Moving Charges and Magnetism Set-1

Q1. A proton of mass $m$ and charge $+e$ and an alpha particle of mass $4m$ and charge $+2e$ are accelerated from rest through the same potential difference $V$. Each then enters a uniform magnetic field $B$ perpendicular to its velocity and moves in a circular path of radius $r_p$ (proton) and $r_\alpha$ (alpha). The ratio $\dfrac{r_\alpha}{r_p}$ is:

Q2. Magnetic field measurements taken radially outward from the axis of a long cylindrical conductor of radius $R$ carrying steady current $I$ show $B(r)\propto r$ for $r\le R$ and $B(r)\propto 1/r$ for $r\ge R$. Which statement is best supported by these measurements?

Q3. A metallic slab of thickness $t=0.50\ \text{mm}$ carries a current $I=5.0\ \text{A}$ along its length. When a uniform magnetic field $B=0.20\ \text{T}$ (normal to the slab) is applied, a Hall voltage $V_H=2.0\ \text{mV}$ appears across its width. Assuming charge carriers have charge magnitude $e=1.6\times10^{-19}\ \text{C}$, the carrier density $n$ (in m$^{-3}$) is approximately:

Q4. Statement I: A charged particle moving with velocity parallel to the local magnetic field experiences no magnetic force. Statement II: If the magnetic field is non-uniform, a particle moving parallel to the field can experience a magnetic force that changes its speed.

Q5. In a velocity selector an electron remains undeflected when its speed satisfies $v=\dfrac{E}{B}$. If $E=2.0\times10^{4}\ \text{V/m}$ and $B=5.0\times10^{-2}\ \text{T}$, the required electron speed is:

Q6. Two very long, parallel straight conductors separated by $d=5.0\ \text{cm}$ carry steady currents $I_1=20\ \text{A}$ and $I_2=10\ \text{A}$ in opposite directions. The magnitude of the magnetic force per unit length between them and its sense (attraction/repulsion) is:

Q7. The axial field of a circular loop of radius $R$ carrying current $I$ is

At what axial distance $z=kR$ does $B(z)$ drop to half its central value $B(0)$? (Numerical value of $k$)

Q8. Statement I: Ampère’s circuital law in the form $\displaystyle\oint\mathbf{B}\cdot d\mathbf{l}=\mu_0 I_{\text{enc}}$ does not give a unique result for a charging capacitor unless modified. Statement II: Maxwell’s addition of the displacement current term $\mu_0\varepsilon_0\,\dfrac{d\Phi_E}{dt}$ to Ampère’s law restores consistency for time-varying fields and preserves charge conservation.

Q9. A proton moves adiabatically along a magnetic field line. At point A the field magnitude is $B_0$ and its velocity makes a pitch angle $\theta_0$ with the field. It moves toward a region where $B$ increases to $B_{\text{max}}=4B_0$. Using conservation of the magnetic moment $\mu=\dfrac{1}{2}m v_\perp^2/B$ (total speed constant in purely magnetic forces), the minimum initial pitch angle $\theta_0$ required for the proton to be reflected before reaching $B_{\text{max}}$ is:

Q10. Statement I: A static magnetic field alone can never change the kinetic energy (speed) of a charged particle. Statement II: A time-varying magnetic field produces an induced electric field which can do work on charges and change their kinetic energy.