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Question #43782

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The Question

Consider a hypothetical atom consisting of a stationary nucleus of mass $M$ and charge $+Ze$ , and an electron of mass $m$ and charge $-e$ . Assume that both the electrostatic and gravitational forces between the nucleus and the electron are significant and act simultaneously. Using the Bohr model of quantization of angular momentum, determine the magnitude of the total energy of the electron in its $n$ -th allowed circular orbit. Let $h$ be Planck's constant, $\hbar = \frac{h}{2\pi}$ , $G$ be the universal gravitational constant, and $\epsilon_0$ be the permittivity of free space. All quantities are in SI units and the final energy is expressed in Joules ( $\text{J}$ ).

Options

A.
$\frac{m}{2n^2\hbar^2} \left( \frac{Ze^2}{4\pi\epsilon_0} + GMm \right)^2 \, \text{J}$
B.
$\frac{m}{2n^2\hbar^2} \left( \left(\frac{Ze^2}{4\pi\epsilon_0}\right)^2 + (GMm)^2 \right) \, \text{J}$
C.
$\frac{m}{2n^2\hbar^2} \left( \frac{Ze^2}{4\pi\epsilon_0} - GMm \right)^2 \, \text{J}$
D.
$\frac{m}{n^2\hbar^2} \left( \frac{Ze^2}{4\pi\epsilon_0} + GMm \right)^2 \, \text{J}$

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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

Quantization of Angular Momentum|Gravitational Potential Energy (System)|Coulomb's Law Vector Form

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the principle of superposition to central forces within the semi-classical Bohr model.

A common trap is treating the total energy of the system as the simple sum of the isolated electrostatic Bohr energy and the isolated gravitational Bohr energy (Option B).

This incorrectly assumes that $E_{total} \propto \alpha_{elec}^2 + \alpha_{grav}^2$ , ignoring the cross-term $2\alpha_{elec}\alpha_{grav}$ that arises because the orbital radius itself depends on the combined effective coupling constant $\alpha = \alpha_{elec} + \alpha_{grav}$ .

Another trap is misapplying the sign of the potential (Option C) or forgetting the virial theorem factor of $1/2$ (Option D).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Quantization of Angular Momentum
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free