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Question #42943

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ is centered at the origin. The sphere has a spherically symmetric, non-uniform volume charge density given by $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$ for $r \le R$ , and $\rho(r) = 0$ for $r > R$ , where $\rho_0$ is a positive constant and $r$ is the radial distance from the center. What is the absolute electric potential $V$ at a distance $r = \frac{R}{2}$ from the center of the sphere? Assume the electric potential at infinity is zero, and $\epsilon_0$ is the permittivity of free space.

Options

A.
$\frac{13 \rho_0 R^2}{96 \epsilon_0}$
B.
$\frac{5 \rho_0 R^2}{96 \epsilon_0}$
C.
$\frac{11 \rho_0 R^2}{96 \epsilon_0}$
D.
$\frac{\rho_0 R^2}{6 \epsilon_0}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Potential Gradient Direction|Charged Conducting Sphere|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the potential gradient $E = -dV/dr$ in a piecewise continuous field generated by a non-uniform charge distribution.

A common trap is to calculate the potential difference $\Delta V$ from the surface to the interior point and mistakenly present it as the absolute potential (Option B), completely ignoring the additive constant $V(R)$ required by the boundary condition $V(\infty) = 0$ .

Furthermore, students often incorrectly apply the shell theorem by assuming all enclosed charge acts as a point charge at the center for potential calculations (Option D), or they substitute the total charge into the formula for a uniform sphere (Option C).

By forcing piecewise integration of Gauss's Law derived fields, the problem ensures rigorous application of fundamental electrostatic principles.

Quick Facts About This Question

Question Information

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