1
JEE Advanced 2024 Paper 1 Online
+3
-1

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
$${{7{\mu _0}I} \over {24}}$$
B
$${{7{\mu _0}I} \over {12}}$$
C
$${{{\mu _0}I} \over {8}}$$
D
$${{{\mu _0}I} \over {6}}$$
2
JEE Advanced 2022 Paper 2 Online
+3
-1

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
$\vec{B}=\frac{-\mu_{o} I}{L}\left(\frac{3}{2}+\frac{1}{4 \sqrt{2} \pi}\right) \hat{k}$
B
$\vec{B}=-\frac{\mu_{o} I}{L}\left(\frac{3}{2}+\frac{1}{2 \sqrt{2} \pi}\right) \hat{k}$
C
$\vec{B}=\frac{-\mu_{o} I}{L}\left(1+\frac{1}{4 \sqrt{2} \pi}\right) \hat{k}$
D
$\vec{B}=\frac{-\mu_{o} I}{L}\left(1+\frac{1}{4 \pi}\right) \hat{k}$
3
JEE Advanced 2022 Paper 1 Online
+3
-1

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 $\alpha=\frac{A^{2} \mu_{0}^{2} m^{2} I^{2} \omega}{2 R}$.

List-I List-II
(I) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\jmath}+\cos \omega t \hat{k})$ (P) 0
(II) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\imath}+\cos \omega t \hat{\jmath})$ (Q) $-\frac{\alpha}{4} \hat{\imath}$
(III) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\imath}+\cos \omega t \hat{k})$ (R) $\frac{3 \alpha}{4} \hat{\imath}$
(IV) $\frac{1}{\sqrt{2}}(\cos \omega t \hat{\jmath}+\sin \omega t \hat{k})$ (S) $\frac{\alpha}{4} \hat{\jmath}$
(T) $-\frac{3 \alpha}{4} \hat{\imath}$

Which one of the following options is correct?

A
I $\rightarrow$ Q, II $\rightarrow$ P, III $\rightarrow$ S, IV $\rightarrow$ T
B
$\mathrm{I} \rightarrow \mathrm{S}, \mathrm{II} \rightarrow \mathrm{T}$, III $\rightarrow \mathrm{Q}$, IV $\rightarrow \mathrm{P}$
C
$\mathrm{I} \rightarrow \mathrm{Q}, \mathrm{II} \rightarrow \mathrm{P}$, III $\rightarrow \mathrm{S}$, IV $\rightarrow \mathrm{R}$
D
$\mathrm{I} \rightarrow \mathrm{T}$, II $\rightarrow \mathrm{Q}$, III $\rightarrow \mathrm{P}$, IV $\rightarrow \mathrm{R}$
4
JEE Advanced 2021 Paper 2 Online
+3
-1
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 $${{{\mu _0}} \over {2\pi }}{m \over {{r^3}}}$$, where $$\mu$$0 is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, m1 and m2, separated by a distance r on the common axis, with their north poles facing each other, is $${{k{m_1}{m_2}} \over {{r^4}}}$$, where k is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

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
m/r3
B
m2/r2
C
m/r2
D
m2/r
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