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Question #43621

Difficulty: 8/10 | 🟢 Free

The Question

In the classical Drude model of a metallic conductor, the free electrons are treated as an ideal gas at absolute temperature $T$ . The relaxation time $\tau$ of the electrons due to lattice scattering is given to be inversely proportional to the temperature ( $\tau \propto T^{-1}$ ). A uniform electric field $E$ is applied across the conductor, establishing a steady drift velocity $v_d$ . Let $\lambda_{dB}$ be the de Broglie wavelength of the electrons, and $\lambda_{mfp}$ be their mean free path between successive collisions. Assuming the thermal velocity $v_{th}$ is much greater than $v_d$ , which of the following correctly represents the temperature dependence of $\lambda_{dB}$ and the ratio $R = \frac{\lambda_{dB}}{\lambda_{mfp}}$ ?

Options

A.
$\lambda_{dB} \propto T^{-1/2}$ and $R \propto T^{0}$
B.
$\lambda_{dB} \propto T$ and $R \propto T^{3/2}$
C.
$\lambda_{dB} \propto T^{-1/2}$ and $R \propto T^{3/2}$
D.
$\lambda_{dB} \propto T$ and $R \propto T^{3}$

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

Wave-Particle Duality Constraint|Relaxation Time Dependency|De Broglie Wavelength Scaling

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between random thermal motion and directed drift motion in the context of wave-particle duality and classical transport.

A common misconception is to use the macroscopic drift velocity ( $v_d$ ) instead of the microscopic thermal velocity ( $v_{th}$ ) to calculate the de Broglie wavelength of charge carriers.

Since $v_d \propto \tau \propto T^{-1}$ , this incorrect assumption leads to $\lambda_{dB} \propto T$ .

Furthermore, students often miscalculate the mean free path by using the drift distance ( $v_d \tau$ ) instead of the actual path length ( $v_{th} \tau$ ).

By requiring the ratio $R$ , the problem elegantly isolates these errors, rewarding only the correct synthesis of the Drude model and quantum duality.

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