Moving Charges and Magnetism Set-2

Q1. A proton (charge $+e$, $e=1.6\times10^{-19}\,\mathrm{C}$) moves with speed $2.0\times10^{6}\,\mathrm{m\,s^{-1}}$ in a uniform magnetic field $B=0.20\,\mathrm{T}$. Its velocity makes an angle $60^\circ$ with the field. The magnitude of the magnetic force on the proton is:

Q2. An electron ($e=1.6\times10^{-19}\,\mathrm{C},\;m_e=9.11\times10^{-31}\,\mathrm{kg}$) with speed $5.0\times10^{6}\,\mathrm{m\,s^{-1}}$ enters a uniform magnetic field $B=0.020\,\mathrm{T}$ at $30^\circ$ to the field. Determine (i) the radius $r$ of the circular component of motion and (ii) the pitch $p$ (distance advanced along the field in one revolution) of the resulting helical path.

Q3. Statement A: A charged particle placed in a region where the magnetic field $\mathbf{B}$ is changing with time will always have its speed unchanged by the electromagnetic forces present in that region.
Reason R: The magnetic part of the Lorentz force is $\mathbf{F}_B=q\,\mathbf{v}\times\mathbf{B}$, which is always perpendicular to velocity and therefore cannot change the speed of the particle.

Q4. A square loop of side $a=5.0\ \mathrm{cm}$ carries current $I=2.0\ \mathrm{A}$ and is placed in a uniform magnetic field $B=0.40\ \mathrm{T}$. The plane of the loop makes an angle $30^\circ$ with the magnetic field (so the loop normal makes $60^\circ$ with $B$). The magnitude of torque on the loop is:

Q5. A long solid cylindrical conductor of radius $R$ carries a total current $I$. The volume current density varies as $J(r)=J_0\,(r/R)$ (where $r$ is radial distance from axis). For $r<R$, the magnitude of magnetic field at radius $r$ is:

Q6. Two very long parallel straight wires are separated by $d=0.20\ \mathrm{m}$. The left wire carries current $I_1=5.0\ \mathrm{A}$ and the right wire carries $I_2=3.0\ \mathrm{A}$. Currents are in opposite directions. At a point along the line joining the wires, the net magnetic field vanishes. The distance of that point from the left wire is:

Q7. A circular coil of radius $R=0.10\ \mathrm{m}$ produces a magnetic field at its centre given by $B=\dfrac{\mu_0 N I}{2R}$, where $N$ is the number of turns. In an experiment the slope of the straight-line graph of $B$ versus $I$ for this coil is found to be $2.0\times10^{-5}\ \mathrm{T/A}$. Using $\mu_0=4\pi\times10^{-7}\ \mathrm{T\,m/A}$, the number of turns $N$ is approximately:

Q8. Statement A: Ampère's circuital law in the form $\displaystyle\oint_C\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{\text{enc}}$ is not valid for a closed path if the electric flux through any surface bounded by the path is changing with time.
Reason R: When the electric flux through a chosen surface bounded by the path changes with time, Ampère's law must be corrected by adding the displacement current term $\mu_0\varepsilon_0\,\dfrac{d\Phi_E}{dt}$ on the right-hand side.

Q9. Singly charged ions (charge $+e$) are accelerated through a potential difference $V=5000\ \mathrm{V}$ and then enter a region of uniform magnetic field $B=0.20\ \mathrm{T}$ where they move in a circular path of radius $r=0.050\ \mathrm{m}$. The mass of the ion is approximately:

Q10. A charged particle moves along magnetic field lines from a region of weak field into a region of stronger magnetic field slowly (adiabatically). Which one of the following statements is correct?