Magnetic Effects of Current

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 $\int \vec{B}\cdot\vec{dl}$ 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 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 P and Q. When a current $I$ is passed through the loop, the loop turns about the line PQ by an angle $\theta$ given by

A positive, singly ionized atom of mass number $A_M$ 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 $= (5/3)\times10^{-27}\,\mathrm{kg}$, charge of the electron $= 1.6\times10^{-19}\,\mathrm{C}$.]

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

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 $\dfrac{\sqrt3}{2}\hat x + \dfrac12\hat y$ with a constant speed v. The wire is carrying a steady current $I = 10$ A in the positive $x$-direction. A current of $10\ \mu$A flows through the loop when it is at a distance $d = 4$ cm from the wire. If the resistance of the loop is $0.1\ \Omega$, then the value of v is _____ $\mathrm{m\,s^{-1}}$.

[Given: The permeability of free space $\mu_0 = 4\pi\times10^{-7}\ \mathrm{N\,A^{-2}}$]

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 $\hat n$ 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 $t = \dfrac{\pi}{6\omega}$. Let $\alpha = \dfrac{A^2\mu_0^2m^2I^2\omega}{2R}$.

Which one of the following options is correct?

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

Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : In an uniform magnetic field, speed and energy remains the same for a moving charged particle.

Reason (R) : Moving charged particle experiences magnetic force perpendicular to its direction of motion.

The magnetic field at the centre of a circular coil of radius $r$, due to current $I$ flowing through it, is $B$. The magnetic field at a point along the axis at a distance $\dfrac r2$ from the centre is :

A proton, a deuteron and an $\alpha$-particle with same kinetic energy enter into a uniform magnetic field at right angle to magnetic field. The ratio of the radii of their respective circular paths is :

Match List - I with List - II.

Choose the correct answer from the options given below :

A long straight wire with a circular cross-section having radius R, is carrying a steady current I. The current I is uniformly distributed across this cross-section. Then the variation of magnetic field due to current I with distance r (r < R) from its centre will be :

A teacher in his physics laboratory allotted an experiment to determine the resistance (G) of a galvanometer. Students took the observations for $\dfrac13$ deflection in the galvanometer. Which of the below is true for measuring value of G ?

A long solenoid carrying a current produces a magnetic field $B$ along its axis. If the current is doubled and the number of turns per cm is halved, the new value of magnetic field will be equal to

If $n$ represents the actual number of deflections in a converted galvanometer of resistance $G$ and shunt resistance $S$. Then the total current $I$ when its figure of merit is $K$ will be :

A proton and an alpha particle of the same velocity enter in a uniform magnetic field which is acting perpendicular to their direction of motion. The ratio of the radii of the circular paths described by the alpha particle and proton is :

A metal surface is illuminated by a radiation of wavelength 4500 Å. The ejected photo-electron enters a constant magnetic field of 2 mT making an angle of 90° with the magnetic field. If it starts revolving in a circular path of radius 2 mm, the work function of the metal is approximately :

Two 10 cm long, straight wires, each carrying a current of 5 A are kept parallel to each other. If each wire experienced a force of $10^{-5}$ N, then separation between the wires is ______ cm.

A $72\ \Omega$ galvanometer is shunted by a resistance of $8\ \Omega$. The percentage of the total current which passes through the galvanometer is :

Two long parallel conductors $S_1$ and $S_2$ are separated by a distance 10 cm and carrying currents of 4A and 2A respectively. The conductors are placed along $x$-axis in X-Y plane. There is a point P located between the conductors (as shown in figure).

A charge particle of $3\pi$ coulomb is passing through the point P with velocity $\vec v=\left(2\hat i+3\hat j\right)$ m/s; where $\hat i$ & $\hat j$ represents unit vector along $x$ & $y$ axis respectively.

The force acting on the charge particle is $4\pi\times10^{-5}\left(-x\hat i+2\hat j\right)$ N. The value of $x$ is :

A deuteron and a proton moving with equal kinetic energy enter into to a uniform magnetic field at right angle to the field. If $r_d$ and $r_p$ are the radii of their circular paths respectively, then the ratio $\dfrac{r_d}{r_p}$ will be $\sqrt x:1$ where $x$ is ______.