Moving Charges and Magnetism Set-6

Q1. A proton and an alpha particle, each having the same kinetic energy $K$, enter a region of uniform magnetic field $\mathbf{B}$ with their velocities perpendicular to $\mathbf{B}$. If $r_p$ and $r_\alpha$ are the radii of their circular trajectories, the ratio $r_p:r_\alpha$ is

Q2. A charged particle moves in a uniform magnetic field $\mathbf{B}$ and its trajectory is a helix. If the pitch (distance moved along $\mathbf{B}$ in one revolution) equals the circumference of the circular projection, the angle $\theta$ between its velocity $\mathbf{v}$ and the field $\mathbf{B}$ is

Q3. A conducting rod of length $l=0.50\ \text{m}$ and mass $m=0.200\ \text{kg}$ slides without friction on conducting rails that make an angle $\alpha=30^\circ$ with the horizontal. The rod and rails form a closed circuit of resistance $R=2.0\ \Omega$. A uniform magnetic field $B=0.80\ \text{T}$ is directed into the plane of the rails. After transients the rod attains a steady terminal speed $v_t$ down the incline when magnetic damping balances the component of gravity. The terminal speed $v_t$ (approximately) is

Q4. A rectangular metallic slab of thickness $t=5.0\times10^{-4}\ \text{m}$ carries a current $I=15.0\ \text{A}$ along its length. A uniform magnetic field $B=0.20\ \text{T}$ is applied perpendicular to the slab, producing a Hall voltage across the thickness. If the charge carrier density is $n=8.5\times10^{28}\ \text{m}^{-3}$ and the electronic charge is $e=1.602\times10^{-19}\ \text{C}$, the magnitude of the Hall voltage $V_H$ across the slab is closest to

Q5. A thin wire consists of a semicircular arc of radius $R$ joined smoothly to a very long straight segment which is tangent to the arc at one end. A steady current $I$ flows through the wire. The magnitude of the magnetic field at the centre of the semicircular arc is

Q6. An electron ($q=1.6\times10^{-19}\ \mathrm{C}$) moves with speed $v=2.0\times10^{6}\ \mathrm{m\,s^{-1}}$ in a uniform magnetic field $B=0.01\ \mathrm{T}$. Its velocity makes an angle $60^\circ$ with $\vec{B}$. The magnitude of the magnetic force on the electron is:

Q7. A rectangular metallic slab carries a current $I=10.0\ \mathrm{A}$ along its length. The distance between the faces where Hall voltage is measured is $w=2.0\ \mathrm{mm}$ and the slab thickness is $t=0.50\ \mathrm{mm}$. A perpendicular magnetic field $B=0.20\ \mathrm{T}$ produces a measured Hall voltage $V_H=0.25\ \mathrm{mV}$. Assuming charge carriers are electrons of charge magnitude $1.6\times10^{-19}\ \mathrm{C}$, the carrier density $n$ is closest to:

Q8. Two identical point charges each of charge $+e$ move in vacuum along two parallel straight lines separated by a distance $d$, both moving in the same direction with speed $v\ll c$. Find the speed $v$ (use $c=3.0\times10^{8}\ \mathrm{m\,s^{-1}}$) for which the magnetic attraction between them has magnitude equal to half the electrostatic repulsion.

Q9. A rectangular loop ABCD of width $a$ and height $b$ lies in the plane of the paper with its centre at the origin; its vertical sides are at $x=\pm a/2$ and horizontal sides at $y=\pm b/2$. A uniform magnetic field $\vec{B}=-B_0\hat{k}$ exists only in the region $x>0$ and is zero for $x\le0$. A current $I$ flows anticlockwise around the loop (from A at $x=-a/2,y=+b/2$ to B at $x=+a/2,y=+b/2$, etc.). If $a=6.0\ \mathrm{cm}$, $b=4.0\ \mathrm{cm}$, $I=5.0\ \mathrm{A}$ and $B_0=0.40\ \mathrm{T}$, the net magnetic force on the loop is:

Q10. A proton ($q=1.6\times10^{-19}\ \mathrm{C}$, $m=1.67\times10^{-27}\ \mathrm{kg}$) has speed $v=2.00\times10^{6}\ \mathrm{m\,s^{-1}}$ and enters a uniform magnetic field $\vec{B}=0.500\ \hat{k}\ \mathrm{T}$ with its velocity making an angle $60^\circ$ with $\vec{B}$. After the proton completes $100$ full revolutions about the field lines, the distance it has moved parallel to $\vec{B}$ is closest to:

Q11. A proton (mass $m_p=1.67\times10^{-27}\,\mathrm{kg}$, charge $q=1.6\times10^{-19}\,\mathrm{C}$) has kinetic energy $K=1.0\ \mathrm{keV}$ and enters a uniform magnetic field $B=0.10\ \mathrm{T}$ with its velocity perpendicular to $\mathbf{B}$. Using the relation $r=\dfrac{\sqrt{2mK}}{qB}$, the radius of its circular path is closest to

Q12. A proton with speed $v=1.00\times10^{6}\ \mathrm{m/s}$ enters a region of uniform magnetic field $B=0.50\ \mathrm{T}$ at an angle $\theta=60^\circ$ to the field direction. The pitch (distance advanced along the field in one full revolution) is $p=v\cos\theta\times T$ with $T=\dfrac{2\pi m}{qB}$. The numerical value of the pitch is about

Q13. A long straight cylindrical conductor of radius $R$ carries total current $I$ with a radially varying current density $J(r)\propto r$ for $0\le r\le R$. Using Ampère's law and the relation between $J(r)$ and $I$, the magnetic field magnitude at a distance $r=\dfrac{R}{2}$ from the axis is

Q14. Consider the two statements:
Statement A: "For a charged particle moving in a static magnetic field the Lorentz force $\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}$ does no work, so the particle's kinetic energy (and speed) cannot change."
Statement B: "If the magnetic field varies with time, Faraday induction produces an electric field which can do work on the charge and change its kinetic energy." Which of the following is correct?

Q15. A singly charged ion moves into a region where the magnetic field magnitude slowly increases from $B_0$ to $B_{\max}=3B_0$ along the field direction. The ion's initial velocity makes an angle $\alpha_0=30^\circ$ with the initial field. Will the ion be reflected (magnetic mirror) by the increasing field?