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Question #43570

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ has a spherically symmetric volume charge density $\rho(r) = \rho_0 \frac{r}{R}$ , where $r$ is the radial distance from its center and $\rho_0$ is a positive constant. The solid sphere is surrounded by a concentric, grounded conducting spherical shell of inner radius $2R$ and outer radius $3R$ . A point charge $q$ is placed at a distance $4R$ from the center of the system. What is the net electrostatic potential at the center of the solid sphere? (Assume the potential at infinity is zero, and $\epsilon_0$ is the permittivity of free space).

Options

A.
$\frac{5\rho_0 R^2}{24\epsilon_0}$
B.
$\frac{5\rho_0 R^2}{24\epsilon_0} + \frac{q}{16\pi\epsilon_0 R}$
C.
$\frac{\rho_0 R^2}{3\epsilon_0}$
D.
$\frac{5\rho_0 R^2}{24\epsilon_0} - \frac{q}{48\pi\epsilon_0 R}$

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

Superposition Principle|Potential of Concentric Shells|Superposition of Potentials|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Superposition Principle and the electrostatic shielding properties of conductors.

A common trap is assuming that a grounded conductor has zero charge on its outer surface, leading students to incorrectly add the potential of the external point charge $q$ to the center.

Another trap is attempting to calculate the non-uniform outer charge distribution directly, which is mathematically intractable here.

The correct logical framework requires recognizing that the potential inside the cavity due to external charges (outer surface charge and point charge) is constant and equal to the conductor's potential (zero).

Thus, the external charge $q$ is perfectly shielded and does not affect the potential at the center, leaving only the contributions from the solid sphere and the uniformly charged inner surface of the shell.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Superposition Principle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free