Before You Begin

Question #42804

Difficulty: 8/10 | 🟢 Free

The Question

A non-conducting solid sphere of radius $R$ is assembled by bringing infinitesimal charges from infinity. The final charge distribution within the sphere is spherically symmetric but non-uniform, with a volume charge density given by $\rho(r) = c r$ , where $c$ is a positive constant and $r$ is the radial distance from the center. If the total charge of the assembled sphere is $Q$ , what is the total work done by external forces in assembling this charge configuration? 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.
$\frac{Q^2}{4\pi\epsilon_0 R}$

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

Work against Conservative Force|Self-Energy of Charged Sphere|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.

A common trap is to blindly apply the standard formula for a uniform solid sphere ( $U = 3Q^2/20\pi\epsilon_0 R$ ) or a spherical shell ( $U = Q^2/8\pi\epsilon_0 R$ ).

Students often fail to recognize that the potential at the surface of the growing sphere, $V(r)$ , depends on the instantaneous enclosed charge $q(r)$ , which scales as $r^4$ for a linear charge density, rather than $r^3$ as in the uniform case.

Correctly applying the conservative work theorem requires integrating $dW = V(r)dq$ with the specific non-uniform $\rho(r)$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Work against Conservative Force
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free