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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 1994

LEVELJEE Advanced

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 $R_{2}$ .

JEE Advanced 1996

LEVELJEE Main

A thin semicircular conducting ring of radius $R$ is falling with its plane vertical in a horizontal magnetic induction $B$ . At the position $M N Q$ the speed of the ring is $v$ and the potential difference developed across the ring is

(A)

zero

(B)

B v π R^{2} /2

and

M

is at higher potential

(C)

π B R v

and

Q

is at higher potential

(D)

2 R B v

and

Q

is at higher potential

LEVELJEE Main

A metal rod moves at a constant velocity in a direction perpendicular to its length. A constant uniform magnetic field exists in space in a direction perpendicular to the rod as well as its velocity. Select the correct statement (s) from the following.

(A)

The entire rod is at the same electric potential.

(B)

There is an electric field in the rod.

(C)

The electric potential is highest at the centre of the rod and decrease towards its ends.

(D)

The electric potential is lowest at the centre of the rod and increases towards its ends.

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.

JEE Advanced 1983

LEVELJEE Main

A square metal wire loop of side 10 cm and resistance 1 $Ω$ is moved with a constant velocity $v_{0}$ in a uniform magnetic field of induction $B = 2 Wb/m^{2}$ as shown in the figure. The magnetic field lines are perpendicular to the plane of the loop (directed into the paper). The loop is connected to a network of resistors each of value 3 $Ω$ . What should be the speed of the loop so as to have a steady current of 1mA in the loop? Give the direction of current in the loop.

LEVELJEE Main

A metallic square loop $A B C D$ is moving in its own plane with velocity $v$ in a uniform magnetic field perpendicular to its plane as shown in the figure. Electric field is induced

(A)

A D

, but not in

B C

(B)

B C

, but not in

A D

(C)

neither in

A D

nor in

B C

(D)

in both

A D

and

B C

JEE Advanced (1997)

LEVELJEE Advanced

A pair of parallel horizontal conducting rails of negligible resistance shorted at one end is fixed on a table. The distance between the rails is $L$ . A conducting massless rod of resistance $R$ can slide on the rails frictionlessly. The rod is tied to a massless string which passes over a pulley fixed to the edge of the table. A mass $m$ tied to the other end of the string hangs vertically. A constant magnetic field $B$ exists perpendicular to the table. If the system is released from rest. Calculate : (a) the terminal velocity achieved by the rod, and (b) the acceleration of the mass at the instant when the velocity of the rod is half the terminal velocity.

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 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

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 $1 cm$ and $4 cm$ from the wire. At time $t = 0$ , the rod starts moving on the rails with a speed $v = 3.0 m/s$ (see the figure). A resistor $R = 1.4 Ω$ and a capacitor $C_{0} = 5.0 μ F$ are connected in series between the rails. At time $t = 0$ , $C_{0}$ is uncharged. Which of the following statement(s) is(are) correct ? [ $μ_{0} = 4 π \times 1 0^{- 7}$ SI units. Take $ln 2 = 0.7$ ]

* Multiple Correct Options

(A)

Maximum current through

R

1.2 \times 1 0^{- 6} ampere

(B)

Maximum current through

R

3.8 \times 1 0^{- 6} ampere

(C)

Maximum charge on capacitor

C_{0}

8.4 \times 1 0^{- 12} coulomb

(D)

Maximum charge on capacitor

C_{0}

2.4 \times 1 0^{- 12} coulomb

JEE Advanced 2019

LEVELJEE Advanced

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

JEE Main 2025

LEVELJEE Advanced

A conducting square loop of side $L$ , mass $M$ and resistance $R$ is moving in the $X Y$ plane with its edges parallel to the $X$ and $Y$ axes. The region $y \geq 0$ has a uniform magnetic field, $B = B_{0} \hat{k}$ . 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 $K = \frac{B _{0}^{2} L ^{2}}{R M}$ in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

* Multiple Correct Options

(A)

v_{0} = 1.5 K L

, the loop will stop before it enters completely inside the region of magnetic field.

(B)

When the complete loop is inside the region of magnetic field, the net force acting on the loop is zero.

(C)

v_{0} = \frac{K L}{10}

, the loop comes to rest at

t = (\frac{1}{K}) ln (\frac{5}{2})

(D)

v_{0} = 3 K L

, the complete loop enters inside the region of magnetic field at time

t = (\frac{1}{K}) ln (\frac{3}{2})

JEE Advanced 2023

LEVELJEE Advanced

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 $\frac{3}{2} \overset{x}{^} + \frac{1}{2} \overset{y}{^}$ with a constant speed $v$ . The wire is carrying a steady current $I = 10 A$ in the positive x-direction. A current of $10 μ A$ flows through the loop when it is at a distance $d = 4 cm$ from the wire. If the resistance of the loop is $0.1 Ω$ , then the value of $v$ is _________ $ms^{- 1}$ . [Given: The permeability of free space $μ_{0} = 4 π \times 1 0^{- 7} NA^{- 2}$ ]

JEE Advanced 2019

LEVELJEE Advanced

A conducting wire of parabolic shape, initially $y = x^{2}$ , is moving with velocity $V = V_{0} \hat{i}$ in a non-uniform magnetic field $B = B_{0} (1 + (\frac{y}{L})^{β}) \hat{k}$ , as shown in figure. If $V_{0}, B_{0}, L$ and $β$ are positive constants and $Δ ϕ$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

* Multiple Correct Options

(A)

∣Δ ϕ ∣

remains the same if the parabolic wire is replaced by a straight wire,

y = x

initially, of length

2 L

(B)

∣Δ ϕ ∣

is proportional to the length of the wire projected on the y-axis.

(C)

∣Δ ϕ ∣ = \frac{1}{2} B_{0} V_{0} L

for

β = 0

(D)

∣Δ ϕ ∣ = \frac{4}{3} B_{0} V_{0} L

for

β = 2

JEE Advanced 2023

LEVELJEE Advanced

A thin conducting rod MN of mass 20 gm, length 25 cm and resistance 10 $Ω$ 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 $g = 10 ms^{- 2}$ and $e^{- 1} = 0.4$ ]

List-I

(P)

t = 0.2 s

, the magnitude of the induced emf in Volt

(Q)

t = 0.2 s

, the magnitude of the magnetic force in Newton

(R)

t = 0.2 s

, the power dissipated as heat in Watt

(S)

The magnitude of terminal velocity of the rod in

m s^{- 1}

List-II

A rectangular loop has a sliding connector $P Q$ of length $l$ and resistance $R Ω$ and it is moving with a speed $v$ as shown. The set-up is placed in a uniform magnetic field going into the plane of the paper. The three currents $I_{1}, I_{2}$ and $I$ are

(A)

I_{1} = - I_{2} = \frac{B l v}{R}, I = \frac{2 B l v}{R}

(B)

I_{1} = I_{2} = \frac{B l v}{3 R}, I = \frac{2 B l v}{3 R}

(C)

I_{1} = I_{2} = I = \frac{B l v}{R}

(D)

I_{1} = I_{2} = \frac{B l v}{6 R}, I = \frac{B l v}{3 R}

JEE Main 2021

LEVELJEE Advanced

A square loop of side $20 cm$ and resistance $1Ω$ is moved towards right with a constant speed $v_{0}$ . The right arm of the loop is in a uniform magnetic field of $5 T$ . The field is perpendicular to the plane of the loop and is going into it. The loop is connected to a network of resistors each of value $4Ω$ . What should be the value of $v_{0}$ , so that a steady current of $2 mA$ flows in the loop ?

(A)

1 m/s

(B)

1 cm/s

(C)

1 0^{2} m/s

(D)

1 0^{- 2} cm/s

JEE Main 2021

LEVELJEE Advanced

The arm PQ of a rectangular conductor is moving from $x = 0$ to $x = 2 b$ outwards and then inwards from $x = 2 b$ to $x = 0$ as shown in the figure. A uniform magnetic field perpendicular to the plane is acting from $x = 0$ to $x = b$ . Identify the graph showing the variation of different quantities with distance.

(A)

A - flux, B - power dissipated, C - emf

(B)

A - power dissipated, B - flux, C - emf

(C)

A - flux, B - emf , C - power dissipated

(D)

A - emf , B - power dissipated, C - flux

LEVELJEE Main

A metal conductor of length $1 m$ rotates vertically about one of its ends at angular velocity $5 rad/s$ . If the horizontal component of earth's magnetic field is $0.2 \times 1 0^{- 4} T$ , then the emf developed between the two ends of the conductor is

(A)

5 μ V

(B)

50 μ V

(C)

5 mV

(D)

50 mV

JEE Main 2013

LEVELJEE Advanced

A metallic rod of length $l$ is tied to a string of length $2 l$ and made to rotate with angular speed $ω$ on a horizontal table with one end of the string fixed. If there is a vertical magnetic field $B$ in the region, the emf induced across the ends of the rod is

(A)

\frac{2 B ω l ^{2}}{2}

(B)

\frac{3 B ω l ^{2}}{2}

(C)

\frac{4 B ω l ^{2}}{2}

(D)

\frac{5 B ω l ^{2}}{2}

JEE Main 2019

LEVELJEE Main

A $10 m$ long horizontal wire extends from North-East to South-West. It is falling with a speed of $5.0 ms^{- 1}$ at right angles to the horizontal component of the earth's magnetic field of $0.3 \times 1 0^{- 4} Wb/m^{2}$ . The value of the induced emf in wire is

(A)

1.5 \times 1 0^{- 3} V

(B)

1.1 \times 1 0^{- 3} V

(C)

0.3 \times 1 0^{- 3} V

(D)

2.5 \times 1 0^{- 3} V

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

A boat is moving due East in a region where the earth's magnetic field is $5.0 \times 1 0^{- 5} N A^{- 1} m^{- 1}$ due North and horizontal. The boat carries a vertical aerial 2m long. If the speed of the boat is $1.50 ms^{- 1}$ , the magnitude of the induced emf in the wire of aerial is