---
title: "A solid conducting sphere of radius \\(R\\) is located in vacuum and carries a net positive charge \\(+Q\\) in electrostatic equilibrium. The electric energy density stored in an electric field of magnitude \\(E\\) is given by \\(u = \\dfrac{1}{2}\\varepsilon_0 E^2\\). Which of the following integral expressions correctly represents the total electrostatic energy \\(U\\) stored in the electric field produced by the sphere?"
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url: "https://nerd-notes.com/ubq/124863/"
date_modified: "2026-09-28T14:11:43+00:00"
---

# A solid conducting sphere of radius \(R\) is located in vacuum and carries a net positive charge \(+Q\) in electrostatic equilibrium. The electric energy density stored in an electric field of magnitude \(E\) is given by \(u = \dfrac{1}{2}\varepsilon_0 E^2\). Which of the following integral expressions correctly represents the total electrostatic energy \(U\) stored in the electric field produced by the sphere?

A solid conducting sphere of radius \(R\) is located in vacuum and carries a net positive charge \(+Q\) in electrostatic equilibrium. The electric energy density stored in an electric field of magnitude \(E\) is given by \(u = \dfrac{1}{2}\varepsilon_0 E^2\). Which of the following integral expressions correctly represents the total electrostatic energy \(U\) stored in the electric field produced by the sphere?

![A circular outline representing a solid spherical conductor of radius \(R\) centered at the origin, filled with light gray shading. From the center to the upper-right edge of the circle at a 45-degree angle, a solid black line segment ends with an arrowhead touching the circle's perimeter, labeled \(R\) near its midpoint. A text label \(+Q\) is located inside the circle near the bottom. Exactly eight radial arrows representing the electric field originate from the outer perimeter of the circle and point symmetrically outward into the surrounding white space at 0, 45, 90, 135, 180, 225, 270, and 315 degrees, each extending a distance equal to the sphere's radius. One radial arrow pointing upward at 90 degrees is labeled \(\vec{E}\) near its tip. The interior of the circle contains no field lines or arrows. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604702-ViwYJF.jpg)

- **A.** \(\dfrac{Q^2}{8\pi\varepsilon_0} \int_0^R \dfrac{1}{r^2} \, dr\)
- **B.** \(\dfrac{Q^2}{4\pi\varepsilon_0} \int_R^\infty \dfrac{1}{r^2} \, dr\)
- **C.** \(\dfrac{Q^2}{8\pi\varepsilon_0} \int_R^\infty \dfrac{1}{r^2} \, dr\)
- **D.** \(\dfrac{Q^2}{8\pi\varepsilon_0} \int_0^\infty \dfrac{1}{r^2} \, dr\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124863/*