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Question #42226

Difficulty: 8/10 | 🟢 Free

The Question

A very large open tank is filled with an incompressible liquid of uniform mass density $\rho$ and a uniform volume charge density $\rho_e$ . An infinite, thin, non-conducting horizontal sheet carrying a uniform surface charge density $+\sigma$ is held fixed at a depth $h$ below the free surface of the liquid. The atmospheric pressure is $P_0$ and the permittivity of the liquid is $\epsilon$ . Assuming the electric field produced by the liquid's own volume charge is negligible, what is the absolute pressure $P$ at a depth $d$ ( $d > h$ ) below the free surface?

Options

A.
$P_0 + \rho g d + \frac{\rho_e \sigma}{2\epsilon}(d - 2h)$
B.
$P_0 + \rho g d + \frac{\rho_e \sigma}{2\epsilon}d$
C.
$P_0 + \rho g d + \frac{\rho_e \sigma}{2\epsilon}(d - h)$
D.
$P_0 + \rho g d$

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

Atmospheric Pressure Reference|Field due to Infinite Sheet|Vector Summation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate electrostatic body forces into the hydrostatic equilibrium equation.

The primary trap lies in the discontinuous nature of the electric field produced by an infinite charged sheet.

Students often incorrectly assume a uniform electric field throughout the entire fluid column (leading to option B) or ignore the field's effect above the sheet (leading to option C).

By requiring the piecewise integration of the modified pressure gradient $\nabla P = \rho \vec{g} + \rho_e \vec{E}$ , the problem evaluates the transition from simple scalar hydrostatics to vector-field-based mechanical equilibrium, highlighting that the electric field breaks the vertical symmetry of Pascal's Law and effectively modifies the weight of the fluid column differently above and below the sheet.

Quick Facts About This Question

Question Information

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