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Question #42701

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ has a spherically symmetric volume charge density given by $\rho(r) = kr$ , where $k$ is a positive constant and $r$ is the radial distance from the center of the sphere. Assuming the electrostatic potential at infinity is zero, what is the electrostatic potential $V(r)$ at a point inside the sphere ( $r < R$ )?

Options

A.
$\frac{k}{12\epsilon_0}(4R^3 - r^3)$
B.
$\frac{k}{12\epsilon_0}(R^3 - r^3)$
C.
$\frac{k R^4}{4\epsilon_0 r}$
D.
$\frac{k}{3\epsilon_0}(R^3 - r^3)$

Check Your Answer

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 of Concentric Shells|Shell Theorem (Internal vs External)|Electric Field Symmetry

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Shell Theorem and the superposition principle for non-uniform continuous charge distributions.

A common trap is for students to calculate the electric field $E(r)$ inside the sphere and integrate it from $r$ to $R$ , mistakenly assuming $V(R) = 0$ instead of calculating the absolute potential by integrating from infinity, which leads to the incorrect option B.

Another lethal error is failing to distinguish between the potential contributions of shells internal to the point (which behave like point charges, $V \propto 1/r$ ) and shells external to the point (which contribute a constant potential everywhere inside them, $V \propto 1/r'$ ).

Students who treat the entire sphere as a point charge at the center will incorrectly choose option C, ignoring the volumetric calculus required for internal points.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Potential of Concentric Shells
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free