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Question #42175

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor has plates of length $L$ , width $b$ , and separation $d$ . Let the left edge of the plates be at $x=0$ and the right edge at $x=L$ . A rigid dielectric slab of width $b$ , thickness $d$ , and length greater than $L$ is inserted between the plates from the right. Its left face is at position $x$ ( $0 < x < L$ ). The permittivity of the slab varies with the distance $s$ from its left face as $\epsilon(s) = \epsilon_0(1 + \alpha s)$ . The region between the plates from $0$ to $x$ contains an ideal monatomic gas (permittivity $\epsilon_0$ ), confined by the slab which acts as a frictionless piston. The plates are maintained at a constant voltage $V$ . If the gas expands reversibly from $x = L/4$ to $x = L/2$ while maintaining mechanical equilibrium, find the total heat $Q$ supplied to the gas. (Ignore fringing fields)

Options

A.
$\frac{\epsilon_0 b \alpha V^2 L^2}{8 d}$
B.
$\frac{3 \epsilon_0 b \alpha V^2 L^2}{16 d}$
C.
$\frac{5 \epsilon_0 b \alpha V^2 L^2}{64 d}$
D.
$\frac{5 \epsilon_0 b \alpha V^2 L^2}{32 d}$

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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

Internal Energy Dependency|Non-Uniform Dielectric Filling|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of electrostatics and thermodynamics, specifically targeting the misconception of how non-uniform dielectrics behave when translated.

The primary trap lies in assuming the dielectric profile $\epsilon(x)$ is fixed in the lab frame rather than moving with the slab.

If a student incorrectly integrates $\epsilon(u) = \epsilon_0(1+\alpha u)$ instead of shifting the coordinate to $\epsilon(u-x)$ , they obtain a fundamentally flawed capacitance gradient, leading to an incorrect pressure profile $P \propto x$ instead of $P \propto (L-x)$ (Option B).

Furthermore, it forces students to derive the local mechanical equilibrium condition from the Maxwell stress tensor's macroscopic equivalent ( $F_e = \frac{1}{2}V^2 \frac{dC}{dx}$ ) and apply it to a thermodynamic cycle, penalizing those who blindly use standard ideal gas formulas or ignore the internal energy change (Option C).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Internal Energy Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free