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Question #42632

Difficulty: 8/10 | 🟢 Free

The Question

In a conductor, free electrons of mass $m$ , charge $-e$ , and number density $n$ are subjected to a high-frequency alternating electric field $E(t) = E_0 \cos(\omega t)$ . The electrons experience a resistive drag force $\vec{F}_d = -m\vec{v}/\tau$ , where $\vec{v}$ is the drift velocity and $\tau$ is the mean relaxation time. If the driving frequency $\omega$ is exactly equal to the collision frequency $\tau^{-1}$ , what is the time-averaged power dissipated per unit volume $\langle P \rangle$ and the phase angle $\phi$ by which the current density lags the electric field?

Options

A.
$\frac{n e^2 \tau E_0^2}{4m}, \; \frac{\pi}{4}$
B.
$\frac{n e^2 \tau E_0^2}{2m}, \; \frac{\pi}{4}$
C.
$\frac{n e^2 \tau E_0^2}{4m}, \; \frac{\pi}{2}$
D.
$\frac{n e^2 \tau E_0^2}{2m}, \; 0$

Check Your Answer

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

Driven Oscillations and Resonance|Drift Velocity Mechanism|Capacitive Reactance Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the breakdown of the quasi-static Drude model at high frequencies.

Students commonly fall into the trap of blindly applying the DC Ohm's law ( $J = \sigma_0 E$ ), ignoring the electron's inertial term ( $m \frac{dv}{dt}$ ) which becomes comparable to the collision drag at high frequencies.

This incorrect assumption leads to the belief that current and electric field are always in phase ( $\phi=0$ ) and results in an overestimation of the dissipated power (Option D).

By solving the driven harmonic oscillator model, students must recognize that the complex conductivity introduces a phase lag and reduces the effective real resistance, simulating a resonant plasma-like response.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Driven Oscillations and Resonance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free