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Question #79027

Difficulty: 5/10 | 🟢 Free

The Question

In a modified semi-classical Bohr model, an electron (mass $m$ , charge $-e$ ) in a hydrogen-like atom (atomic number $Z$ ) is in the $n$ -th circular orbit. The unperturbed centripetal acceleration of the electron is $a_0$ . Suppose a hypothetical isotropic radiation field exerts a constant outward radial force $F$ on the electron. The electron's angular momentum remains strictly quantized as $L = n\hbar$ . As $F$ is slowly increased from zero, the orbit radius adjusts to maintain equilibrium. However, if $F$ exceeds a critical maximum value $F_{\text{max}}$ , classical equilibrium fails and the quantized orbit is destroyed, forcing a quantum jump. What is the centripetal acceleration of the electron exactly at the critical limit $F = F_{\text{max}}$ ?

Options

A.
$\frac{8}{27} a_0$
B.
$\frac{4}{9} a_0$
C.
$\frac{16}{81} a_0$
D.
$\frac{1}{8} a_0$

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

Centripetal Acceleration Vector|Quantization of Angular Momentum|Radiation Pressure

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to apply the Bohr quantization condition under a non-standard perturbation (a constant outward radial force).

The primary trap lies in the kinematic scaling of centripetal acceleration.

Students often incorrectly assume that centripetal acceleration scales as $1/r^2$ (mirroring the Coulomb force) and select $4/9 a_0$ .

However, they must recognize that the velocity is not constant; the quantization constraint $v \propto 1/r$ dictates that $a_c = v^2/r \propto 1/r^3$ .

Furthermore, the problem requires analyzing the stability of the effective potential, where the critical force corresponds to the inflection point (disappearance of the local minimum), forcing a quantum jump.

This elegantly bridges classical mechanics stability with semi-classical quantum constraints.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Centripetal Acceleration Vector
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free