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Question #43650

Difficulty: 8/10 | 🟢 Free

The Question

Consider a hypothetical hydrogen-like atom where the nucleus is a uniformly charged solid sphere of radius $R_0$ and total charge $+Ze$ . An electron of mass $m$ and charge $-e$ revolves in circular orbits around the center of the nucleus. Assume that $R_0$ is sufficiently large such that the first few allowed Bohr orbits lie entirely inside the nucleus ( $r < R_0$ ). If $\lambda_n$ represents the de Broglie wavelength of the electron in the $n^{\text{th}}$ allowed orbit, what is the ratio $\frac{\lambda_2}{\lambda_1}$ ?

Options

A.
$\frac{1}{\sqrt{2}}$
B.
$2$
C.
$\sqrt{2}$
D.
$\frac{1}{2}$

Check Your Answer

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

Radius and Velocity Scaling|Conservative Force Field|Shell Theorem (Internal vs External)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to adapt Bohr's quantization to a non-standard conservative force field, specifically exploiting the Shell Theorem.

A common trap is the reflexive application of standard Bohr model results ( $v \propto 1/n$ and $\lambda \propto n$ ), which are only valid for a point-charge $1/r^2$ force field.

By placing the electron inside a uniformly charged sphere, the force becomes linear ( $F \propto r$ ), fundamentally altering the scaling laws of radius and velocity.

Students who fail to derive the new velocity dependence from first principles will incorrectly choose $\lambda_2/\lambda_1 = 2$ .

Quick Facts About This Question

Question Information

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