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Question #88262

Difficulty: 5/10 | 🟢 Free

The Question

A parallel plate capacitor has plate area $A \text{ m}^2$ and separation $d \text{ m}$ . The space between the plates ( $0 \le x \le d$ ) is filled with a non-uniform dielectric whose relative permittivity varies as $\kappa(x) = 1 + \frac{x}{d}$ . A constant potential difference $V_0 \text{ V}$ is applied across the plates. What is the magnitude of the bound volume charge density (polarization charge density) $\rho_p$ within the dielectric at the plane $x = \frac{d}{2}$ ?

Options

A.
$\frac{4 \epsilon_0 V_0}{9 d^2 \ln 2} \text{ C/m}^3$
B.
$\frac{\epsilon_0 V_0}{d^2} \text{ C/m}^3$
C.
$\frac{2 \epsilon_0 V_0}{3 d^2} \text{ C/m}^3$
D.
$\frac{\epsilon_0 V_0}{4 d^2 \ln 2} \text{ C/m}^3$

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

Dielectric Polarization Logic|Gauss Law|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the fundamental distinction between the displacement field $D$ and the electric field $E$ in non-uniform dielectrics.

A common trap is assuming the electric field remains uniform ( $E = V_0/d$ ) despite the gradient dielectric.

If a student makes this error, they calculate $P = \epsilon_0 (\kappa(x)-1) E = \epsilon_0 (x/d) (V_0/d)$ , leading to a constant bound charge density $\rho_p = -\nabla \cdot P = -\epsilon_0 V_0 / d^2$ (Option B).

The correct physical approach recognizes that $D$ is uniform (driven by free charge on the plates), while $E$ varies inversely with $\kappa(x)$ .

Furthermore, finding the true free charge requires integrating the infinitesimal elastances ( $1/dC$ ) to find the equivalent capacitance, testing the transition from algebraic formula-memory to fundamental electrostatics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Dielectric Polarization Logic
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free