Embibe Experts Solutions for Chapter: Electromagnetic Induction and Alternating Currents, Exercise 1: JEE Advanced Paper 1 - 2019

The inductors of two $LR$ circuits are placed next to each other, as shown in the figure. The values of the self-inductance of the inductors, resistances, mutual-inductance and applied voltages are specified in the given circuit. After both the switches are closed simultaneously, the total work done by the batteries against the induced $EMF$ in the inductors by the time the currents reach their steady-state values is_______$\mathrm{mJ}$.

A $10\mathrm{ }\mathrm{c}\mathrm{m}$ long perfectly conducting wire $\mathrm{P}\mathrm{Q}$ is moving, with a velocity $1\mathrm{ }\mathrm{c}\mathrm{m}/\mathrm{s}$ on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor $\mathrm{L}=1\mathrm{ }\mathrm{m}\mathrm{H}$ and a resistance $\mathrm{R}=1\mathrm{\Omega }$ as shown in figure. The horizontal rails, $\mathrm{L}$ and $\mathrm{R}$ lie in the same plane with a uniform magnetic field $\mathrm{B}=1\mathrm{ }\mathrm{T}$ perpendicular to the plane. If the key $\mathrm{S}$ is closed at certain instant, the current in the circuit after $1$ milli second is $\mathrm{x}\times {10}^{-3}\mathrm{A},$ where the value of $\mathrm{x}$ is_______.
[Assume the velocity of wire $\mathrm{P}\mathrm{Q}$ remains constant $\left(1\mathrm{c}\mathrm{m}/\mathrm{s}\right)$ after key $\mathrm{S}$ is closed. Given: ${\mathrm{e}}^{-1}=0.37,$ where $\mathrm{e}$ is base of the natural logarithm]

The instantaneous voltages at three terminals marked $X, Y$ and $Z$ are given by

$V_X=V_0\sin \omega t$

$V_Y=V_0\sin \left(\omega t+\frac{2\pi }{3}\right)$ and

$V_Z=V_0\sin \left(\omega t+\frac{4\pi }{3}\right).$

An ideal voltmeter is configured to read rms value of the potential difference between its terminals. It is connected between points $X$ and $Y$ and then between $Y$ and $Z$ . The reading(s) of the voltmeter will be

A long straight wire carries a current, $I=2$ ampere. A semi-circular conducting rod is placed beside it on two conducting parallel rails of negligible resistance. Both the rails are parallel to the wire. The wire, the rod and the rails lie in the same horizontal plane, as shown in the figure. Two ends of the semi-circular rod are at distances $1 \mathrm{cm}$ and $4 \mathrm{cm}$ from the wire.  At time $t=0$, the rod starts moving on the rails with a speed $v=3.0 \mathrm{m} {\mathrm{s}}^{-1}$ (see the figure).

A resistor $R=1.4 \mathrm{\Omega }$ and a capacitor $C_0=5.0 \mathrm{\mu F}$ are connected in series between the rails. At time $t=0$, $C_0$ is uncharged. Which of the following statement(s) is (are) correct? [${\mu }_0=4\pi \times {10}^{-7}$ SI units. Take $\ln 2=0.7$]

A light disc made of aluminium (a nonmagnetic material) is kept horizontally and is free to rotate about its axis as shown in the figure. A strong magnet is held vertically at a point above the disc away from its axis. On revolving the magnet about the axis of the disc, the disc will (figure is schematic and not drawn to scale)

A conducting wire of parabolic shape, initially $\mathrm{y}={\mathrm{x}}^2,$ is moving with velocity $\overrightarrow{\mathrm{V}}={\mathrm{V}}_0\widehat{\mathrm{i}}$ in a non-uniform magnetic field $\overrightarrow{\mathrm{B}}={\mathrm{B}}_0\left(1+{\left(\frac{\mathrm{y}}{\mathrm{L}}\right)}^{\mathrm{\beta }}\right)\widehat{\mathrm{k}},$ as shown in figure. If ${\mathrm{V}}_0,{\mathrm{B}}_0,\mathrm{L}$ and $\mathrm{\beta }$ are positive constants and $\mathrm{\Delta }\mathrm{\varphi }$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

In the circuit shown, initially there is no charge on capacitors and keys ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ are open. The values of the capacitors are ${\mathrm{C}}_1=10\mathrm{\mu }\mathrm{F},{\mathrm{C}}_2=30\mathrm{\mu }\mathrm{F}$ and ${\mathrm{C}}_3={\mathrm{C}}_4=80\mathrm{\mu }\mathrm{F}.$


Which of the statement(s) is/are correct?

In the figure below, the switches $S_1 \mathrm{a}\mathrm{n}\mathrm{d} S_2$ are closed simultaneously at $t = 0$ and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current I in the middle wire reaches its maximum magnitude $I_{max}$ time $t =\tau$. Which of the following statements is (are) true?