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Question #43159

Difficulty: 8/10 | 🟢 Free

The Question

An AC voltage source $V(t) = V_0 \cos(\omega t)$ with a constant internal resistance $R_i$ is connected in series with a capacitor of capacitance $C$ and a special load wire. The load wire is a cylindrical conductor of length $l$ , cross-sectional area $A$ , electron number density $n$ , and electron relaxation time $\tau$ . The effective mass of an electron is $m$ and its charge is $-e$ . According to the Drude model, the drift velocity $v_d$ of electrons under an electric field $E$ obeys $m \frac{dv_d}{dt} + \frac{m}{\tau} v_d = -e E$ . At what angular frequency $\omega$ is the average power dissipated in the load wire maximized?

Options

A.
$\sqrt{\frac{n e^2 A}{m l C}}$
B.
$\frac{1}{\tau}$
C.
$\sqrt{\frac{n e^2 A}{m l C} + \frac{1}{\tau^2}}$
D.
$\infty$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Frequency Sensitivity of Z|Relaxation Time Dependency|Maximum Power Transfer

Why This Question Is Tricky

Common Mistakes

Students typically model a conducting wire purely as a macroscopic resistor, ignoring the microscopic inertia of charge carriers.

If the electron mass $m$ is ignored, the wire is treated as purely resistive, leading to a total circuit impedance of $Z = R_{total} - j/\omega C$ .

Under this false assumption, students would attempt to maximize power by minimizing the capacitive reactance, incorrectly concluding that $\omega \to \infty$ (Option D).

This question forces students to bridge microscopic transport (the Drude model) with macroscopic AC circuit theory, revealing that electron inertia manifests as a 'kinetic inductance'.

This inductive component shifts the maximum power transfer point to a finite resonant frequency.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Frequency Sensitivity of Z
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free