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Question #43644

Difficulty: 8/10 | 🟢 Free

The Question

Consider a hypothetical hydrogen-like atom where the nucleus is not a point charge, but a non-conducting solid sphere of radius $R$ . The volume charge density of the nucleus is given by $\rho(r) = \rho_0 \left(\frac{r}{R}\right)$ for $r \le R$ , where $\rho_0$ is a positive constant and $r$ is the radial distance from the center. The total charge of the nucleus is $+e$ . An electron of mass $m$ and charge $-e$ revolves around the center of the nucleus in a circular Bohr orbit. Assuming the $n^{\text{th}}$ Bohr orbit lies entirely inside the nucleus ( $r_n < R$ ), which of the following expressions represents the radius $r_n$ of this orbit? (Here $\epsilon_0$ is the permittivity of free space and $\hbar = \frac{h}{2\pi}$ is the reduced Planck constant).

Options

A.
$\left( \frac{4\pi \epsilon_0 R^4 n^2 \hbar^2}{m e^2} \right)^{\frac{1}{5}}$
B.
$\left( \frac{4\pi \epsilon_0 R^2 n^2 \hbar^2}{m e^2} \right)^{\frac{1}{3}}$
C.
$\left( \frac{4\pi \epsilon_0 R^3 n^2 \hbar^2}{m e^2} \right)^{\frac{1}{4}}$
D.
$\left( \frac{2\pi \epsilon_0 R^4 n^2 \hbar^2}{m e^2} \right)^{\frac{1}{5}}$

Check Your Answer

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

Coulombic Central Force|Principle of Superposition|Non-conducting Charged Sphere|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize electrostatics (Gauss's Law for non-uniform continuous charge distributions) with modern physics (Bohr model).

The primary trap lies in the assumption of the effective central force.

Students often blindly apply the standard inverse-square Coulomb force ( $F \propto 1/r^2$ ) or assume a uniform charge density ( $F \propto r$ ) inside the nucleus.

By introducing a linearly varying charge density $\rho(r) \propto r$ , the enclosed charge scales as $r^4$ , leading to an electric field that scales as $r^2$ .

This fundamentally alters the centripetal force balance and the resulting dependence of the orbital radius on the principal quantum number $n$ .

Furthermore, the options are carefully constructed to be dimensionally identical, neutralizing dimensional analysis as a guessing strategy and forcing rigorous integration of the spherical volume element $4\pi r^2 dr$ .

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