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Question #43254

Difficulty: 8/10 | 🟢 Free

The Question

An ideal parallel plate capacitor of plate area $A$ is at rest in an inertial frame $S$ , with its plates parallel to the $x-z$ plane. It is being charged by a constant conduction current $I_0$ . An observer in an inertial frame $S'$ moves with a constant velocity $v \hat{i}$ relative to $S$ . Neglecting the induced magnetic field between the plates, the observer in $S'$ calculates the total displacement current $I'_d = \int \epsilon_0 \frac{\partial \vec{E}'}{\partial t'} \cdot d\vec{A}'$ through a surface fixed in $S'$ that exactly coincides with the electric field region between the plates at a given instant $t'$ . Which of the following expressions gives the correct value of $I'_d$ ? (Here $c$ is the speed of light in vacuum)

Options

A.
$I_0$
B.
$I_0 \sqrt{1 - \frac{v^2}{c^2}}$
C.
$\frac{I_0}{\sqrt{1 - \frac{v^2}{c^2}}}$
D.
$I_0 \left(1 - \frac{v^2}{c^2}\right)$

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

Ampere’s Law Displacement Current|Instantaneous Nature|Frame of Reference

Why This Question Is Tricky

Common Mistakes

This question tests the transformation of electromagnetic fields and the relativity of simultaneity applied to Maxwell's displacement current.

Students typically fall into the trap of assuming displacement current is a relativistic invariant (like electric charge) or incorrectly applying simple time dilation to the rate of change of charge ( $I' = I_0/\gamma$ ).

They fail to recognize that the partial time derivative of the electric field in the moving frame introduces a $\gamma^2$ factor due to the relativity of simultaneity ( $t = \gamma(t' + vx'/c^2)$ ).

When combined with the length-contracted area ( $A/\gamma$ ), the perceived displacement current evaluates to $\gamma I_0$ , demonstrating that displacement current and conduction current do not transform identically when evaluated over a fixed spatial surface in a dynamic frame.

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