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Question #43617

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ is released from rest at the origin $x = 0$ . It moves along the positive $x$ -axis under the influence of a position-dependent force $F(x) = \alpha x^n$ , where $\alpha$ and $n$ are positive constants. If $\lambda(x)$ represents the de Broglie wavelength of the particle at a coordinate $x > 0$ , which of the following expressions correctly gives the spatial derivative of the de Broglie wavelength, $\frac{d\lambda}{dx}$ ?

Options

A.
$-\frac{(n+1)\lambda}{2x}$
B.
$-\frac{n\lambda}{2x}$
C.
$-\frac{(n+1)\lambda}{x}$
D.
$-\frac{n\lambda}{x}$

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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

De Broglie Wavelength Scaling|Variable Force Integration|Work, Energy, and Power

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of classical variable-force kinematics with quantum wave-particle duality.

The primary trap lies in the dimensional and functional relationship between force, potential energy, and kinetic energy.

Students frequently misapply the power-law by assuming kinetic energy scales directly with the force's exponent ( $K \propto x^n$ ), effectively confusing force with potential energy, which leads to the incorrect gradient $-\frac{n\lambda}{2x}$ .

A secondary trap involves neglecting the square root in the momentum-kinetic energy relation ( $p = \sqrt{2mK}$ ), leading to an incorrect missing factor of 2 in the denominator.

The correct approach requires rigorous integration of the force to find the kinetic energy scaling ( $x^{n+1}$ ), followed by direct differentiation of the de Broglie wavelength.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	De Broglie Wavelength Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free