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Question #42685

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor has plate area $A$ (in $\text{m}^2$ ) and separation $d$ (in $\text{m}$ ). The space between the plates, from $x = 0$ to $x = d$ , is filled with a non-uniform dielectric whose relative permittivity varies as $\epsilon_r(x) = 1 + \frac{x}{d}$ . The plate at $x = 0$ is given a free charge $+Q$ (in $\text{C}$ ) and the plate at $x = d$ is given a free charge $-Q$ (in $\text{C}$ ). Let $x_0$ be the distance from the positive plate such that the electrostatic energy stored in the region $0 \leq x \leq x_0$ is exactly half of the total electrostatic energy stored in the entire capacitor. Find the value of $x_0$ .

Options

A.
$(\sqrt{2} - 1)d$
B.
$\frac{d}{2}$
C.
$\frac{d}{3}$
D.
$\frac{\sqrt{10} - 2}{2}d$

Check Your Answer

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

Energy Density Locality|Medium Dependency|Non-Uniform Dielectric Filling

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the locality of electrostatic energy density in a non-uniform medium.

A common trap is assuming that the electric field $E$ is constant and calculating energy density as $u \propto \epsilon_r(x)$ , which leads to an incorrect quadratic integration.

Another lethal mistake is forgetting the dielectric constant in the energy density formula and using $u = \frac{1}{2}\epsilon_0 E^2$ , which yields an inverse-square integration.

Students must recognize that the displacement field $D$ is the conserved quantity across the dielectric layers, making $E(x)$ inversely proportional to $\epsilon_r(x)$ , and correctly integrate the local energy density $u(x) = \frac{1}{2} D E(x)$ to find the spatial distribution of stored energy.

Quick Facts About This Question

Question Information

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