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Question #43799

Difficulty: 8/10 | 🟢 Free

The Question

In a hypothetical hydrogen-like atom, the electrostatic potential energy of the electron (mass $m$ ) is modified to $V(r) = -\frac{k}{r^{3/2}}$ , where $k$ is a positive constant and $r$ is the orbital radius. Assuming the Bohr model of angular momentum quantization is valid, let $\lambda_{L}$ be the longest wavelength and $\lambda_{S}$ be the shortest wavelength of the spectral lines emitted in the Lyman-like series (transitions ending at the ground state $n=1$ ). What is the ratio $\frac{\lambda_{L}}{\lambda_{S}}$ ?

Options

A.
$\frac{64}{63}$
B.
$\frac{4}{3}$
C.
$\frac{8}{7}$
D.
$\frac{16}{15}$

Check Your Answer

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

Spectral Series Limits|Potential vs Field Relationship|Calculus (Integration)

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental origin of the Rydberg formula by placing the electron in a non-Coulombic central potential $V(r) \propto r^{-3/2}$ .

The primary trap is the rote application of the standard Rydberg formula, which assumes energy levels scale as $1/n^2$ , leading students to incorrectly choose $4/3$ .

By forcing the derivation of the energy level dependence ( $E_n \propto 1/n^6$ ) via force balance combined with Bohr's quantization condition, the problem discriminates between students who memorize formulas and those who understand the underlying quantum mechanical principles of atomic spectra.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Spectral Series Limits
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free