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Question #42714

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor with circular plates of radius $R$ is being charged by a long cylindrical copper wire of radius $r$ (where $r < R$ ). The free electron density in the copper wire is $n$ , and the magnitude of the elementary charge is $e$ . At a given instant, the electrons in the wire are moving with a drift velocity of magnitude $v_d$ . Assuming the electric field $E$ between the capacitor plates is uniform and neglecting fringe effects, what is the rate of change of the electric field $\frac{\partial E}{\partial t}$ in the region between the plates at this instant? Let $\epsilon_0$ be the permittivity of free space.

Options

A.
$\frac{n e r^2 v_d}{\epsilon_0 R^2}$
B.
$\frac{n e v_d}{\epsilon_0}$
C.
$\frac{n e R^2 v_d}{\epsilon_0 r^2}$
D.
$\frac{n e r v_d}{\epsilon_0 R}$

Check Your Answer

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

Drift Velocity Mechanism|Displacement Current Source|Continuous Charge Distribution

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge microscopic conduction kinetics with macroscopic Maxwellian electrodynamics.

The primary trap lies in the misapplication of the continuity equation by incorrectly equating the conduction current density ( $J_c$ ) in the wire directly to the displacement current density ( $J_d$ ) between the plates, which leads to the erroneous answer $\frac{n e v_d}{\epsilon_0}$ .

Students often forget that while total current ( $I$ ) is continuous across the capacitor boundary due to the conservation of charge, the current density ( $J$ ) is not, because the cross-sectional area of the wire ( $r$ ) differs from the area of the capacitor plates ( $R$ ).

The correct approach requires equating the total conduction current to the total displacement current, thereby incorporating the geometric area ratio $(r/R)^2$ .

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