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Question #42768

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor consists of two square conductive plates of side length $L$ separated by a distance $d$ . The region between the plates from $x = 0$ to $x = d/2$ is vacuum. The region from $x = d/2$ to $x = d$ is filled with a non-uniform dielectric material whose relative permittivity varies spatially as $\epsilon_r(x) = \frac{2x}{d}$ . If a constant charge $Q$ is supplied to the positive plate at $x=0$ and $-Q$ to the negative plate at $x=d$ , what is the total electrostatic energy stored in the capacitor? (Assume edge effects are negligible and $\epsilon_0$ is the permittivity of free space).

Options

A.
$\frac{Q^2 d}{4 \epsilon_0 L^2} (1 + \ln 2)$
B.
$\frac{5 Q^2 d}{12 \epsilon_0 L^2}$
C.
$\frac{5 Q^2 d}{8 \epsilon_0 L^2}$
D.
$\frac{Q^2 d}{4 \epsilon_0 L^2} \ln 2$

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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

Capacitance Independence|Medium Dependency|Energy Density Locality|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the principle of energy density locality in non-uniform dielectrics.

Students frequently fall into the trap of calculating an average dielectric constant ( $\epsilon_{avg} = 1.5$ ) and using the macroscopic formula $U = Q^2 / 2C_{eq}$ , which incorrectly assumes the electric field responds to the bulk average rather than the local permittivity (leading to Option B).

Another common trap is assuming the electric field $E$ remains constant across the varying medium, rather than the displacement field $D$ (leading to Option C).

By forcing the integration of the local energy density $u = \frac{1}{2} D \cdot E$ , the problem discriminates between rote application of capacitance formulas and a fundamental understanding of Maxwell's equations in matter.

Quick Facts About This Question

Question Information

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