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title: "A long, solid insulating cylinder of radius \\(R\\) has a non-uniform volume charge density given by \\(\\rho(r) = \\rho_0 \\left(1 – \\dfrac{r}{R}\\right)\\) for \\(r \\le R\\), where \\(\\rho_0\\) is a positive constant and \\(r\\) is the radial distance from the central axis. The cylinder is coaxially surrounded by a thin conducting cylindrical shell of radius \\(3R\\). Which of the following expressions gives the electric field magnitude \\(E(r)\\) as a function of distance \\(r\\) in the region \\(R < r < 3R\\)?"
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url: "https://nerd-notes.com/ubq/118004/"
date_modified: "2026-08-04T08:02:51+00:00"
---

# A long, solid insulating cylinder of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = \rho_0 \left(1 – \dfrac{r}{R}\right)\) for \(r \le R\), where \(\rho_0\) is a positive constant and \(r\) is the radial distance from the central axis. The cylinder is coaxially surrounded by a thin conducting cylindrical shell of radius \(3R\). Which of the following expressions gives the electric field magnitude \(E(r)\) as a function of distance \(r\) in the region \(R < r < 3R\)?

A long, solid insulating cylinder of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = \rho_0 \left(1 - \dfrac{r}{R}\right)\) for \(r \le R\), where \(\rho_0\) is a positive constant and \(r\) is the radial distance from the central axis. The cylinder is coaxially surrounded by a thin conducting cylindrical shell of radius \(3R\). Which of the following expressions gives the electric field magnitude \(E(r)\) as a function of distance \(r\) in the region \(R < r < 3R\)?

![A cross-sectional diagram showing a solid circle of radius R centered at the origin, filled with light gray shading. Surrounding this solid circle is a concentric thin outer circle of radius 3R drawn with a thin black line. A dashed radial line segment extends from the center to the boundary of the solid circle labeled R. A second dashed radial line segment extends from the center to the outer circle labeled 3R. In the unshaded space between the two circles, a single point is shown at a radial distance from the center, connected to the center by a dotted line segment labeled r. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830571-marOxg.jpg)

- **A.** \(\dfrac{\rho_0 R^2}{2\varepsilon_0 r}\)
- **B.** \(\dfrac{\rho_0 R^2}{3\varepsilon_0 r}\)
- **C.** \(\dfrac{5\rho_0 R^2}{12\varepsilon_0 r}\)
- **D.** \(\dfrac{\rho_0 R^2}{6\varepsilon_0 r}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118004/*