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Question #43048

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The Question

A hydrogen-like heavy ion of atomic number $Z$ has an electron (rest mass $m_0$ , charge $-e$ ) in a circular orbit corresponding to the principal quantum number $n$ . The Bohr quantization condition applies to the relativistic angular momentum, i.e., $L = \gamma m_0 v r = n\hbar$ , where $\gamma = (1-v^2/c^2)^{-1/2}$ and $c$ is the speed of light. If the binding energy of the electron in this orbit is $B$ (defined such that the total relativistic energy including rest mass is $E = m_0 c^2 - B$ ), which of the following expressions correctly represents the orbital magnetic dipole moment $\mu$ of the electron? (Here $\mu_B = \frac{e\hbar}{2m_0}$ is the classical Bohr magneton).

Options

A.
$n \mu_B \left( 1 - \frac{B}{m_0 c^2} \right)$
B.
$n \mu_B \left( 1 + \frac{B}{m_0 c^2} \right)$
C.
$n \mu_B \left( 1 - \frac{B}{m_0 c^2} \right)^{-1}$
D.
$n \mu_B$

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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|Mass Defect vs. Binding Energy|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the classical gyromagnetic ratio when an electron is in a deep potential well (high $Z$ ), requiring a semi-relativistic treatment.

Students commonly fall into the trap of blindly applying the classical formula $\mu = n \mu_B$ (Option D), ignoring that the effective relativistic mass $m = \gamma m_0$ attenuates the magnetic moment.

Furthermore, it tests the correct application of mass-energy equivalence: students often misinterpret relativistic mass increase, failing to realize that the localized binding energy $B$ actually reduces the total energy of the bound system ( $E = m_0 c^2 - B$ ), which elegantly links the Lorentz factor $\gamma$ directly to the binding energy via $E = m_0 c^2 / \gamma$ .

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