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Question #42791

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ has a non-uniform volume charge density given by $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$ , where $r$ is the radial distance from the center of the sphere. A spherical cavity of radius $R/2$ , centered at a distance $R/2$ from the center of the original sphere, is completely hollowed out. A point test charge $q$ is placed exactly at the center of the original sphere. What is the magnitude of the net electric force experienced by the test charge?

Options

A.
$\frac{5 q \rho_0 R}{48 \epsilon_0}$
B.
$\frac{q \rho_0 R}{12 \epsilon_0}$
C.
$\frac{q \rho_0 R}{6 \epsilon_0}$
D.
$\frac{q \rho_0 R}{15 \epsilon_0}$

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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

Symmetry and Virtual Work|Gravitational Potential Reference|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the misapplication of Gauss's Law and the uniform cavity theorem ( $\vec{E} = \frac{\rho}{3\epsilon_0}\vec{a}$ ).

Students reflexively assume that the electric field inside any spherical cavity is uniform and depends only on the density at the cavity's center or the origin, leading to incorrect answers like $\frac{q \rho_0 R}{12 \epsilon_0}$ or $\frac{q \rho_0 R}{6 \epsilon_0}$ .

However, because the original charge distribution $\rho(r)$ is non-uniform, the removed charge lacks the necessary symmetry for the standard shell theorem to apply.

To solve this, one must use the principle of virtual work (or direct potential gradient integration), setting up the integral in spherical coordinates from the origin to the cavity's boundary $r = R\cos\theta$ .

Failing to account for the variable density within the integration limits exposes a fundamental misunderstanding of when symmetry-based shortcuts are valid.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Symmetry and Virtual Work
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free