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Question #42327

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor with plates of width $b$ , length $L$ , and separation $d$ is connected to a constant voltage source $V$ . A dielectric slab of mass $m$ , dielectric constant $K$ , and dimensions matching the capacitor's interior is initially placed just at the edge of the plates ( $x=0$ ). The slab is pulled into the capacitor by electrostatic forces. As it enters, it experiences a position-dependent resistive friction force $f = \alpha x$ , where $\alpha$ is a positive constant and $x$ is the length of the slab inside the capacitor. Assuming gravity is negligible and the plates are long enough, what is the maximum kinetic energy acquired by the dielectric slab?

Options

A.
$\frac{\epsilon_0^2 b^2 (K-1)^2 V^4}{8 d^2 \alpha}$
B.
$\frac{\epsilon_0^2 b^2 (K-1)^2 V^4}{2 d^2 \alpha}$
C.
$\frac{\epsilon_0^2 b^2 (K-1)^2 V^4}{4 d^2 \alpha}$
D.
$\frac{\epsilon_0^2 b^2 (K-1)^2 V^4}{d^2 \alpha}$

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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

Mechanical Energy Conservation|Dielectric Slab Displacement|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy Theorem in an electro-mechanical system with non-uniform acceleration.

A common lethal trap for JEE Advanced students is incorrectly equating the electrostatic force to the spatial derivative of the battery's work ( $F = dW_{batt}/dx$ ), leading to a force twice as large ( $F = 2F_e$ ).

Another trap is assuming simple energy conservation ( $\Delta K = \Delta U$ ) without accounting for the work done by the battery and the non-conservative friction force.

By requiring the integration of a position-dependent net force, students must correctly identify $F_e = \frac{1}{2}V^2 \frac{dC}{dx}$ and apply $\Delta K = \int F_{net} dx$ , avoiding the false assumption of constant acceleration or isolated system energy balances.

Quick Facts About This Question

Question Information

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