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Question #43256

Difficulty: 8/10 | 🟢 Free

The Question

Consider a hydrogen-like atom consisting of a nucleus of mass $M$ and charge $+Ze$ , and an electron of mass $m$ and charge $-e$ . In the Bohr model, both the nucleus and the electron revolve in circular orbits about their common center of mass. Let $L$ be the magnitude of the total orbital angular momentum of the system and $\mu$ be the magnitude of the total magnetic dipole moment of the system. Assuming $M > \sqrt{Z}m$ , which of the following expressions correctly represents the gyromagnetic ratio $\gamma = \frac{\mu}{L}$ of this atom?

Options

A.
$\frac{e(M^2 - Zm^2)}{2mM(m+M)}$
B.
$\frac{e}{2m}$
C.
$\frac{e(m+M)}{2mM}$
D.
$\frac{eM}{2m(m+M)}$

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 Dipole Moment of Revolving Electron|Reduced Mass Effect|Radius and Velocity Scaling

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to move beyond the standard approximation of an infinitely massive nucleus in the Bohr model.

The common trap is to either blindly use the standard gyromagnetic ratio $\gamma = e/2m$ or to incorrectly substitute the electron mass with the reduced mass $\mu_{red} = mM/(m+M)$ yielding $\gamma = e/2\mu_{red}$ .

A deeper misconception is ignoring the magnetic moment contribution of the revolving nucleus itself.

By analyzing the system from the center of mass frame, students must realize that both particles contribute to the total angular momentum and total magnetic moment, but with different weightings and opposite signs for the magnetic moment due to their opposite charges.

Quick Facts About This Question

Question Information

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