---
title: "A long, solid, non-conducting cylinder of radius \\(R\\) carries a volume charge density given by \\(\\rho(r) = \\rho_0 \\dfrac{r}{R}\\) for \\(r \\le R\\), where \\(\\rho_0\\) is a positive constant and \\(r\\) is the radial distance from the central axis of the cylinder. Which of the following expressions gives the magnitude of the electric field at a distance \\(r < R\\) from the central axis?"
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url: "https://nerd-notes.com/ubq/117956/"
date_modified: "2026-08-04T08:02:41+00:00"
---

# A long, solid, non-conducting cylinder of radius \(R\) carries a volume charge density given by \(\rho(r) = \rho_0 \dfrac{r}{R}\) for \(r \le R\), where \(\rho_0\) is a positive constant and \(r\) is the radial distance from the central axis of the cylinder. Which of the following expressions gives the magnitude of the electric field at a distance \(r < R\) from the central axis?

A long, solid, non-conducting cylinder of radius \(R\) carries a volume charge density given by \(\rho(r) = \rho_0 \dfrac{r}{R}\) for \(r \le R\), where \(\rho_0\) is a positive constant and \(r\) is the radial distance from the central axis of the cylinder. Which of the following expressions gives the magnitude of the electric field at a distance \(r < R\) from the central axis?

![Cross-sectional view of a long solid cylinder of outer radius R. A centered concentric dashed circle of radius r represents a Gaussian surface inside the cylinder, with r less than R. A radial vector arrow extends from the center origin to the inner dashed circle labeled r, and another radial vector arrow extends from the center to the outer solid circle labeled R. A label \(\rho(r) = \rho_0 \dfrac{r}{R}\) is positioned near the top of the outer cylinder. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830561-Ccfw5P.jpg)

- **A.** \(\dfrac{\rho_0 r^2}{2 \varepsilon_0 R}\)
- **B.** \(\dfrac{\rho_0 r^2}{3 \varepsilon_0 R}\)
- **C.** \(\dfrac{\rho_0 r^2}{4 \varepsilon_0 R}\)
- **D.** \(\dfrac{\rho_0 r^3}{4 \varepsilon_0 R^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117956/*