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Question #43786

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The Question

In a theoretical model of a hydrogen-like atom, the nucleus is not a point charge but is modeled as a solid sphere of radius $R$ with a spherically symmetric, non-uniform volume charge density $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$ for $r \le R$ , where $r$ is the radial distance from the center. The total charge of the nucleus is $+Ze$ . An electron of mass $m$ and charge $-e$ is released from rest at a very small distance $r_0$ ( $r_0 \ll R$ ) from the center of the nucleus. Assuming no energy is lost to radiation, the electron undergoes simple harmonic motion. Which of the following correctly gives the angular frequency $\omega$ of this motion and the electrostatic potential $V_0$ at the exact center of the nucleus? (Here $\epsilon_0$ is the vacuum permittivity).

Options

A.
$\omega = \sqrt{\frac{Z e^2}{\pi \epsilon_0 m R^3}}, \; V_0 = \frac{Z e}{2 \pi \epsilon_0 R}$
B.
$\omega = \sqrt{\frac{Z e^2}{4 \pi \epsilon_0 m R^3}}, \; V_0 = \frac{3 Z e}{8 \pi \epsilon_0 R}$
C.
$\omega = \sqrt{\frac{Z e^2}{4 \pi \epsilon_0 m r_0^3}}, \; V_0 = \frac{Z e}{4 \pi \epsilon_0 R}$
D.
$\omega = \sqrt{\frac{Z e^2}{\pi \epsilon_0 m R^3}}, \; V_0 = \frac{Z e}{4 \pi \epsilon_0 R}$

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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

Coulombic Central Force|Continuous Charge Distribution|Gauss Law|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the transition from a singular point-charge Coulomb potential to a continuous charge distribution model.

The primary trap lies in students blindly applying the inverse-square law $F = kQq/r^2$ or memorizing the standard results for a uniform spherical charge distribution (which yields $\omega \propto 1/\sqrt{4\pi\epsilon_0}$ and $V_0 = 3Ze/8\pi\epsilon_0 R$ , matching Option B).

By introducing a non-uniform density $\rho(r) = \rho_0(1 - r/R)$ , students are forced to apply Gauss's Law from first principles, perform a small $r$ approximation to find the linear restoring force for SHM, and integrate the electric field (or use superposition) to find the exact central potential.

Failing to account for the non-uniformity or assuming the nucleus acts as a conductor (constant potential inside, Option D) leads directly to the distractor options.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Coulombic Central Force
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free