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Question #42222

Difficulty: 8/10 | 🟢 Free

The Question

To analyze the Hydrostatic Paradox, a physicist models the gauge pressure $P$ of a fluid as the energy density of a fictitious electric field $\vec{E} = -E(z)\hat{k}$ , such that $P(z) = \frac{1}{2}\epsilon_0 |\vec{E}|^2$ . A conical container with a flat horizontal base of radius $R$ at $z=0$ and apex at $z=H$ is completely filled with an incompressible fluid of uniform density $\rho$ . The container is in a uniform downward gravitational field $g$ , and the pressure at the apex is zero. By applying Gauss's Law to the fictitious field $\vec{E}$ , determine the total fictitious charge $Q_{enc}$ enclosed within the fluid volume.

Options

A.
$\frac{\pi R^2}{5} \sqrt{2 \epsilon_0 \rho g H}$
B.
$\frac{2\pi R^2}{3} \sqrt{\epsilon_0 \rho g H}$
C.
$\frac{\pi R^2}{3} \sqrt{2 \epsilon_0 \rho g H}$
D.
$\frac{4\pi R^2}{5} \sqrt{\epsilon_0 \rho g H}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Hydrostatic Paradox|Gauss Law Applicability|Energy Density Locality

Why This Question Is Tricky

Common Mistakes

This question tests the 'Energy Density Locality' trap by forcing students to map a global macroscopic property (fluid pressure) to a local field property (energy density and divergence).

Students often incorrectly assume that a uniform fluid density implies a uniform fictitious source density.

However, the non-linear mapping ( $P \propto E^2$ ) results in a non-uniform, divergent (yet integrable) charge density $\rho_e \propto 1/\sqrt{H-z}$ .

By requiring the integration of this density over a variable cross-section, the problem bypasses intuitive shortcuts (like simply multiplying volume by a constant) and demands rigorous application of Gauss's Law and volume integration, perfectly highlighting the mathematical structure underlying the Hydrostatic Paradox.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Hydrostatic Paradox
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free