A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass $m$ and radius $r$ and it is in a uniform vertical magnetic field $B_0$, as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity $g$, on two conducting supports at $\mathrm{P}$ and $\mathrm{Q}$. When a current $I$ is passed through the loop, the loop turns about the line $\mathrm{PQ}$ by an angle $\theta$ given by

An infinitely long wire, located on the $z$-axis, carries a current $I$ along the $+z$-direction and produces the magnetic field $\vec{B}$. The magnitude of the line integral $\int \vec{B} \cdot \overrightarrow{d l}$ along a straight line from the point $(-\sqrt{3} a, a, 0)$ to $(a, a, 0)$ is given by

[ $\mu_0$ is the magnetic permeability of free space.]

A positive, singly ionized atom of mass number $A_{\mathrm{M}}$ is accelerated from rest by the voltage $192 \mathrm{~V}$. Thereafter, it enters a rectangular region of width $w$ with magnetic field $\vec{B}_0=0.1 \hat{k}$ Tesla, as shown in the figure. The ion finally hits a detector at the distance $x$ below its starting trajectory.

[Given: Mass of neutron/proton $=(5 / 3) \times 10^{-27} \mathrm{~kg}$, charge of the electron $=1.6 \times 10^{-19} \mathrm{C}$.]

Which of the following option(s) is(are) correct?

Which one of the following options represents the magnetic field $\vec{B}$ at $\mathrm{O}$ due to the current flowing in the given wire segments lying on the $x y$ plane?

A small circular loop of area $A$ and resistance $R$ is fixed on a horizontal $x y$-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 $t=\frac{\pi}{6 \omega}$. Let $\alpha=\frac{A^{2} \mu_{0}^{2} m^{2} I^{2} \omega}{2 R}$.

Which one of the following options is correct?

When d $\approx$ a but wires are not touching the loop, it is found that the net magnetic field on the axis of the loop is zero at a height h above the loop. In that case

Consider d >> a, and the loop is rotated about its diameter parallel to the wires by 30$^\circ$ from the position shown in the below figure. If the currents in the wires are in the opposite directions, the torque on the loop at its new position will be (assume that the net field due to the wires is constant over the loop)

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 steady current I flows along an infinitely long hollow cylindrical conductor of radius R. This cylinder is placed coaxially inside an infinite solenoid of radius 2R. The solenoid has n turns per unit length and carries a steady current I. Consider a point P at a distance r from the common axis. The correct statement(s) is(are)

A particle of mass M and positive charge Q, moving with a constant velocity ${\overrightarrow u _1} = 4\widehat i$ ms^$-$1 enters a region of uniform static magnetic field, normal to the xy plane. The region of the magnetic field extends from x = 0 to x = L for all values of y. After passing through this region, the particle emerges on the other side after 10 ms with a velocity ${\overrightarrow u _2} = 2\left( {\sqrt 3 \widehat i + \widehat j} \right)$ ms^$-$1. The correct statement(s) is(are)

A loop carrying current $l$ lies in the xy-plane as shown in the figure. The unit vector $\widehat k$ is coming out of the plane of the paper. The magnetic moment of the current loop is

An infinite long hollow conducting cylinder with inner radius R/2 and outer radius R carries a uniform current density along its length. The magnitude of the magnetic field, $\left| {\overrightarrow B } \right|$ as a function of the radial distance r from the axis is best represented by

Consider the motion of a positive point charge in a region, there are simultaneous uniform electric and magnetic fields $\overrightarrow E = {E_0}\widehat j$ and $\overrightarrow B = {B_0}\widehat j$. At time t = 0, this charge has velocity $\overrightarrow v $ in the xy-plane, making an angle $\theta$ with the x-axis. Which of the following option(s) is(are) correct for time t > 0 ?

Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons N $ \approx $ 4 $ \times $ 10²⁷ m^-3. Taking ${{\varepsilon _0}}$ = 10^{- 11} and m $ \approx $ 10^{- 30}, where these quantities are in proper SI units.

A long insulated copper wire is closely wound as a spiral of N turns. The spiral has inner radius a and outer radius b. The spiral lies in the xy-plane and a steady current I flows through the wire. The z-component of the magnetic field at the centre of the spiral is

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is/are true?

A thin flexible wire of length L is connected to two adjacent fixed points and carries a current I in the clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength B going into the plane of the paper, the wire takes the shape of a circle. The tension in the wire is

In the graph below, the resistance R of a superconductor is shown as a friction of its temperature T for two different magnetic fields B₁ (solid line) and B₂ (dashed line). If B₂ is larger than B₁ which of the following graphs shows the correct variation of R with T in these fields?

A superconductor has T_c(0) = 100 K. When a magnetic field of 7.5 T is applied, its T_c decreases to 75 K. For this material, one can definitely say that when

STATEMENT 1 : The sensitivity of a moving coil galvanometer is increased by placing a suitable magnetic material as a core inside the coil.

and

STATEMENT 2 : Soft iron has a high magnetic permeability and cannot be easily magnetized or demagnetized.

A particle of mass m and charge q, moving with velocity v enters Region II normal to the boundary as shown in the figure. Region II has a uniform magnetic field B perpendicular to the plane of the paper. The length of the Region II is $l$. Choose the correct choice (s).

A magnetic field $\overrightarrow{\mathrm{B}}=\mathrm{B}_{0} \hat{j}$ exists in the region $a < x < 2 a$ and $\overrightarrow{\mathrm{B}}=-\mathrm{B}_{0} \hat{j}$, in the region $2 a < x < 3 a$, where $\mathrm{B}_{0}$ is a positive constant. A positive point charge moving with a velocity $\vec{v}=v_{0} \hat{i}$, where $v_{0}$ is a positive constant, enters the magnetic field at $x=a$. The trajectory of the charge in this region can be like,

Two wires each carrying a steady current I are shown in four configurations in Column I. Some of the resulting effects are described in Column II. Match the statements in Column I with the statements in Column II and indicate your answer by darkening appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

	Column I		Column II
(A)	Point P is situated midway between the wires.	(P)	The magnetic fields (B) at P due to the currents in the wire are in same direction.
(B)	Point P is situated at the mid-point of the line joining the centers of the circular wires, which have same radii.	(Q)	The magnetic fields (B) at P due to the currents in the wires are in opposite directions.
(C)	Point P is situated at the mid-point of the line joining the centers of the circular wires, which have same radii.	(R)	There is no magnetic field at P.
(D)	Point P is situated at the common center of the wires.	(S)	The wires repel each other.

An infinite current-carrying wire passes through point O and in perpendicular to the plane containing a current-carrying loop ABCD as shown in the figure. Choose the correct option(s):

$ \text { Match the following Columns. } $

In a moving coil galvanometer, torque on the coil can be expressed as $\tau=k i$, where $i$ is current through the wire and $k$ is constant. The rectangular coil of the galvanometer having numbers of turns $\mathrm{N}$, area $\mathrm{A}$ and moment of inertia I is placed in magnetic field B. Find

(A) $k$ in terms of given parameters $\mathrm{N}, \mathrm{I}, \mathrm{A}$ and B;

(B) The torsional constant of the spring, if a current $i_{0}$ produces a deflection of $\frac{\pi}{2}$ in the coil;

(C) The maximum angle through which coil is deflected, if charge $\mathrm{Q}$ is passed through the coil almost instantaneously. (Ignore the damping in mechanical oscillations.)