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Question #42771

Difficulty: 8/10 | 🟢 Free

The Question

A parallel-plate capacitor has plate area $A$ and separation $d$ . It is completely filled with a semiconducting material of dielectric constant $K$ . The material exhibits non-linear conduction, such that the conduction current density $J$ is related to the local electric field $E$ by $J = \beta E^2$ , where $\beta$ is a positive constant with appropriate units. At $t = 0$ , the capacitor is charged to an initial potential difference $V_0$ and then isolated from the voltage source. Find the time $t$ at which the magnitude of the potential gradient inside the capacitor reduces to half of its initial value. Assume edge effects are negligible and the permittivity of free space is $\epsilon_0$ .

Options

A.
$\frac{K\epsilon_0 d}{\beta V_0}$
B.
$\frac{K\epsilon_0 d}{2\beta V_0}$
C.
$\frac{K\epsilon_0 d \ln 2}{\beta V_0}$
D.
$\frac{K\epsilon_0 d \ln 2}{2\beta V_0}$

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

Capacitance Independence|Potential Gradient Direction|Dynamic Resistance (AC Resistance)|Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between geometric capacitance and non-linear circuit impedance.

A common trap is to incorrectly apply the concept of 'dynamic resistance' ( $R_{dyn} = dV/dI$ ) to the discharge equation, writing $I = V/R_{dyn}$ instead of the fundamental charge conservation $I = -dQ/dt$ .

Because the semiconductor is non-linear ( $J \propto E^2$ ), students often mistakenly assume the capacitance $C$ itself becomes a function of voltage, or they attempt to use an exponential decay formula $t = RC \ln(2)$ with an initial or dynamic resistance.

The correct approach requires recognizing that the geometric capacitance remains strictly constant (Capacitance Independence) while setting up a non-linear differential equation based on the instantaneous conduction current.

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