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Question #43622

Difficulty: 8/10 | 🟢 Free

The Question

A beam of non-relativistic electrons is accelerated from rest through a time-varying potential difference $V(t) = V_0 e^{-\alpha t}$ , where $V_0$ and $\alpha$ are positive constants. The electron beam is incident normally on a single slit of width $a$ and produces a Fraunhofer diffraction pattern on a distant screen. What is the magnitude of the angular velocity of the first diffraction minimum at the instant when its angular position is $\theta = 30^\circ$ ?

Options

A.
$\frac{\alpha}{2\sqrt{3}}$
B.
$\frac{\alpha}{\sqrt{3}}$
C.
$\frac{\sqrt{3}\alpha}{2}$
D.
$\frac{\alpha}{4}$

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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

Wave-Particle Duality Constraint|Single Slit Diffraction Minima|Calculus (Differentiation)

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of wave-particle duality with wave optics under dynamic conditions.

Students often memorize the static diffraction condition $a \sin\theta = n\lambda$ and the de Broglie relation $\lambda = h/\sqrt{2mK}$ independently.

The trap lies in treating the diffraction pattern as stationary or incorrectly linking the rate of change of the accelerating potential to the angular shift.

A common mistake is forgetting the square root dependence of momentum on potential, leading to an incorrect factor of 2 in the chain rule differentiation (yielding $\alpha/\sqrt{3}$ instead of $\alpha/(2\sqrt{3})$ ), or failing to apply implicit differentiation correctly to $\sin\theta$ , resulting in a missing $\sec\theta$ factor.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Wave-Particle Duality Constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free