Before You Begin

Question #43424

Difficulty: 8/10 | 🟢 Free

The Question

A free atom of mass $M$ , initially at rest in an excited state with principal quantum number $n$ , emits a single photon of frequency $\nu$ and transitions directly to its ground state ( $n=1$ ). The energy levels of the atom are given by $E_k = -\frac{E_0}{k^2}$ , where $E_0$ is the ionization energy from the ground state. Which of the following expressions correctly represents the magnitude of the change in the orbital magnetic moment of the electron during this transition? (Here $e$ is the elementary charge, $m_e$ is the mass of the electron, $h$ is Planck's constant, and $c$ is the speed of light in vacuum).

Options

A.
$\frac{eh}{4\pi m_e} \left[ \left( 1 - \frac{h\nu}{E_0} - \frac{h^2\nu^2}{2Mc^2 E_0} \right)^{-1/2} - 1 \right]$
B.
$\frac{eh}{4\pi m_e} \left[ \left( 1 - \frac{h\nu}{E_0} \right)^{-1/2} - 1 \right]$
C.
$\frac{eh}{4\pi m_e} \left[ \left( 1 - \frac{h\nu}{E_0} + \frac{h^2\nu^2}{2Mc^2 E_0} \right)^{-1/2} - 1 \right]$
D.
$\frac{eh}{4\pi M} \left[ \left( 1 - \frac{h\nu}{E_0} - \frac{h^2\nu^2}{2Mc^2 E_0} \right)^{-1/2} - 1 \right]$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Magnetic Moment of Orbiting Electron|Quantization of Angular Momentum|Recoil of Atom

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple atomic transitions with macroscopic conservation laws.

The primary trap is treating the transition energy as purely photonic ( $\Delta E = h\nu$ ), which ignores the recoil kinetic energy of the atom ( $K = p^2/2M$ ).

By requiring the calculation of the change in the magnetic moment, the problem forces students to accurately determine the initial principal quantum number $n$ .

If the recoil energy is neglected, the calculated energy gap is too small, leading to an incorrect value for $n$ (Option B).

Additionally, Option D tests the confusion between the mass of the recoiling atom ( $M$ ) and the mass of the electron ( $m_e$ ) responsible for the magnetic moment.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Magnetic Moment of Orbiting Electron
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free