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title: "A solid insulating sphere of radius \\(R\\) has a non-uniform volume charge density \\(\\rho(r) = \\rho_0 \\left(\\dfrac{r}{R}\\right)\\) for \\(r \\le R\\). The sphere is concentric with a thin conducting spherical shell of radius \\(3R\\) that carries a net surface charge \\(Q_{\\text{shell}}\\). Across the surface of the outer shell at \\(r = 3R\\), the step change in the magnitude of the electric field, \\(\\Delta E = |E((3R)^+) – E((3R)^-)|\\), is equal to the magnitude of the electric field at the surface of the inner solid sphere, \\(E(R)\\). In terms of \\(\\rho_0\\) and \\(R\\), what is the magnitude of the net charge \\(|Q_{\\text{shell}}|\\) on the outer spherical shell?"
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url: "https://nerd-notes.com/ubq/118018/"
date_modified: "2026-08-04T08:02:58+00:00"
---

# A solid insulating sphere of radius \(R\) has a non-uniform volume charge density \(\rho(r) = \rho_0 \left(\dfrac{r}{R}\right)\) for \(r \le R\). The sphere is concentric with a thin conducting spherical shell of radius \(3R\) that carries a net surface charge \(Q_{\text{shell}}\). Across the surface of the outer shell at \(r = 3R\), the step change in the magnitude of the electric field, \(\Delta E = |E((3R)^+) – E((3R)^-)|\), is equal to the magnitude of the electric field at the surface of the inner solid sphere, \(E(R)\). In terms of \(\rho_0\) and \(R\), what is the magnitude of the net charge \(|Q_{\text{shell}}|\) on the outer spherical shell?

A solid insulating sphere of radius \(R\) has a non-uniform volume charge density \(\rho(r) = \rho_0 \left(\dfrac{r}{R}\right)\) for \(r \le R\). The sphere is concentric with a thin conducting spherical shell of radius \(3R\) that carries a net surface charge \(Q_{\text{shell}}\). Across the surface of the outer shell at \(r = 3R\), the step change in the magnitude of the electric field, \(\Delta E = |E((3R)^+) - E((3R)^-)|\), is equal to the magnitude of the electric field at the surface of the inner solid sphere, \(E(R)\). In terms of \(\rho_0\) and \(R\), what is the magnitude of the net charge \(|Q_{\text{shell}}|\) on the outer spherical shell?

![Cross-sectional view of a central solid sphere of radius R surrounded coaxially by a larger thin circular shell of radius 3R. The inner solid sphere is shaded with a light gray gradient that darkens toward its outer edge at radius R, with a radial dashed line labeled R extending from the center origin to its outer boundary. A larger concentric outer circle represents a thin spherical shell, with a radial dashed line extending from the center origin to the outer circle labeled 3R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830578-jHnmj5.jpg)

- **A.** \(\dfrac{\pi}{3} \rho_0 R^3\)
- **B.** \(\pi \rho_0 R^3\)
- **C.** \(3\pi \rho_0 R^3\)
- **D.** \(9\pi \rho_0 R^3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118018/*