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Question #42713

Difficulty: 8/10 | 🟢 Free

The Question

A uniform conducting wire of constant cross-sectional area $A$ and free electron density $n$ is bent into a semi-circle of radius $R$ . A time-varying voltage is applied across its ends such that the tangential electric field inside the wire is $E(t) = E_0 \sin(\omega t)$ for $0 \leq t \leq \pi/\omega$ , where $E_0$ is a positive constant. An average conduction electron enters one end of the semi-circle at $t=0$ and exactly reaches the other end at $t = \pi/\omega$ . Let $\mu$ be the electron mobility and $e$ be the elementary charge. Which of the following gives the magnitude of the average drift velocity vector $|\langle \vec{v}_d \rangle|$ of this electron during this time interval, and the total charge $Q$ that flows through any cross-section of the wire?

Options

A.
$|\langle \vec{v}_d \rangle| = \frac{4\mu E_0}{\pi^2}, \; Q = \pi R n e A$
B.
$|\langle \vec{v}_d \rangle| = \frac{2\mu E_0}{\pi}, \; Q = \pi R n e A$
C.
$|\langle \vec{v}_d \rangle| = \frac{4\mu E_0}{\pi^2}, \; Q = 2 R n e A$
D.
$|\langle \vec{v}_d \rangle| = \frac{2\mu E_0}{\pi}, \; Q = 2 R n e A$

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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

Drift Velocity Mechanism|Displacement vs. Total Path Length|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between microscopic path length and macroscopic displacement in the context of drift kinematics and the continuity equation.

Students often erroneously calculate the average drift velocity vector using the total path length along the curved conductor (yielding $2\mu E_0/\pi$ , which is actually the average speed) instead of the net straight-line displacement ( $2R$ ).

Conversely, when calculating the total charge flow, students might incorrectly use the displacement instead of the path length.

The correct physical model requires using displacement for the average velocity vector, but the path length (arc length) for the charge flux, as current is constrained to flow along the wire's geometry.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Drift Velocity Mechanism
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free