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Question #42800

Difficulty: 8/10 | 🟢 Free

The Question

A spherical body of radius $R$ has a total mass $M$ uniformly distributed throughout its volume and a total charge $Q$ uniformly distributed over its surface. The energy density of the electric field just outside the surface of the sphere is equal to the magnitude of the energy density of the gravitational field just outside the surface. What is the total work $W$ required by external agents to assemble this configuration from infinitesimally dispersed matter and charge at rest at infinity? (Here, $G$ is the universal gravitational constant and $\epsilon_0$ is the vacuum permittivity.)

Options

A.
$-\frac{GM^2}{10R}$
B.
$\frac{11GM^2}{10R}$
C.
$-\frac{GM^2}{2R}$
D.
$0$

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

Self-Energy of Charged Sphere|Gravitational Potential Energy (System)|Coulombic Central Force

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize electrostatic and gravitational self-energies while navigating critical sign conventions and distribution dependencies.

The primary trap lies in confusing the self-energy of a surface charge distribution ( $U_E = Q^2/8\pi\epsilon_0 R$ ) with that of a volume charge distribution ( $3Q^2/20\pi\epsilon_0 R$ ), which leads to the incorrect answer of zero.

Additionally, it probes the fundamental difference in the nature of the fields: electrostatic assembly requires positive work against repulsion, whereas gravitational assembly releases energy (negative work).

Students who incorrectly assign a positive sign to gravitational self-energy will erroneously add the magnitudes, leading to $11GM^2/10R$ .

The problem mandates a rigorous application of field energy densities ( $u_E \propto E^2$ and $u_G \propto -g^2$ ) to establish the coupling between mass and charge.

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