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Question #42486

Difficulty: 8/10 | 🟢 Free

The Question

An isolated solid conducting sphere of radius $R$ carries a total charge $Q$ . The electrostatic potential $V(\vec{r})$ is defined such that it vanishes at the surface of the sphere ( $V=0$ at $r=R$ ). Let $U_E$ be the total electrostatic energy stored in the electric field over all space, and $W_\rho = \frac{1}{2} \int \rho(\vec{r}) V(\vec{r}) d^3r$ be the volume integral over all space, where $\rho(\vec{r})$ is the charge density distribution. Which of the following gives the correct values for $U_E$ and $W_\rho$ ?

Options

A.
$U_E = \frac{Q^2}{8\pi\epsilon_0 R}, \quad W_\rho = 0$
B.
$U_E = \frac{Q^2}{8\pi\epsilon_0 R}, \quad W_\rho = \frac{Q^2}{8\pi\epsilon_0 R}$
C.
$U_E = 0, \quad W_\rho = 0$
D.
$U_E = \frac{Q^2}{4\pi\epsilon_0 R}, \quad W_\rho = -\frac{Q^2}{8\pi\epsilon_0 R}$

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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 Energy Reference|Self-Energy of Charged Sphere|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental distinction between the physical energy stored in an electric field and the mathematical formula $W_\rho = \frac{1}{2}\int \rho V d\tau$ .

Students often blindly apply $U = \frac{1}{2}QV$ (which derives from $W_\rho$ ) to find the self-energy of a sphere.

By shifting the gauge such that $V=0$ at the surface, the formula yields zero, creating a lethal trap.

The true electrostatic energy is the volume integral of the energy density ( $\frac{1}{2}\epsilon_0 E^2$ ), which depends only on the electric field and remains invariant under gauge transformations.

The identity $U_E = W_\rho$ requires a boundary term at infinity to vanish, which fails when $V(\infty) \neq 0$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Potential Energy Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free