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Question #43795

Difficulty: 8/10 | 🟢 Free

The Question

An electron (mass $m$ , charge $-e$ ) in a hydrogen atom is in a circular Bohr orbit of principal quantum number $n$ . A weak uniform magnetic field $B$ is applied perpendicular to the orbital plane, such that the magnetic Lorentz force on the electron points towards the nucleus. The generalized Bohr-Sommerfeld quantization rule states that the canonical angular momentum is quantized: $\oint \vec{p}_{\text{can}} \cdot d\vec{r} = nh$ , where $\vec{p}_{\text{can}} = m\vec{v} - e\vec{A}$ and the vector potential is $\vec{A} = \frac{1}{2}(\vec{B} \times \vec{r})$ . What is the change in the kinetic energy of the electron to first order in $B$ ?

Options

A.
$\frac{e n \hbar B}{2m}$
B.
$\frac{e n \hbar B}{m}$
C.
$0$
D.
$-\frac{e n \hbar B}{2m}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Rydberg Constant Sensitivity|Magnetic Dipole Moment of Revolving Electron|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Bohr model in the presence of an external magnetic field (Zeeman effect) and targets a profound misconception regarding quantization.

Students commonly trap themselves by applying the standard mechanical quantization $mvr = n\hbar$ instead of the canonical momentum quantization.

If the mechanical momentum is incorrectly quantized, the force equation implies a fictitious first-order change in the orbital radius ( $\Delta r \neq 0$ ), which leads to an erroneous kinetic energy shift of $\frac{en\hbar B}{m}$ (Option B).

By using the correct canonical momentum, the $O(B)$ terms in the radial equation cancel, proving the radius is invariant to first order, and yielding the correct kinetic energy shift of exactly half the trapped value.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Rydberg Constant Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free