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Question #42263

Difficulty: 8/10 | 🟢 Free

The Question

A uniform solid sphere of initial mass $M$ (in $\text{kg}$ ) and radius $R$ (in $\text{m}$ ) has a spherical cavity of radius $R/2$ created within it. The center of the cavity is located at a distance $R/2$ from the center of the original solid sphere. Let $V_1$ and $V_2$ (in $\text{J}/\text{kg}$ ) be the gravitational potentials at the center of the original sphere and the center of the cavity, respectively. If $G$ is the universal gravitational constant, determine the potential difference $V_2 - V_1$ .

Options

A.
$\frac{GM}{4R}$
B.
0
C.
$\frac{3GM}{8R}$
D.
$\frac{GM}{2R}$

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

Gravitational Potential Reference|Potential of Concentric Shells|Cavity Field Shielding

Why This Question Is Tricky

Common Mistakes

This question tests the 'Equipotential Cavity' trap.

Students often conflate the gravitational field inside a cavity with the electrostatic field inside a conducting shell.

In electrostatics, a cavity inside a conductor is field-free ( $E=0$ ), making it an equipotential volume.

However, in gravitation, a non-concentric cavity inside a uniform mass distribution has a uniform, non-zero gravitational field ( $\vec{g} = -\frac{4}{3}\pi G \rho \vec{a}$ ).

Because the field is non-zero, the potential must vary linearly along the field direction, shifting relative to the reference at infinity.

Students who assume $V_1 = V_2$ fall for this false equivalence.

Other traps include sign errors during superposition or incorrectly treating the removed mass as a point mass for internal points.

Quick Facts About This Question

Question Information

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