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Question #42107

Difficulty: 8/10 | 🟢 Free

The Question

A grounded spherical conducting liquid drop of radius $R$ and surface tension $T$ is situated at the center of a rigid, isolated spherical conducting shell of radius $kR$ ( $k > 1$ ). The outer shell carries a total charge $Q$ . The space between them is vacuum. Assuming the effect of the grounding wire is negligible, what is the excess pressure $\Delta P$ inside the liquid drop?

Options

A.
$\frac{2T}{R} - \frac{Q^2}{32\pi^2 \epsilon_0 k^2 R^4}$
B.
$\frac{2T}{R} - \frac{Q^2 (k-1)}{32\pi^2 \epsilon_0 k^2 R^4}$
C.
$\frac{2T}{R} + \frac{Q^2}{32\pi^2 \epsilon_0 k^2 R^4}$
D.
$\frac{2T}{R} - \frac{Q^2}{32\pi^2 \epsilon_0 R^4}$

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

Molecular Forces of Cohesion|Energy Density Locality|Self-Energy of Charged Sphere|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of electrostatics and continuum mechanics, specifically focusing on the locality of energy density and the correct application of boundary conditions.

A lethal trap awaits students who use the global energy method: they might calculate the total electrostatic energy $U_E = \frac{Q^2(k-1)}{8\pi\epsilon_0 k^2 R}$ and incorrectly differentiate it with respect to $R$ while treating $k$ as a constant ratio.

This yields the erroneous term $\frac{Q^2(k-1)}{32\pi^2 \epsilon_0 k^2 R^4}$ (Option B).

Students must recognize that the outer shell is rigid, so the derivative $\frac{\partial U_E}{\partial R}$ must be taken with the shell radius $R_{shell}$ held constant.

Using the local energy density $P_E = \frac{1}{2}\epsilon_0 E^2$ elegantly bypasses this calculus trap.

Another conceptual trap is adding the electrostatic pressure instead of subtracting it, failing to realize that self-repulsion acts outward against the cohesive surface tension.

Quick Facts About This Question

Question Information

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