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Question #43811

Difficulty: 8/10 | 🟢 Free

The Question

Consider a muonic atom where an electron is replaced by a muon of mass $m_\mu$ and charge $-e$ . The nucleus has mass $M_N$ , atomic number $Z$ , and is modeled as a uniformly charged sphere of radius $R$ . For a sufficiently heavy nucleus, the muon's lowest energy orbits lie entirely inside the nuclear volume. Assuming the Bohr quantization rule for angular momentum holds, which of the following expressions gives the total energy $E_n$ of the muon in its $n^{\text{th}}$ orbit?

(Assume the orbit is entirely inside the nucleus, $\hbar = \frac{h}{2\pi}$ , and $\epsilon_0$ is the vacuum permittivity.)

Options

A.
$n\hbar \sqrt{\frac{Z e^2 (m_\mu + M_N)}{4\pi\epsilon_0 m_\mu M_N R^3}} - \frac{3 Z e^2}{8\pi\epsilon_0 R}$
B.
$n\hbar \sqrt{\frac{Z e^2}{4\pi\epsilon_0 m_\mu R^3}} - \frac{3 Z e^2}{8\pi\epsilon_0 R}$
C.
$n\hbar \sqrt{\frac{Z e^2 (m_\mu + M_N)}{4\pi\epsilon_0 m_\mu M_N R^3}} - \frac{Z e^2}{4\pi\epsilon_0 R}$
D.
$-\frac{m_\mu M_N}{(m_\mu + M_N)} \left( \frac{Z^2 e^4}{32 \pi^2 \epsilon_0^2 \hbar^2 n^2} \right)$

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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

Nuclear Radius Scaling|Coulombic Central Force|Reduced Mass Effect

Why This Question Is Tricky

Common Mistakes

This question tests the transition from the standard Coulombic $1/r$ potential to the harmonic oscillator-like potential inside a finite nucleus, a critical concept in muonic atoms where the Bohr radius scales down by $m_e/m_\mu$ and penetrates the nucleus.

The primary trap is twofold: first, students often blindly apply the standard Bohr energy formula (Option D) ignoring the finite nuclear size.

Second, even if they derive the new potential, they frequently forget to use the reduced mass $\mu$ of the two-body system (Option B), or they miscalculate the constant potential energy offset at the center of the uniformly charged sphere (Option C).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Radius Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free