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Question #42318

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The Question

A solid non-conducting sphere of radius $R$ has a spherically symmetric volume charge density given by $\rho(r) = \rho_0 \frac{r}{R}$ for $r \le R$ , where $r$ is the radial distance from the center and $\rho_0$ is a positive constant. The electrostatic potential at the surface of the sphere is explicitly defined to be zero ( $V(R) = 0$ ). Which of the following correctly gives the electrostatic potential at the center of the sphere ( $V_c$ ) and at a radial distance $r = 2R$ ( $V_{2R}$ )? Let $\epsilon_0$ be the permittivity of free space.

Options

A.
$V_c = \frac{\rho_0 R^2}{12 \epsilon_0}, \quad V_{2R} = -\frac{\rho_0 R^2}{8 \epsilon_0}$
B.
$V_c = \frac{\rho_0 R^2}{3 \epsilon_0}, \quad V_{2R} = \frac{\rho_0 R^2}{8 \epsilon_0}$
C.
$V_c = \frac{\rho_0 R^2}{4 \epsilon_0}, \quad V_{2R} = -\frac{\rho_0 R^2}{4 \epsilon_0}$
D.
$V_c = \frac{\rho_0 R^2}{12 \epsilon_0}, \quad V_{2R} = \frac{\rho_0 R^2}{8 \epsilon_0}$

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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

Potential Energy Reference|Reference Potential Convention|Charged Conducting Sphere

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental definition of electrostatic potential as the line integral of the electric field, specifically targeting the trap of the reference potential convention.

Students often blindly memorize formulas like $V = kQ/r$ for regions outside a sphere, which implicitly assumes $V(\infty) = 0$ .

By explicitly setting the boundary condition $V(R) = 0$ , the problem forces students to abandon memorized formulas and integrate $\vec{E} \cdot d\vec{r}$ from the new reference point.

If a student incorrectly assumes $V(\infty) = 0$ , they will calculate $V_{2R} = \frac{\rho_0 R^2}{8 \epsilon_0}$ and $V_c = \frac{\rho_0 R^2}{3 \epsilon_0}$ , leading directly to the primary distractor.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Potential Energy Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free