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Question #42816

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The Question

Consider a classical electron gas of mass $m$ and charge $-e$ at temperature $T$ in a uniform electric field $E$ . In the microscopic Drude-Lorentz model, the drift velocity of an electron with thermal speed $v$ is given by $v_d(v) = \frac{eE}{m}\tau(v)$ , where $\tau(v)$ is the relaxation time. Assume the electrons scatter off neutral impurities such that their mean free path $\lambda$ is constant and independent of $v$ . The macroscopic drift velocity $\langle v_d \rangle$ is defined as the average of $v_d(v)$ over the 3D Maxwell-Boltzmann speed distribution $f(v)$ . Which of the following expressions gives the correct macroscopic drift velocity? (Here $k_B$ is the Boltzmann constant).

Options

A.
$eE\lambda \sqrt{\frac{2}{\pi m k_B T}}$
B.
$eE\lambda \sqrt{\frac{\pi}{8 m k_B T}}$
C.
$eE\lambda \sqrt{\frac{1}{3 m k_B T}}$
D.
$eE\lambda \sqrt{\frac{2}{m k_B T}}$

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

Relaxation Time Dependency|Maxwell-Boltzmann Distribution|Temperature and Kinetic Energy

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply statistical averaging over a velocity distribution rather than relying on lumped macroscopic parameters.

A common trap (related to Jensen's Inequality) is assuming that the average of an inverse quantity equals the inverse of the average, i.e., $\langle \tau(v) \rangle = \lambda / \langle v \rangle$ .

Students falling for this will incorrectly substitute the mean speed $\langle v \rangle = \sqrt{8k_B T / \pi m}$ into the drift velocity formula, leading to Option B.

The correct physical approach requires integrating the speed-dependent drift velocity $v_d(v) \propto v^{-1}$ over the 3D Maxwell-Boltzmann probability density function, which yields a different numerical prefactor and correctly captures the microscopic transport physics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Relaxation Time Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free