Moving Charges and Magnetism Set-3

Q1. An electron (mass $m_e=9.11\times10^{-31}\ \text{kg}$, charge $e=1.60\times10^{-19}\ \text{C}$) moves perpendicular to a uniform magnetic field $B=5.0\times10^{-4}\ \text{T}$ with speed $v=2.0\times10^{6}\ \text{m/s}$. The radius $r$ of its circular trajectory is

Q2. A coil has $N=50$ turns. The magnetic flux through each turn varies with time as $\Phi(t)=2.0\times10^{-3}\,t-5.0\times10^{-4}\,t^{2}$ Wb for $0\le t\le4\ \text{s}$. The magnitude of the induced emf in the coil at $t=2.0\ \text{s}$ is

Q3. Electrons are accelerated from rest through a potential difference $V=150\ \text{V}$ and enter a region with perpendicular electric field $E=5.0\times10^{4}\ \text{V/m}$ and a magnetic field $B$ (both fields perpendicular to the electron velocity and to each other). What magnitude of $B$ is required so that electrons pass undeflected through the region?

Q4. A very long cylindrical conductor of radius $R$ carries steady total current $I$. A measured plot of magnetic field magnitude $B$ versus radial distance $r$ from the axis shows $B\propto r$ for $r<R$ and $B\propto 1/r$ for $r>R$. Which current distribution inside the conductor is consistent with this plot?

Q5. A long solid cylindrical conductor of radius $R$ carries total current $I$. The volume current density varies as $J(r)=k\,r$ (proportional to radial distance $r$ from axis). Using Ampère's law, the magnetic field at a distance $r$ inside the conductor is $B(r)=\dfrac{\mu_{0}I}{2\pi R^{3}}\,r^{2}$. The magnitude of $B$ at $r=\dfrac{R}{2}$ equals

Q6. Assertion (A): Ampère's circuital law written as $\displaystyle\oint_{C}\vec{B}\cdot d\vec{\ell}=\mu_{0}I_{\text{enc}}$ is valid without modification for any time-dependent current distribution.
Reason (R): To make Ampère's law valid for time-varying situations one must add the displacement current term $\mu_{0}\varepsilon_{0}\dfrac{d\Phi_{E}}{dt}$; the general form is $\displaystyle\oint_{C}\vec{B}\cdot d\vec{\ell}=\mu_{0}\bigl(I_{\text{enc}}+\varepsilon_{0}\dfrac{d\Phi_{E}}{dt}\bigr)$.

Q7. A spatially uniform magnetic field $\vec{B}(t)=B_{0}+\alpha t$ (directed along $+\hat{z}$) increases linearly with time. A positive ion initially at rest at the origin will

Q8. A thin wire carries current $I=5.0\ \text{A}$ and is bent into a semicircle of radius $R=0.10\ \text{m}$ whose ends are joined to two semi-infinite straight wires that extend radially outward from the ends of the semicircle to infinity. The magnitude of the net magnetic field at the centre of the semicircle (due to the entire wire) is

Q9. Assertion (A): A charged particle moving in a region where the magnetic field is spatially non-uniform but time-independent can have its speed increased purely by the magnetic force.
Reason (R): The magnetic part of the Lorentz force is $q\vec{v}\times\vec{B}$ which is always perpendicular to $\vec{v}$, so it does no work and cannot change the particle's kinetic energy.

Q10. A small circular loop (area $A=2.0\times10^{-4}\ \text{m}^2$) carries current $I=1.0\ \text{A}$. Its magnetic moment $\vec{\mu}$ points along $+\hat{x}$. The loop sits near $x=0$ in a magnetic field $\vec{B}(x)=0.2\,(1+50x)\,\hat{x}$ (tesla, with $x$ in metres). Treating the loop as a magnetic dipole, the net force on the loop is approximately