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Question #42277

Difficulty: 8/10 | 🟢 Free

The Question

A small uncharged satellite orbits a spherical planet of rest mass $M$ and radius $R$ in a circular orbit. The initial orbital period is $T_0$ . The planet is then given a total charge $Q$ distributed uniformly throughout its volume. According to mass-energy equivalence, the electrostatic self-energy of the planet contributes to its effective gravitational mass. If $c$ is the speed of light in vacuum, $\epsilon_0$ is the vacuum permittivity, and the added mass equivalent is much smaller than $M$ , which of the following expressions best represents the new orbital period $T$ of the satellite?

Options

A.
$T_0 \left( 1 - \frac{3Q^2}{40\pi\epsilon_0 R M c^2} \right)$
B.
$T_0 \left( 1 - \frac{Q^2}{16\pi\epsilon_0 R M c^2} \right)$
C.
$T_0 \left( 1 + \frac{3Q^2}{40\pi\epsilon_0 R M c^2} \right)$
D.
$T_0 \left( 1 - \frac{3Q^2}{20\pi\epsilon_0 R M c^2} \right)$

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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

Kepler’s Third Law (Time Period)|Self-Energy of Charged Sphere|Approximation (Binomial)

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of electrostatics, gravitation, and relativity.

The primary trap is failing to recognize that electrostatic self-energy contributes to the effective inertial and gravitational mass of a system via $E=mc^2$ .

Students often either ignore this mass increment, use the incorrect self-energy formula (e.g., for a spherical shell instead of a solid volume), or incorrectly apply the binomial expansion to Kepler's Third Law ( $T \propto M^{-1/2}$ ), leading to sign errors or missing factors of $1/2$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Kepler’s Third Law (Time Period)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free