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Question #42802

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly reflecting, flexible single-surface spherical membrane of radius $R$ and constant surface tension $S$ carries a uniformly distributed total charge $Q$ . The membrane encloses an isotropic radiation field. The entire system is in mechanical and thermal equilibrium in a vacuum. If the total internal energy of the system is defined as the sum of the radiation energy, the electrostatic self-energy of the charge distribution, and the surface energy of the membrane, what is the total internal energy of the system? (Here $\epsilon_0$ is the permittivity of free space)

Options

A.
$12\pi S R^2$
B.
$12\pi S R^2 + \frac{Q^2}{8\pi\epsilon_0 R}$
C.
$8\pi S R^2$
D.
$12\pi S R^2 + \frac{Q^2}{4\pi\epsilon_0 R}$

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

Self-Energy of Charged Sphere|Internal Energy Dependency|Radiation Pressure

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of thermodynamics, electrostatics, and fluid mechanics.

The primary trap lies in treating the electrostatic self-energy merely as a static scalar to be added at the end, neglecting its radial gradient which manifests as an outward mechanical pressure (Maxwell stress).

Students who fail to include the electrostatic pressure in the mechanical equilibrium equation will incorrectly evaluate the radiation energy, leading to an uncancelled charge-dependent term in the final total energy (Option B).

Additionally, confusing the direction of the electrostatic pressure leads to another predictable error (Option D).

The elegant cancellation of the charge term rewards a rigorous application of mechanical equilibrium and energy density principles.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Self-Energy of Charged Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free