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Question #43522

Difficulty: 8/10 | 🟢 Free

The Question

A solid spherical planet of mass $M$ and radius $R$ initially has a uniform mass density. It carries a total charge $Q$ uniformly distributed over its surface. A concentric thin spherical shell of mass $m$ and radius $2R$ carries a total charge $-Q$ uniformly distributed over its surface. The shell slowly collapses under the combined action of gravity and electrostatic forces until it rests on the surface of the planet. During this process, the planet undergoes a structural phase transition, changing its mass density profile from uniform to $\rho(r) = \rho_0 (1 - r/R)$ while maintaining its total mass $M$ , radius $R$ , and surface charge $Q$ . What is the total work done by the conservative forces (gravitational and electrostatic) of the system during this entire process?

Options

A.
$\frac{GM^2}{7R} + \frac{Gm^2}{4R} + \frac{GMm}{2R} + \frac{Q^2}{16\pi\epsilon_0 R}$
B.
$\frac{GM^2}{7R} + \frac{Gm^2}{4R} + \frac{Q^2}{16\pi\epsilon_0 R}$
C.
$\frac{26GM^2}{35R} + \frac{Gm^2}{2R} + \frac{GMm}{R} + \frac{Q^2}{8\pi\epsilon_0 R}$
D.
$\frac{5GM^2}{21R} + \frac{Gm^2}{4R} + \frac{GMm}{2R} + \frac{Q^2}{32\pi\epsilon_0 R}$

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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

Gravitational Self-Energy|Electric Field Symmetry|Integration

Why This Question Is Tricky

Common Mistakes

This problem tests the rigorous application of field energy and superposition in hybrid systems.

A common trap is assuming the total energy of a system is merely the sum of the self-energies of its components, neglecting the cross-interaction energy ( $U_{int}$ ) between the shell and the planet.

Furthermore, students often misapply the work-energy theorem by failing to account for the internal structural energy change of the planet during its phase transition.

By requiring the integration of a non-uniform mass distribution alongside a capacitor-like electrostatic setup, the question forces candidates to abandon simple formula-plugging and construct the total energy functional from first principles, distinguishing between average macroscopic changes and instantaneous calculus-based field energies.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Gravitational Self-Energy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free