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Question #42711

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor with plate area $A$ and separation $d$ is filled with a non-ideal, leaky dielectric. The permittivity of the dielectric varies as $\epsilon(x) = \epsilon_0 e^{\alpha x}$ and its electrical conductivity varies as $\sigma(x) = \sigma_0 e^{\beta x}$ , where $x$ is the distance from the positive plate ( $0 \le x \le d$ ), and $\alpha, \beta$ are positive constants with units of $\text{m}^{-1}$ . A constant DC voltage $V_0$ is applied across the plates. After a long time, the system reaches a steady state. What is the volume density of free charge $\rho_f(x)$ in the bulk of the dielectric at a distance $x$ from the positive plate?

Options

A.
$\frac{\epsilon_0 V_0 \beta (\alpha - \beta)}{1 - e^{-\beta d}} e^{(\alpha - \beta) x}$
B.
0
C.
$\frac{\epsilon_0 V_0 \alpha (\beta - \alpha)}{1 - e^{-\alpha d}} e^{(\beta - \alpha) x}$
D.
$-\frac{\epsilon_0 V_0 \beta^2}{1 - e^{-\beta d}} e^{(\alpha - \beta) x}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Non-Uniform Dielectric Filling|Kirchhoff’s Loop Rule|Relaxation Time Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between an ideal electrostatic capacitor and a leaky dielectric in a DC circuit.

Students commonly fall into the trap of assuming electrostatic conditions where the displacement field $D$ is uniform, leading to $E \propto 1/\epsilon(x)$ and a bulk free charge of zero (Option B).

However, a leaky dielectric allows a steady-state current, meaning the system acts as a distributed resistor.

The electric field is instead governed by Ohm's Law ( $E \propto 1/\sigma(x)$ ).

Because the spatial variations of $\epsilon(x)$ and $\sigma(x)$ do not match, Gauss's Law ( $\nabla \cdot D = \rho_f$ ) forces a non-zero free charge to accumulate in the bulk to simultaneously satisfy current continuity and electrostatic boundary conditions.

Option D traps students who incorrectly apply Gauss's law as $\rho_f = \epsilon(x) \frac{dE}{dx}$ instead of differentiating the entire product $\epsilon(x)E(x)$ .

Quick Facts About This Question

Question Information

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