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Question #42706

Difficulty: 8/10 | 🟢 Free

The Question

A uniform solid sphere of radius $R$ is assembled from a dust of infinitesimal particles. The sum of the rest masses of these particles when infinitely separated is $M_0$ . The sphere also carries a total charge $Q$ uniformly distributed throughout its volume. According to the mass-energy equivalence principle ( $E=mc^2$ ), the total energy of the assembled stationary system determines its effective inertial mass $M_{eff}$ . Assuming the charge and mass distributions remain perfectly uniform, which of the following expressions correctly represents the effective mass $M_{eff}$ of the sphere? (Here $G$ is the universal gravitational constant, $\epsilon_0$ is the permittivity of free space, and $c$ is the speed of light in vacuum).

Options

A.
$M_0 + \frac{3}{5Rc^2} \left( \frac{Q^2}{4\pi\epsilon_0} - GM_0^2 \right)$
B.
$M_0 + \frac{3}{5Rc^2} \left( \frac{Q^2}{4\pi\epsilon_0} + GM_0^2 \right)$
C.
$M_0 + \frac{1}{2Rc^2} \left( \frac{Q^2}{4\pi\epsilon_0} - GM_0^2 \right)$
D.
$M_0 - \frac{3}{5Rc^2} \left( \frac{Q^2}{4\pi\epsilon_0} - GM_0^2 \right)$

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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|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of mass-energy equivalence applied to classical self-energies.

The primary trap is the sign convention and physical nature of the self-energies: gravitational self-energy is a negative binding energy (which reduces the effective mass), while electrostatic self-energy for a single sign of charge is positive (which increases the effective mass).

Students frequently misapply the superposition principle by either adding both terms as positive contributions (Option B), confusing the geometric prefactors for solid spheres ( $3/5$ ) with those of spherical shells ( $1/2$ ) (Option C), or incorrectly assigning the overall sign of the energy correction (Option D).

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