---
title: "A solid insulating sphere of radius \\(R\\) carries a net charge \\(Q\\) distributed uniformly throughout its volume. The electric field energy density stored in space is defined by \\(u = \\dfrac{1}{2}\\varepsilon_0 E^2\\). Integrating this energy density over all space, which of the following expressions represents the total electrostatic potential energy \\(U\\) stored in the electric field of the sphere?"
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url: "https://nerd-notes.com/ubq/118121/"
date_modified: "2026-08-04T08:05:13+00:00"
---

# A solid insulating sphere of radius \(R\) carries a net charge \(Q\) distributed uniformly throughout its volume. The electric field energy density stored in space is defined by \(u = \dfrac{1}{2}\varepsilon_0 E^2\). Integrating this energy density over all space, which of the following expressions represents the total electrostatic potential energy \(U\) stored in the electric field of the sphere?

A solid insulating sphere of radius \(R\) carries a net charge \(Q\) distributed uniformly throughout its volume. The electric field energy density stored in space is defined by \(u = \dfrac{1}{2}\varepsilon_0 E^2\). Integrating this energy density over all space, which of the following expressions represents the total electrostatic potential energy \(U\) stored in the electric field of the sphere?

![A solid circle of radius R centered at the origin, filled with light grey shading and small plus sign symbols distributed evenly inside to indicate a uniform volume charge Q. A solid radial arrow from the origin to the sphere's surface is labeled R. Outside the sphere, thin dashed concentric circles centered at the origin depict integration regions for radius r less than R and radius r greater than R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830713-Q6pMsQ.jpg)

- **A.** \(\dfrac{3 Q^2}{20 \pi \varepsilon_0 R}\)
- **B.** \(\dfrac{Q^2}{8 \pi \varepsilon_0 R}\)
- **C.** \(\dfrac{3 Q^2}{10 \pi \varepsilon_0 R}\)
- **D.** \(\dfrac{Q^2}{40 \pi \varepsilon_0 R}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118121/*