Before You Begin

Question #43917

Difficulty: 8/10 | 🟢 Free

The Question

Consider a non-degenerate semiconductor plasma with an electron number density $n$ . The electrons are in thermal equilibrium at temperature $T$ and follow a 3D Maxwell-Boltzmann velocity distribution. The collision frequency of an electron with the crystal lattice depends on its thermal speed $v$ as $\nu(v) = \alpha v$ , where $\alpha$ is a positive constant. A high-frequency alternating electric field $\vec{E}(t) = \vec{E}_0 \cos(\omega t)$ is applied to the semiconductor. Assuming the frequency is extremely high such that $\omega \gg \nu(v)$ for all statistically significant electron speeds, determine the exact expression for the time-averaged power dissipated per unit volume, $\langle P \rangle$ , due to Joule heating.

(Given: $m$ is the mass of an electron, $e$ is the elementary charge, and $k_B$ is the Boltzmann constant)

Options

A.
$\frac{n e^2 E_0^2 \alpha}{m \omega^2} \sqrt{\frac{2 k_B T}{\pi m}}$
B.
$\frac{n e^2 E_0^2 \alpha}{2 m \omega^2} \sqrt{\frac{3 k_B T}{m}}$
C.
$\frac{n e^2 E_0^2 \alpha}{2 m \omega^2} \sqrt{\frac{2 k_B T}{m}}$
D.
$\frac{2 n e^2 E_0^2 \alpha}{m \omega^2} \sqrt{\frac{2 k_B T}{\pi m}}$

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

Maxwell-Boltzmann Distribution|Dynamic Resistance (AC Resistance)|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge kinetic theory with AC electrodynamics, specifically targeting the dynamic resistance regime where the relaxation time is not constant.

The primary trap lies in the statistical averaging of the collision frequency.

Students commonly misapply the Drude model by either assuming a constant relaxation time and using the RMS velocity ( $v_{rms} = \sqrt{3k_B T/m}$ ) or the most probable velocity ( $v_p = \sqrt{2k_B T/m}$ ) to find an 'effective' collision frequency.

Furthermore, they might incorrectly average the conductivity $\langle 1/\nu \rangle$ instead of $\langle \nu \rangle$ for the high-frequency regime ( $\omega \gg \nu$ ).

The correct physical approach requires deriving the velocity-dependent AC conductivity and integrating the collision frequency $\nu(v)$ over the 3D Maxwell-Boltzmann distribution to find the true mean $\langle v \rangle$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	Maxwell-Boltzmann Distribution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free