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Question #42029

Difficulty: 8/10 | 🟢 Free

The Question

Initially at $t=0$ , a system comprising a large parallel-plate capacitor and an uncharged, perfectly conducting solid block placed entirely between its plates is completely field-free. The block contains an internal, empty, eccentric spherical cavity of radius $r_0$ . For $t > 0$ , the capacitor is charged such that a uniform displacement current density $\vec{J}_d = J_0 \hat{k}$ exists in the vacuum region between the plates. An infinitesimal test charge $+q_0$ is moved from the center of the cavity to its inner surface along a complex path of total length $L$ at a constant speed $v$ . What is the magnitude of the total work done by the electromagnetic forces on the test charge during this process?

Options

A.
$0$
B.
$\frac{1}{2} q_0 \mu_0 J_0 v r_0 L$
C.
$\frac{q_0 J_0 L^2}{2 \epsilon_0 v}$
D.
$q_0 \mu_0 J_0 v L^2$

Check Your Answer

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

Displacement vs. Total Path Length|Cavity Field Shielding|Equipotential Surface Geometry|Vector Integration

Why This Question Is Tricky

Common Mistakes

This question tests the concept of electromagnetic shielding in perfect conductors, specifically targeting the misconception that external displacement currents or time-varying fields can penetrate an enclosed cavity.

Students often erroneously calculate the induced magnetic field using the Ampere-Maxwell law or the electric field from the displacement current, and then integrate these non-zero forces over the path length $L$ .

The trap exploits the failure to recognize that a perfect conductor shields not only electrostatic fields (via surface charges) but also time-varying magnetic fields (via induced surface currents), rendering the entire cavity strictly field-free.

Consequently, the Lorentz force is zero everywhere along the path, making the work done identically zero.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Displacement vs. Total Path Length
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free