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Question #42689

Difficulty: 8/10 | 🟢 Free

The Question

Two large parallel conducting plates of area $A$ are placed at $x = 0$ and $x = d$ in vacuum. Both plates are grounded ( $V = 0$ ). The space between the plates is filled with a frozen-in, non-uniform volume charge density $\rho(x) = \rho_0 \frac{x}{d}$ , where $\rho_0$ is a positive constant. Neglecting edge effects, what is the total electrostatic energy stored in the region between the plates?

Options

A.
$\frac{\rho_0^2 A d^3}{90 \epsilon_0}$
B.
$\frac{\rho_0^2 A d^3}{40 \epsilon_0}$
C.
$\frac{\rho_0^2 A d^3}{96 \epsilon_0}$
D.
$\frac{\rho_0^2 A d^3}{8 \epsilon_0}$

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

Energy Density Locality|Conservative Field Path Independence|Continuous Charge Distribution

Why This Question Is Tricky

Common Mistakes

This question tests the application of Poisson's equation and the locality of energy density in a non-uniform charge distribution.

A common 'Lethal Collision' (trap) is assuming the electric field is simply the integral of the charge density starting from zero (i.e., $E(0)=0$ ), which ignores the uniform background field generated by the induced surface charges on the grounded plates.

Students falling for this will incorrectly select the option with denominator 40.

Another trap is treating the system as a standard capacitor with the total integrated charge, or using an average uniform charge density, which fails because electrostatic energy depends non-linearly on the local electric field ( $u \propto E^2$ ).

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