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title: "An uncharged, thin spherical shell of radius \\(R\\) is assembled by an external agent that brings infinitesimal charge elements from infinity and deposits them uniformly onto the shell until a final net charge \\(Q\\) is reached. The work required to assemble the shell equals the total electrostatic potential energy \\(U\\) stored in the resulting electric field. This stored energy can be determined by integrating the electric field energy density, \\(u_E = \\dfrac{1}{2}\\varepsilon_0 E^2\\), over all space, where \\(E\\) is the electric field magnitude at a distance \\(r\\) from the center of the shell. Which of the following integral expressions correctly represents \\(U\\)?"
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url: "https://nerd-notes.com/ubq/124715/"
date_modified: "2026-09-28T14:09:16+00:00"
---

# An uncharged, thin spherical shell of radius \(R\) is assembled by an external agent that brings infinitesimal charge elements from infinity and deposits them uniformly onto the shell until a final net charge \(Q\) is reached. The work required to assemble the shell equals the total electrostatic potential energy \(U\) stored in the resulting electric field. This stored energy can be determined by integrating the electric field energy density, \(u_E = \dfrac{1}{2}\varepsilon_0 E^2\), over all space, where \(E\) is the electric field magnitude at a distance \(r\) from the center of the shell. Which of the following integral expressions correctly represents \(U\)?

An uncharged, thin spherical shell of radius \(R\) is assembled by an external agent that brings infinitesimal charge elements from infinity and deposits them uniformly onto the shell until a final net charge \(Q\) is reached. The work required to assemble the shell equals the total electrostatic potential energy \(U\) stored in the resulting electric field. This stored energy can be determined by integrating the electric field energy density, \(u_E = \dfrac{1}{2}\varepsilon_0 E^2\), over all space, where \(E\) is the electric field magnitude at a distance \(r\) from the center of the shell. Which of the following integral expressions correctly represents \(U\)?

![A cross-sectional diagram showing a thin spherical shell centered at the origin. A circle of radius R is drawn with a solid line, labeled +Q on its upper boundary. A straight arrow extends from the center of the circle to the shell boundary at an angle of 45 degrees above the horizontal, labeled with the variable R. A horizontal dashed ray extends from the center of the shell to the right, labeled r at its arrow tip. At a radial distance greater than R, a thin concentric circular band of radius r and thickness dr is indicated with dashed boundaries. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604555-1HfliK.jpg)

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

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124715/*