Before You Begin

Question #42315

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor with plates of length $L$ , width $w$ , and separation $d$ is connected to a battery of constant voltage $V$ . A solid dielectric slab of dielectric constant $K$ and dimensions $L \times w \times d$ is inserted a distance $x$ into the capacitor along its length. The $+x$ direction is defined along the inward direction of insertion, with unit vector $\hat{i}$ . Which of the following correctly represents the electrostatic force $\vec{F}_e$ exerted by the capacitor on the slab, and the external work $W_{ext}$ required to slowly pull the slab completely out from a fully inserted state?

Options

A.
$\vec{F}_e = \frac{\epsilon_0 w V^2}{2d}(K-1) \hat{i}, \quad W_{ext} = \frac{\epsilon_0 L w V^2}{2d}(K-1)$
B.
$\vec{F}_e = -\frac{\epsilon_0 w V^2}{2d}(K-1) \hat{i}, \quad W_{ext} = -\frac{\epsilon_0 L w V^2}{2d}(K-1)$
C.
$\vec{F}_e = \frac{\epsilon_0 w V^2}{d}(K-1) \hat{i}, \quad W_{ext} = \frac{\epsilon_0 L w V^2}{d}(K-1)$
D.
$\vec{F}_e = -\frac{\epsilon_0 w V^2}{2d}(K-1) \hat{i}, \quad W_{ext} = \frac{\epsilon_0 L w V^2}{2d}(K-1)$

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

Conservative Force Field|Dielectric Slab Displacement|Work-Energy Theorem|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of conservative forces in systems connected to an external energy source.

A common lethal mistake is blindly applying the relation $F = -dU/dx$ (valid only for isolated, constant-charge systems) to a constant-potential system.

When connected to a battery, the battery does work $dW_{batt} = V dQ = 2dU$ .

Consequently, the electrostatic force is $F = +dU/dx$ .

Failing to account for the battery's work leads to an incorrect sign for the force and equates the external work directly to the change in potential energy, trapping the student in the incorrect options.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Conservative Force Field
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free