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Question #42812

Difficulty: 8/10 | 🟢 Free

The Question

An uncharged parallel plate capacitor has plate area $A$ and separation $d$ . The space between the plates is filled with a leaky dielectric material. The permittivity of the material varies linearly with the distance $x$ from the positive plate as $\epsilon(x) = \epsilon_0 \left(1 + \frac{x}{d}\right)$ , while its resistivity is uniform at $\rho(x) = \rho_0$ . At $t = 0$ , the capacitor is instantaneously charged to a voltage $V_0$ (such that no free charge has time to accumulate in the bulk of the dielectric) and immediately isolated. What is the magnitude of the initial rate of change of voltage across the capacitor, $\left| \frac{dV}{dt} \right|_{t=0}$ ?

Options

A.
$\frac{V_0}{2 \rho_0 \epsilon_0 \ln 2}$
B.
$\frac{V_0 \ln 2}{\rho_0 \epsilon_0}$
C.
$\frac{V_0}{\rho_0 \epsilon_0}$
D.
$\frac{2 V_0}{3 \rho_0 \epsilon_0}$

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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

Non-Uniform Dielectric Filling|Resistivity vs Resistance|Current Electricity

Why This Question Is Tricky

Common Mistakes

This question tests the misconception that a leaky capacitor with non-uniform dielectric properties can be modeled as a simple lumped RC circuit with a single time constant $\tau = R_{eq} C_{eq}$ .

Students often incorrectly apply this lumped model, calculating equivalent resistance and capacitance to find the initial voltage decay rate (leading to Option B).

However, when $\rho(x)\epsilon(x)$ is not constant, the local relaxation time varies across the dielectric.

Instantaneous charging ensures the initial displacement field $D$ is uniform, but the varying local decay rate $\frac{\partial E}{\partial t} = -\frac{E}{\rho\epsilon}$ causes charge to accumulate in the bulk, making the voltage decay non-exponential.

The true initial decay rate must be found by integrating the local rate of change of the electric field.

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