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Question #43431

Difficulty: 8/10 | 🟢 Free

The Question

Consider a homogeneous, isotropic conductor with static conductivity $\sigma_0$ , electric permittivity $\epsilon$ , and charge carrier relaxation time $\tau$ . The conductor is subjected to a uniform, time-harmonic electric field $\mathbf{E}(t) = \mathbf{E}_0 e^{i\omega t}$ . According to the Drude model, the AC conductivity of the material is given by $\sigma(\omega) = \frac{\sigma_0}{1 + i\omega\tau}$ . In the Ampere-Maxwell law, the total current density is the sum of the conduction current density and the displacement current density. At what angular frequency $\omega_c$ does the amplitude of the displacement current density exactly equal the amplitude of the conduction current density?

Options

A.
$\frac{1}{\tau} \left[ \frac{1}{2} \left( \sqrt{1 + 4\left(\frac{\sigma_0\tau}{\epsilon}\right)^2} - 1 \right) \right]^{1/2}$
B.
$\frac{\sigma_0}{\epsilon}$
C.
$\frac{1}{\tau} \left[ \sqrt{1 + \left(\frac{\sigma_0\tau}{\epsilon}\right)^2} - 1 \right]^{1/2}$
D.
$\frac{\sigma_0}{\epsilon \sqrt{1 + \left(\frac{\sigma_0\tau}{\epsilon}\right)^2}}$

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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

Ampere’s Law Displacement Current|Current Density Vector|Relaxation Time Dependency

Why This Question Is Tricky

Common Mistakes

Students incorrectly apply the static conductivity approximation ( $\sigma \approx \sigma_0$ ) at high frequencies, leading them to equate $\sigma_0 = \omega\epsilon$ and incorrectly choose $\omega = \sigma_0/\epsilon$ .

This neglects the inertia of charge carriers represented by the relaxation time $\tau$ .

The trap tests the understanding that in the high-frequency regime near the collision frequency ( $1/\tau$ ), the conduction current lags the electric field and its amplitude drops.

Students must correctly apply the complex Drude conductivity within the Ampere-Maxwell law and equate the magnitudes of the complex current density vectors to find the true transition frequency.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Ampere’s Law Displacement Current
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free