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Question #42801

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ (in meters) contains a total positive charge $Q$ (in Coulombs). The volume charge density $\rho(r)$ within the sphere is not uniform, but varies directly with the radial distance $r$ from the center, such that $\rho(r) = c r$ for $r \le R$ , where $c$ is a positive constant. For $r > R$ , the charge density is zero. What is the total electrostatic self-energy of this spherical charge distribution? (Assume the permittivity of free space is $\epsilon_0$ )

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|Field due to Infinite Sheet|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to calculate the electrostatic self-energy of a non-uniform charge distribution from first principles.

The primary trap is the reliance on the memorized formula for a uniform solid sphere ( $U = \frac{3Q^2}{20\pi\epsilon_0 R}$ ).

Students must recognize that a spatially varying charge density $\rho(r)$ alters the internal electric field profile, necessitating a fresh derivation using Gauss's Law.

Furthermore, a common mistake is to only integrate the energy density within the sphere ( $U_{in}$ ) or outside the sphere ( $U_{out}$ ), forgetting that the total self-energy is the volume integral of $\frac{1}{2}\epsilon_0 E^2$ over all space from $r=0$ to $r=\infty$ .

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