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Question #42889

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The Question

A particle of mass $m$ and charge $q = Ne$ (where $N$ is a positive integer and $e$ is the elementary charge) is accelerated from rest through a potential difference $V$ . It then enters a vacuum chamber containing a cylindrically symmetric magnetic field given by $\vec{B} = \frac{\alpha}{r} \hat{k}$ , where $r$ is the radial distance from the $z$ -axis and $\alpha$ is a positive constant. The particle moves in a stable circular orbit of radius $R$ in the $xy$ -plane, centered at the origin.

Assume the particle's orbit satisfies the Bohr-Sommerfeld quantization condition, meaning the circumference of the orbit is an integer multiple $k$ ( $k = 1, 2, 3, \dots$ ) of the particle's de Broglie wavelength. Which of the following pairs $(V, R)$ correctly gives the required accelerating potential and the allowed orbital radii?

Options

A.
$\left( \frac{N e \alpha^2}{2m}, \frac{k \hbar}{N e \alpha} \right)$
B.
$\left( \frac{k e \alpha^2}{2m}, \frac{N \hbar}{k e \alpha} \right)$
C.
$\left( \frac{N^2 e \alpha^2}{2m}, \frac{k \hbar}{e \alpha} \right)$
D.
$\left( \frac{N e \alpha^2}{m}, \frac{k \hbar}{2 N e \alpha} \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Quantization of Charge|De Broglie Wavelength Scaling|Trajectory of Charged Particle

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of classical electrodynamics and semiclassical quantum mechanics.

In a $1/r$ magnetic field, the Lorentz force dictates that the momentum required for a circular orbit is strictly constant ( $p = q\alpha$ ), regardless of the radius.

Consequently, the classical radius is arbitrary (neutral equilibrium), and the accelerating potential $V$ depends only on the charge quantization $N$ , not on the orbital radius.

Students typically fall into the trap of blindly applying the uniform magnetic field formula ( $R = mv/qB$ ), incorrectly assuming that the accelerating potential must be quantized with the quantum number $k$ or depend on $R$ .

The problem forces students to realize that $V$ acts only as a velocity selector, while the Bohr-Sommerfeld condition independently discretizes the allowed radii.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Quantization of Charge
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free