Core Syllabus Module

Electromagnetic Induction

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JEE Advanced 1995

LEVELJEE Advanced

A metal rod $O A$ and mass $m$ and length $r$ kept rotating with a constant angular speed $ω$ in a vertical plane about horizontal axis at the end $O$ . The free end $A$ is arranged to slide without friction along a fixed conducting circular ring in the same plane as that of rotation. A uniform and constant magnetic induction $B$ is applied perpendicular and into the plane of rotation as shown in figure. An inductor $L$ and an external resistance $R$ are connected through a switch $S$ between the point $O$ and a point $C$ on the ring to form an electrical circuit. Neglect the resistance of the ring and the rod. Initially, the switch is open. (a) What is the induced emf across the terminals of the switch? (b) The switch $S$ is closed at time $t = 0$ . (i) Obtain an expression for the current as a function of time. (ii) In the steady state, obtain the time dependence of the torque required to maintain the constant angular speed. Given that the rod $O A$ was along the positive $x$ -axis at $t = 0$ .

JEE Advanced 2002

LEVELJEE Advanced

A metal bar $A B$ can slide on two parallel thick metallic rails separated by a distance $l$ . A resistance $R$ and an inductance $L$ are connected to the rails as shown in the figure. A long straight wire, carrying a constant current $I_{0}$ is placed in the plane of the rails as shown. The bar $A B$ is held at rest at a distance $x_{0}$ from the long wire. At $t = 0$ , it made to slide on the rails away from the wire. Answer the following questions. (a) Find a relation among $i$ , $\frac{d i}{d t}$ and $\frac{d ϕ}{d t}$ , where $i$ is the current in the circuit and $ϕ$ is the flux of the magnetic field due to the long wire through the circuit. (b) It is observed that at time $t = T$ , the metal bar $A B$ is at a distance of $2 x_{0}$ from the long wire and the resistance $R$ carries a current $i_{1}$ . Obtain an expression for the net charge that has flown through resistance $R$ from $t = 0$ to $t = T$ . (c) The bar is suddenly stopped at time $T$ . The current through resistance $R$ is found to be $i_{1} /4$ at time $2 T$ . Find the value of $L / R$ in terms of the other given quantities.

JEE Advanced 1991

LEVELJEE Advanced

A circuit containing a two position switch S is shown in figure. (a) The switch S is in position 1. Find the potential difference $V_{A} - V_{B}$ and the rate of production of joule heat in $R_{1}$ . (b) If now the switch S is put in position 2 at $t = 0$ . Find (i) steady current in $R_{4}$ and (ii) the time when current in $R_{4}$ is half the steady value. Also, calculate the energy stored in the inductor $L$ at that time.

JEE Advanced 1997

LEVELJEE Advanced

An infinitesimally small bar magnet of dipole moment $M$ is pointing and moving with the speed $v$ in the positive $x$ -direction. A small closed circular conducting loop of radius $a$ and negligible self-inductance lies in the $y$ - $z$ plane with its centre at $x = 0$ , and its axis coinciding with the $X$ -axis. Find the force opposing the motion of the magnet, if the resistance of the loop is $R$ . Assume that the distance $x$ of the magnet from the centre of the loop is much greater than $a$ .

JEE Advanced 2013

LEVELJEE Advanced

Comprehension Passage

A point charge

Q

is moving in a circular orbit of radius

R

in the

x

y

plane with an angular velocity

ω

. This can be considered as equivalent to a loop carrying a steady current

\frac{Q ω}{2 π}

. A uniform magnetic field along the positive

z

-axis is now switched on, which increases at a constant rate from

0

B

in one second. Assume that the radius of the orbit remains constant. The applications of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant

γ

Question 1:

The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is

(A)

\frac{B R}{4}

(B)

\frac{B R}{2}

(C)

B R

(D)

2 B R

Question 2:

The change in the magnetic dipole moment associated with the orbit, at the end of the time interval of the magnetic field change, is

(A)

γ B Q R^{2}

(B)

- γ \frac{B Q R ^{2}}{2}

(C)

γ \frac{B Q R ^{2}}{2}

(D)

- γ B Q R^{2}

JEE Advanced 2003

LEVELJEE Advanced

Two infinitely long parallel wires carrying currents $I = I_{0} sin ω t$ in opposite directions are placed a distance $3 a$ apart. A square loop of side $a$ of negligible resistance with a capacitor of capacitance $C$ is placed in the plane of wires as shown. Find the maximum current in the square loop. Also sketch the graph showing the variation of charge on the upper plate of the capacitor as a function of time for one complete cycle taking anti-clockwise direction for the current in the loop as positive.

JEE Advanced 1988

LEVELJEE Advanced

Two long parallel horizontal rails, a distance $d$ apart and each having a resistance $λ$ per unit length, are joined at one end by a resistance $R$ . A perfectly conducting rod $M N$ of mass $m$ is free to slide along the rails without friction (see figure). There is a uniform magnetic field of induction $B$ normal to the plane of the paper and directed into the paper. A variable force $F$ is applied to the rod $M N$ such that, as the rod moves, a constant current $i$ flows through $R$ . (i) Find the velocity of the rod and the applied force $F$ as functions of the distance $x$ of the rod from $R$ . (ii) What fraction of the work done per second is converted into heat.

JEE Advanced 2014

LEVELJEE Advanced

At time $t = 0$ , terminal $A$ in the circuit shown in the figure is connected to $B$ by a key and an alternating current $I (t) = I_{0} cos (ω t)$ , with $I_{0} = 1 A$ and $ω = 500 rad s^{- 1}$ starts flowing in it with the initial direction shown in the figure. At $t = \frac{7 π}{6 ω}$ , the key is switched from $B$ to $D$ . Now onwards only $A$ and $D$ are connected. A total charge $Q$ flows from the battery to charge the capacitor fully. If $C = 20 μ F$ , $R = 10 Ω$ and the battery is ideal with emf of $50 V$ , identify the correct statement(s).

* Multiple Correct Options

(A)

Magnitude of the maximum charge on the capacitor before

t = \frac{7 π}{6 ω}

1 \times 1 0^{- 3} C

(B)

The current in the left part of the circuit just before

t = \frac{7 π}{6 ω}

is clockwise

(C)

Immediately after

A

is connected to

D

, the current in

R

10 A

(D)

Q = 2 \times 1 0^{- 3} C

JEE Advanced 2016

LEVELJEE Advanced

A rigid wire loop of square shape having side of length $L$ and resistance $R$ is moving along the $x$ -axis with a constant velocity $v_{0}$ in the plane of the paper. At $t = 0$ , the right edge of the loop enters a region of length $3 L$ , where there is a uniform magnetic field $B_{0}$ into the plane of the paper, as shown in the figure. For sufficiently large $v_{0}$ , the loop eventually crosses the region. Let $x$ be the location of the right edge of the loop. Let $v (x)$ , $I (x)$ and $F (x)$ represent the velocity of the loop, current in the loop, and force on the loop, respectively, as a function of $x$ . Counter-clockwise current is taken as positive. Which of the following schematic plot(s) is (are) correct? (Ignore gravity)

* Multiple Correct Options

Comprehension Passage

A special metal S conducts electricity without any resistance. A closed wire loop, made of S, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius a, with its center at the origin. A magnetic dipole of moment m is brought along the axis of this loop from infinity to a point at distance r (>> a) from the center of the loop with its north pole always facing the loop, as shown in the figure below. The magnitude of magnetic field of a dipole m, at a point on its axis at distance r, is

\frac{μ _{0}}{2 π} \frac{m}{r ^{3}}

, where

μ_{0}

is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments,

m_{1}

and

m_{2}

, separated by a distance r on the common axis, with their north poles facing each other, is

\frac{k m _{1} m _{2}}{r ^{4}}

, where k is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question 1:

When the dipole m is placed at a distance r from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to

(A)

\frac{m}{r ^{3}}

(B)

\frac{m ^{2}}{r ^{2}}

(C)

\frac{m}{r ^{2}}

(D)

\frac{m ^{2}}{r}

Question 2:

The work done in bringing the dipole from infinity to a distance r from the center of the loop by the given process is proportional to

(A)

\frac{m}{r ^{5}}

(B)

\frac{m ^{2}}{r ^{5}}

(C)

\frac{m ^{2}}{r ^{6}}

(D)

\frac{m ^{2}}{r ^{7}}

JEE Advanced 2024

LEVELJEE Advanced

A region in the form of an equilateral triangle (in $x - y$ plane) of height $L$ has a uniform magnetic field $B$ 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 $v$ along the $+ x$ -direction. The plane of the loop and its orientation remain unchanged throughout its motion.

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 $d^{2} ≫ S$ . There is another coil of length $d /2$ , cross sectional area $S /2$ and $2 N$ 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:

(A)

\frac{4}{15 L C}

(B)

\frac{6}{5 L C}

(C)

\frac{2}{3 L C}

(D)

\frac{2}{3 L C}

JEE Advanced 2023

LEVELJEE Advanced

A series LCR circuit is connected to a $45 sin (ω t)$ Volt source. The resonant angular frequency of the circuit is $1 0^{5} rad s^{- 1}$ and current amplitude at resonance is $I_{0}$ . When the angular frequency of the source is $ω = 8 \times 1 0^{4} rad s^{- 1}$ , the current amplitude in the circuit is $0.05 I_{0}$ . If $L = 50 mH$ , match each entry in List-I with an appropriate value from List-II and choose the correct option.

List-I

(P)

I_{0}

in mA

(Q)

The quality factor of the circuit

(R)

The bandwidth of the circuit in

rad s^{- 1}

(S)

The peak power dissipated at resonance in Watt

List-II

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 $\overset{n}{^}$ 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 $\overset{n}{^}$ direction. List-I gives time dependences of $\overset{n}{^}$ in terms of a constant angular frequency $ω$ . List-II gives the torques experienced by the circular loop at time $t = \frac{π}{6 ω}$ , Let $α = \frac{A ^{2} μ _{0}^{2} m ^{2} I ^{2} ω}{2 R}$ .

List-I

(P)

\frac{1}{2} (sin ω t \hat{j} + cos ω t \hat{k})

(Q)

\frac{1}{2} (sin ω t \hat{i} + cos ω t \hat{j})

(R)

\frac{1}{2} (sin ω t \hat{i} + cos ω t \hat{k})

(S)

\frac{1}{2} (cos ω t \hat{i} + sin ω t \hat{k})

List-II

(1)

0

(2)

- \frac{α}{4} \hat{i}

(3)

\frac{3 α}{4} \hat{i}

(4)

\frac{α}{4} \hat{j}

(5)

- \frac{3 α}{4} \hat{i}

JEE Main 2019

LEVELJEE Advanced

A thin strip 10 cm long is on an U-shaped wire of negligible resistance and it is connected to a spring of spring constant $0.5 Nm^{- 1}$ (see figure). The assembly is kept in a uniform magnetic field of $0.1 T$ . If the strip is pulled from its equilibrium position and released, the number of oscillations it performs before its amplitude decreases by a factor of $e$ is $N$ . If the mass of the strip is 50 grams, its resistance $10 Ω$ and air drag negligible, $N$ will be close to

Two parallel vertical metallic rails $A B$ and $C D$ are separated by $1 m$ . They are connected at two ends by resistances $R_{1}$ and $R_{2}$ as shown in figure. A horizontal metallic bar of mass $0.2 kg$ slides without friction vertically down the rails under the action of gravity. There is a uniform horizontal magnetic field of $0.6 T$ perpendicular to the plane of the rails. It is observed that when the terminal velocity is attained, the powers dissipated in $R_{1}$ and $R_{2}$ are $0.76 W$ and $1.2 W$ respectively. Find the terminal velocity of the bar $L$ and the values of $R_{1}$ and $R_{2}$ .

LEVELJEE Advanced

An inductor of inductance $2.0 mH$ is connected across a charged capacitor of capacitance $5.0 μ F$ and the resulting $L - C$ circuit is set oscillating at its natural frequency. Let $Q$ denote the instantaneous charge on the capacitor and $I$ , the current in the circuit. It is found that the maximum value of $Q$ is $200 μ C$ . (a) When $Q = 100 μ C$ , what is the value of $\frac{d I}{d t}$ ? (b) When $Q = 200 μ C$ , what is the value of $I$ ? (c) Find the maximum value of $I$ . (d) When $I$ is equal to one-half its maximum value, what is the value of $∣ Q ∣$ ?

JEE Advanced 2016

LEVELJEE Advanced

A conducting loop in the shape of a right angled isosceles triangle of height 10 cm is kept such that the $9 0^{\circ}$ vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. The current in the triangular loop is in counterclockwise direction and increased at a constant rate of $10 As^{- 1}$ . Which of the following statement(s) is (are) true? (2016 Adv.)

* Multiple Correct Options

(A)

There is a repulsive force between the wire and the loop.

(B)

If the loop is rotated at a constant angular speed about the wire, an additional emf of

(\frac{μ _{0}}{π})

volt is induced in the wire.

(C)

The magnitude of induced emf in the wire is

(\frac{μ _{0}}{π})

volt.

(D)

The induced current in the wire is in opposite direction to the current along the hypotenuse.

JEE Advanced 1999

LEVELJEE Advanced

A magnetic field $B = (B_{0} y / a) \hat{k}$ is acting into the paper in the $+ z$ direction. $B_{0}$ and $a$ are positive constants. A square loop $E F G H$ of side $a$ , mass $m$ and resistance $R$ in $x - y$ plane starts falling under the influence of gravity. Note the directions of $x$ and $y$ in the figure. Find (a) the induced current in the loop and indicate its direction. (b) the total Lorentz force acting on the loop and indicate its direction. (c) an expression for the speed of the loop $v (t)$ and its terminal velocity.

LEVELJEE Advanced

A thermocol vessel contains $0.5 kg$ of distilled water at $3 0^{\circ} C$ . A metal coil of area $5 \times 1 0^{- 3} m^{2}$ , number of turns $100$ , mass $0.06 kg$ and resistance $1.6 Ω$ is lying horizontally at the bottom of the vessel. A uniform time varying magnetic field is setup to pass vertically through the coil at time $t = 0$ . The field is first increased from $0$ to $0.8 T$ at a constant rate between $0$ and $0.2 s$ and then decreased to zero at the same rate between $0.2$ and $0.4 s$ . The cycle is repeated $12000$ times. Make sketches of the current through the coil and the power dissipated in the coil as a function of time for the first two cycles. Clearly indicate the magnitudes of the quantities on the axes. Assume that no heat is lost to the vessel or the surroundings. Determine the final temperature of the water under thermal equilibrium. Specific heat of metal $= 500 J/kg-K$ and the specific heat of water $= 4200 J/kg-K$ . Neglect the inductance of coil.

Micro-Topic Strategy Matrix

Faraday's Laws of Electromagnetic Induction

Lenz's Law

Motional EMF

Self and Mutual Inductance

Alternating Current (AC) and Voltage

Chapter Elite-20 Challenge Zone

Comprehension Passage

The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is

The change in the magnetic dipole moment associated with the orbit, at the end of the time interval of the magnetic field change, is

Comprehension Passage

When the dipole m is placed at a distance r from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to

The work done in bringing the dipole from infinity to a distance r from the center of the loop by the given process is proportional to

List-I

List-II

List-I

List-II