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Question #42705

Difficulty: 8/10 | 🟢 Free

The Question

A total charge $Q$ is distributed within a non-conducting solid sphere of radius $R$ situated in a vacuum. The volume charge density $\rho(r)$ is not uniform but varies with the radial distance $r$ from the center as $\rho(r) = kr$ , where $k$ is a positive constant. What is the total electrostatic self-energy of this charge configuration? (Here, $\epsilon_0$ is the permittivity of free space).

Options

A.
$\frac{Q^2}{7\pi \epsilon_0 R}$
B.
$\frac{3Q^2}{20\pi \epsilon_0 R}$
C.
$\frac{Q^2}{8\pi \epsilon_0 R}$
D.
$\frac{Q^2}{56\pi \epsilon_0 R}$

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

Self-Energy of Charged Sphere|Electric Charges and Fields|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental derivation of electrostatic self-energy for non-uniform charge distributions.

Students often fall into the trap of memorizing standard formulas, such as $U = 3Q^2 / 20\pi\epsilon_0 R$ for a uniformly charged solid sphere or $U = Q^2 / 8\pi\epsilon_0 R$ for a spherical shell, and incorrectly applying them here.

By defining $\rho(r) = kr$ , the problem forces students to abandon rote formulas and return to first principles: applying Gauss's Law to find the piecewise electric field and integrating the energy density $u = \frac{1}{2}\epsilon_0 E^2$ over all space.

A secondary trap is forgetting to include the internal energy contribution $U_{in}$ and only calculating the external field energy, which leads to the shell formula.

Quick Facts About This Question

Question Information

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