---
title: "A thin, nonconducting spherical shell of radius \\(R\\) carries a total positive charge \\(Q\\) distributed uniformly over its surface. The shell is allowed to expand radially outward under its own electrostatic repulsion until its radius becomes \\(2R\\), while its total charge remains constant. Which of the following correctly pairs the work done by the electrostatic forces during the expansion, \\(W_{\\text{field}}\\), with the final electrostatic potential energy of the system, \\(U_f\\)?"
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url: "https://nerd-notes.com/ubq/118115/"
date_modified: "2026-08-04T08:05:13+00:00"
---

# A thin, nonconducting spherical shell of radius \(R\) carries a total positive charge \(Q\) distributed uniformly over its surface. The shell is allowed to expand radially outward under its own electrostatic repulsion until its radius becomes \(2R\), while its total charge remains constant. Which of the following correctly pairs the work done by the electrostatic forces during the expansion, \(W_{\text{field}}\), with the final electrostatic potential energy of the system, \(U_f\)?

A thin, nonconducting spherical shell of radius \(R\) carries a total positive charge \(Q\) distributed uniformly over its surface. The shell is allowed to expand radially outward under its own electrostatic repulsion until its radius becomes \(2R\), while its total charge remains constant. Which of the following correctly pairs the work done by the electrostatic forces during the expansion, \(W_{\text{field}}\), with the final electrostatic potential energy of the system, \(U_f\)?

![A two-panel schematic showing a spherical shell before and after expansion, labeled 'Initial State' on the left and 'Final State' on the right. In the left panel, a dashed circle represents a spherical shell of radius R centered at the origin, with small positive plus signs distributed along its perimeter. A solid black vector arrow extends from the center to the perimeter, labeled R. In the right panel, a larger concentric dashed circle represents the expanded shell of radius 2R, also with small positive plus signs distributed along its perimeter. A solid black vector arrow extends from the center to the outer perimeter, labeled 2R. A horizontal grey dashed arrow points from the left sphere toward the right sphere to indicate expansion. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830712-FM5bxB.jpg)

- **A.** \(W_{\text{field}} = -\dfrac{Q^2}{16\pi\varepsilon_0 R}\), \(U_f = 2 U_i\)
- **B.** \(W_{\text{field}} = +\dfrac{Q^2}{16\pi\varepsilon_0 R}\), \(U_f = \dfrac{1}{2} U_i\)
- **C.** \(W_{\text{field}} = +\dfrac{Q^2}{8\pi\varepsilon_0 R}\), \(U_f = \dfrac{1}{2} U_i\)
- **D.** \(W_{\text{field}} = -\dfrac{Q^2}{8\pi\varepsilon_0 R}\), \(U_f = 2 U_i\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118115/*