Faraday's Laws of Electromagnetic Induction

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JEE Advanced 2017

LEVELJEE Advanced

A circular insulated copper wire loop is twisted to form two loops of area $A$ and $2 A$ as shown in the figure. At the point of crossing, the wires remain electrically insulated from each other. The entire loop lies in the plane (of the paper). A uniform magnetic field $B$ points into the plane of the paper. At $t = 0$ , the loop starts rotating about the common diameter as axis with a constant angular velocity $ω$ in the magnetic field. Which of the following options is/are correct?

* Multiple Correct Options

(A)

The emf induced in the loop is proportional to the sum of the areas of the two loops.

(B)

The rate of change of the flux is maximum when the plane of the loops is perpendicular to plane of the paper.

(C)

The net emf induced due to both the loops is proportional to

cos ω t

(D)

The amplitude of the maximum net emf induced due to both the loops is equal to the amplitude of maximum emf induced in the smaller loop alone.

JEE Advanced 2000

LEVELJEE Advanced

A coil of wire having finite inductance and resistance has a conducting ring placed co-axially within it. The coil is connected to a battery at time $t = 0$ so that a time dependent current $I_{1} (t)$ starts flowing through the coil. If $I_{2} (t)$ is the current induced in the ring and $B (t)$ is the magnetic field at the axis of the coil due to $I_{1} (t)$ , then as a function of time ( $t > 0$ ), the product $I_{2} (t) B (t)$

does not vary with time

(D)

passes through a maximum

JEE Advanced 2007

LEVELJEE Main

Assertion: A vertical iron rod has a coil of wire wound over it at the bottom end. An alternating current flows in the coil. There is a conducting ring round the rod as shown in the figure. The ring can float at a certain height above the coil. Reason: In the above situation, a current is induced in the ring which interacts with the horizontal component of the magnetic field to produce an average force in the upward direction.

(A)

If Assertion is true Reason is true; Reason is the correct explanation for Assertion

(B)

If Assertion is true Reason is true; Reason is not a correct explanation for Assertion

(C)

If Assertion is true; Reason is false

(D)

If Assertion is false; Reason is true

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 $a$ .

LEVELJEE Main

The variation of induced emf (e) with time (t) in a coil if a short bar magnet is moved along its axis with a constant velocity is best represented as

Space is divided by the line $A D$ into two regions. Region I is field free and the region II has a uniform magnetic field $B$ directed into the plane of the paper. $A C D$ is a semicircular conducting loop of radius $r$ with centre at $O$ , the plane of the loop being in the plane of the paper. The loop is now made to rotate with a constant angular velocity $ω$ about an axis passing through $O$ and perpendicular to the plane of the paper. The effective resistance of the loop is $R$ . (a) Obtain an expression for the magnitude of the induced current in the loop. (b) Show the direction of the current when the loop is entering into the region II. (c) Plot a graph between the induced current and the time of rotation for two periods of rotation.

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.

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 Main 2005

LEVELJEE Advanced

A long solenoid of radius $a$ and number of turns per unit length $n$ is enclosed by cylindrical shell of radius $R$ , thickness $d$ ( $d << R$ ) and length $L$ . A variable current $i = i_{0} sin ω t$ flows through the solenoid. If the resistivity of the material of cylindrical shell is $ρ$ , find the induced current in the shell.

LEVELJEE Main

As shown in the figure, $P$ and $Q$ are two coaxial conducting loops separated by some distance. When the switch $S$ is closed, a clockwise current $I_{p}$ flows in $P$ (as seen by $E$ ) and an induced current $I_{Q_{1}}$ flows in $Q$ . The switch remains closed for a long time. When $S$ is opened, a current $I_{Q_{2}}$ flows in $Q$ . Then the direction $I_{Q_{1}}$ and $I_{Q_{2}}$ (as seen by $E$ ) are

(A)

respectively clockwise and anti-clockwise

respectively anti-clockwise and clockwise

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 2012

LEVELJEE Main

A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it. The correct statement(s) is(are)

* Multiple Correct Options

(A)

The emf induced in the loop is zero if the current is constant.

(B)

The emf induced in the loop is finite if the current is constant.

(C)

The emf induced in the loop is zero if the current decreases at a steady rate.

(D)

The emf induced in the loop is finite if the current decreases at a steady rate.

JEE Advanced 2011

LEVELJEE Advanced

A long circular tube of length 10 m and radius 0.3 m carries a current $I$ along its curved surface as shown. A wire-loop of resistance $0.005 Ω$ and of radius 0.1 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 $N μ_{0} I_{0} sin (300 t)$ , then $N$ is

LEVELJEE Main

A uniform but time-varying magnetic field $B (t)$ exists in a circular region of radius $a$ and is directed into the plane of the paper as shown. The magnitude of the induced electric field at point $P$ at a distance $r$ from the centre of the circular region

1/ r

(C)

increases as

r

(D)

decreases as

1/ r^{2}

JEE Advanced 1989

LEVELJEE Main

A conducting square loop of side $L$ and resistance $R$ moves in its plane with a uniform velocity $v$ perpendicular to one of its sides. A magnetic induction $B$ , constant in time and space, pointing perpendicular to and into the plane of the loop exists everywhere. The current induced in the loop is

(A)

B Lv / R

clockwise

(B)

B Lv / R

anti-clockwise

(C)

2 B Lv / R

A field line is shown in the figure. This field cannot represent

* Multiple Correct Options

induced electric field

(D)

gravitational field

LEVELBoard

An infinitely long cylinder is kept parallel to a uniform magnetic field $B$ directed along positive Z-axis. The direction of induced current as seen from the Z-axis will be

(A)

clockwise of the +ve Z-axis

(B)

anti-clockwise of the +ve Z-axis

(C)

zero

(D)

along the magnetic field

LEVELJEE Main

A short-circuited coil is placed in a time varying magnetic field. Electrical power is dissipated due to the current induced in the coil. If the number of turns were to be quadrupled (four times) and the wire radius halved, the electrical power dissipated would be

A rectangular frame $A B C D$ , made of a uniform metal wire, has a straight connection between $E$ and $F$ made of the same wire, as shown in figure. $A E F D$ is a square of side 1 m and $E B = F C = 0.5$ m. The entire circuit is placed in a steadily increasing uniform magnetic field directed into the plane of the paper and normal to it. The rate of change of the magnetic field is 1 T/s. The resistance per unit length of the wire is 1 $Ω$ /m. Find the magnitudes and directions of the currents in the segments $A E$ , $B E$ and $E F$ .

JEE Advanced 2009

LEVELJEE Advanced

Two metallic rings $A$ and $B$ , identical in shape and size but having different resistivities $ρ_{A}$ and $ρ_{B}$ , are kept on top of two identical solenoids as shown in the figure. When current $I$ is switched on in both the solenoids in identical manner, the rings $A$ and $B$ jump to heights $h_{A}$ and $h_{B}$ , respectively, with $h_{A} > h_{B}$ . The possible relation(s) between their resistivities and their masses $m_{A}$ and $m_{B}$ is (are)

* Multiple Correct Options

(A)

ρ_{A} > ρ_{B}

and

m_{A} = m_{B}

(B)

ρ_{A} < ρ_{B}

and

m_{A} = m_{B}

(C)

ρ_{A} > ρ_{B}

and

m_{A} > m_{B}

(D)

ρ_{A} < ρ_{B}

and

m_{A} < m_{B}

JEE Main 2011

LEVELJEE Main

Which of the field patterns given in the figure is valid for electric field as well as for magnetic field?

Which of the field patterns given in the figure is valid for electric field as well as for magnetic field?

Some laws/processes are given in Column I. Match these with the physical phenomena given in Column II.

List-I

(P)

Dielectric ring uniformly charged

(Q)

Dielectric ring uniformly charged rotating with angular velocity

ω

(R)

Constant current in ring

i_{0}

(S)

i = i_{0} cos ω t

List-II

(1)

Time independent electrostatic field out of system

(2)

Magnetic field

(3)

Induced electric field

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 Main 2025

LEVELJEE Advanced

A conducting square loop initially lies in the $X Z$ plane with its lower edge hinged along the $X$ -axis. Only in the region $y \geq 0$ , there is a time dependent magnetic field pointing along the $Z$ -direction, $B (t) = B_{0} (cos ω t) \hat{K}$ , 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 $ω$ 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:

The variation of induced emf (e) with time (t) in a coil if a short bar magnet is moved along its axis with a constant velocity is best represented as

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

A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it. The correct statement(s) is(are)

A field line is shown in the figure. This field cannot represent

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } An infinitely long cylinder is kept parallel to a uniform magnetic field B directed along positive Z-axis. The direction of induced current as seen from the Z-axis will be

A short-circuited coil is placed in a time varying magnetic field. Electrical power is dissipated due to the current induced in the coil. If the number of turns were to be quadrupled (four times) and the wire radius halved, the electrical power dissipated would be

Which of the field patterns given in the figure is valid for electric field as well as for magnetic field?

Which of the field patterns given in the figure is valid for electric field as well as for magnetic field?

Some laws/processes are given in Column I. Match these with the physical phenomena given in Column II.

List-I

List-II

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

An infinitely long cylinder is kept parallel to a uniform magnetic field $B$ directed along positive Z-axis. The direction of induced current as seen from the Z-axis will be