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Question #42693

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The Question

A parallel plate capacitor has rectangular plates of length $L$ , width $w$ , and separation $d$ . It is connected to an ideal battery of electromotive force $V$ . A solid dielectric slab of length $L$ , width $w$ , thickness $d/2$ , and dielectric constant $K$ is initially outside the capacitor. The following sequence of operations is performed:

The slab is slowly inserted into the gap, parallel to the plates, until it completely covers the area $L \times w$ .
The battery is disconnected.
The slab is slowly pulled completely out of the capacitor. Find the total work done by the external agent during this entire process. (Ignore edge effects, $\epsilon_0$ is the permittivity of free space)

Options

A.
$\frac{\epsilon_0 w L V^2}{2d} \left( \frac{K-1}{K+1} \right)^2$
B.
$\frac{\epsilon_0 w L V^2}{2d} \frac{(K-1)(3K+1)}{(K+1)^2}$
C.
$\frac{\epsilon_0 w L V^2}{2d} \frac{2K(K-1)}{(K+1)^2}$
D.
$\frac{\epsilon_0 w L V^2}{2d} \left( \frac{K-1}{K+1} \right)$

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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

Common Potential Redistribution|Symmetry and Virtual Work|Dielectric Constant (K) Shift|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between constant potential and constant charge systems, specifically focusing on the Virtual Work principle.

Students commonly fall into the trap of equating external work directly to the change in electrostatic potential energy ( $\Delta U$ ), ignoring the work done by the battery ( $W_{batt} = V \Delta Q = 2\Delta U$ ) during the insertion phase.

This leads to a sign error in the first step.

Furthermore, it tests the ability to correctly transition to a constant charge state, where the external work to remove the dielectric is purely the change in isolated system energy.

Failing to account for the battery's energy injection results in incorrect summation of the thermodynamic cycle's work.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Common Potential Redistribution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free