\,s^{-1}}$.

[Given: The permeability of free space $\mu_0 = 4\pi\times10^{-7}\ \mathrm{N\,A^{-2}}$]

Consider an LC circuit, with inductance $L = 0.1$ H and capacitance $C = 10^{-3}$ F, kept on a plane. The area of the circuit is $1\ \mathrm m^2$. It is placed in a constant magnetic field of strength $B_0$ which is perpendicular to the plane of the circuit. At time $t = 0$, the magnetic field strength starts increasing linearly as $B = B_0 + \beta t$ with $\beta = 0.04\ \mathrm{T\,s^{-1}}$. The maximum magnitude of the current in the circuit is _____ mA.

A small circular loop of area $A$ and resistance $R$ is fixed on a horizontal $xy$-plane with the center of the loop always on the axis $\hat n$ of a long solenoid. The solenoid has $m$ turns per unit length and carries current $I$ counterclockwise as shown in the figure. The magnetic field due to the solenoid is in $\hat n$ direction. List-I gives time dependences of $\hat n$ in terms of a constant angular frequency $\omega$. List-II gives the torques experienced by the circular loop at time $t = \dfrac{\pi}{6\omega}$. Let $\alpha = \dfrac{A^2\mu_0^2m^2I^2\omega}{2R}$.

Which one of the following options is correct?

A circular coil of 1000 turns each with area $1\ \mathrm m^2$ is rotated about its vertical diameter at the rate of one revolution per second in a uniform horizontal magnetic field of 0.07 T. The maximum voltage generation will be ______ V.