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Question #88355

Difficulty: 5/10 | 🟢 Free

The Question

Two identical cylindrical metallic wires, A and B, have the same free electron number density $n$ . They are connected in parallel across an ideal alternating voltage source $V(t) = V_0 \cos(\omega t)$ . Due to different operating temperatures, the electron relaxation time in wire A is $\tau_A = \tau_0$ , while in wire B it is $\tau_B = 2\tau_0$ . The angular frequency of the voltage source is tuned to $\omega = \frac{1}{\tau_0}$ . Assume the drift velocity $v_d$ of the electrons satisfies the equation of motion $m \frac{dv_d}{dt} + \frac{m}{\tau} v_d = -e E(t)$ , where $m$ and $-e$ are the mass and charge of an electron, and $E(t)$ is the instantaneous uniform electric field inside the wire. What is the ratio of the time-averaged power dissipated in wire A to that in wire B, $\frac{\langle P_A \rangle}{\langle P_B \rangle}$ ?

Options

A.
1.25
B.
0.50
C.
2.00
D.
0.80

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

Power Dissipation Ratio|Relaxation Time Dependency|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the limits of Ohm's Law under high-frequency alternating fields.

Students commonly fall into the trap of assuming the DC power relation $\langle P \rangle \propto \sigma_0 \propto \tau$ holds instantaneously, which would incorrectly yield a ratio of 0.50.

However, at high frequencies where $\omega \tau \sim 1$ , the inertia of the electrons ( $m \frac{dv}{dt}$ ) becomes comparable to the scattering term.

This introduces a phase lag between the electric field and the drift velocity, making the conductivity complex.

The real power dissipation depends on the real part of this complex conductivity, which scales as $\frac{\tau}{1 + \omega^2 \tau^2}$ .

The problem forces students to derive or apply the AC Drude model, recognizing that increased relaxation time can actually decrease power dissipation at high frequencies due to the dominant inertial reactance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Power Dissipation Ratio
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free