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title: "A solid conducting sphere of radius \\(R\\) carries a net positive charge \\(Q\\) and is in electrostatic equilibrium in vacuum. Four concentric, thin spherical shells, each of identical infinitesimal thickness \\(dr\\), are centered on the sphere at radial distances \\(r_1 = 0.25R\\), \\(r_2 = 0.50R\\), \\(r_3 = 2.0R\\), and \\(r_4 = 4.0R\\). Let \\(U_1\\), \\(U_2\\), \\(U_3\\), and \\(U_4\\) represent the electrostatic energy stored within the volume of Shell 1, Shell 2, Shell 3, and Shell 4, respectively. Which of the following correctly ranks the electrostatic energies stored within the shells?"
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url: "https://nerd-notes.com/ubq/124882/"
date_modified: "2026-09-28T14:11:48+00:00"
---

# A solid conducting sphere of radius \(R\) carries a net positive charge \(Q\) and is in electrostatic equilibrium in vacuum. Four concentric, thin spherical shells, each of identical infinitesimal thickness \(dr\), are centered on the sphere at radial distances \(r_1 = 0.25R\), \(r_2 = 0.50R\), \(r_3 = 2.0R\), and \(r_4 = 4.0R\). Let \(U_1\), \(U_2\), \(U_3\), and \(U_4\) represent the electrostatic energy stored within the volume of Shell 1, Shell 2, Shell 3, and Shell 4, respectively. Which of the following correctly ranks the electrostatic energies stored within the shells?

A solid conducting sphere of radius \(R\) carries a net positive charge \(Q\) and is in electrostatic equilibrium in vacuum. Four concentric, thin spherical shells, each of identical infinitesimal thickness \(dr\), are centered on the sphere at radial distances \(r_1 = 0.25R\), \(r_2 = 0.50R\), \(r_3 = 2.0R\), and \(r_4 = 4.0R\). Let \(U_1\), \(U_2\), \(U_3\), and \(U_4\) represent the electrostatic energy stored within the volume of Shell 1, Shell 2, Shell 3, and Shell 4, respectively. Which of the following correctly ranks the electrostatic energies stored within the shells?

![A cross-sectional view in grayscale showing a solid circle centered at a central point. The solid circle has light gray shading and a radius labeled \(R\) via a radial line from the center to its outer boundary. Inside the shaded circle, two concentric circular dashed rings are centered at the origin, one at radius \(0.25R\) with a callout line pointing to label 1, and the other at radius \(0.50R\) with a callout line pointing to label 2. Outside the shaded circle, two larger concentric circular dashed rings are centered at the origin, one at radius \(2.0R\) with a callout line pointing to label 3, and one at radius \(4.0R\) with a callout line pointing to label 4. Each dashed ring represents a shell of uniform radial thickness \(dr\). A label \(+Q\) is placed adjacent to the outer boundary of the shaded circle. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604708-TYEsX7.jpg)

- **A.** \(U_3 > U_4 > U_1 = U_2\)
- **B.** \(U_1 > U_2 > U_3 > U_4\)
- **C.** \(U_3 = U_4 > U_1 = U_2\)
- **D.** \(U_3 > U_4 > U_2 > U_1\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124882/*