---
title: "An infinitely long, solid insulating cylinder of radius \\(R\\) carries 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. Which of the following expressions gives the magnitude of the electric field \\(E(r)\\) inside the cylinder as a function of \\(r\\) for \\(r < R\\)?"
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url: "https://nerd-notes.com/ubq/117968/"
date_modified: "2026-08-04T08:02:43+00:00"
---

# An infinitely long, solid insulating cylinder of radius \(R\) carries 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. Which of the following expressions gives the magnitude of the electric field \(E(r)\) inside the cylinder as a function of \(r\) for \(r < R\)?

An infinitely long, solid insulating cylinder of radius \(R\) carries 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. Which of the following expressions gives the magnitude of the electric field \(E(r)\) inside the cylinder as a function of \(r\) for \(r < R\)?

![A circular cross section of a solid cylinder with outer radius R centered at the origin. A concentric inner dashed circle of radius r represents a coaxial cylindrical Gaussian surface, where r is less than R. A radial arrow points from the origin to the dashed circle labeled r. Another radial arrow points from the origin to the outer circular boundary labeled R. Shading inside the outer circle decreases linearly in density from the center toward the outer boundary to represent non-uniform volume charge density rho(r). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830563-tfuwyQ.jpg)

- **A.** \(\dfrac{\rho_0 r}{\varepsilon_0} \left(1 - \dfrac{r}{R}\right)\)
- **B.** \(\dfrac{\rho_0 r}{2\varepsilon_0} \left(1 - \dfrac{r}{R}\right)\)
- **C.** \(\dfrac{\rho_0 r}{2\varepsilon_0} \left(1 - \dfrac{2r}{3R}\right)\)
- **D.** \(\dfrac{\rho_0 r}{\varepsilon_0} \left(1 - \dfrac{2r}{3R}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117968/*