---
title: "An infinitely long non-conducting solid cylinder of radius \\(R\\) has a non-uniform 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. A particle of mass \\(m\\) and negative charge \\(-q\\) (where \\(q > 0\\)) moves in a stable circular orbit of radius \\(r < R\\) inside the cylinder, centered on and perpendicular to the central axis. Which of the following expressions correctly represents the orbital speed \\(v\\) of the particle in terms of \\(r\\), \\(R\\), \\(\\rho_0\\), \\(q\\), \\(m\\), and fundamental constants?"
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url: "https://nerd-notes.com/ubq/117982/"
date_modified: "2026-08-04T08:02:46+00:00"
---

# An infinitely long non-conducting solid cylinder of radius \(R\) has a non-uniform 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. A particle of mass \(m\) and negative charge \(-q\) (where \(q > 0\)) moves in a stable circular orbit of radius \(r < R\) inside the cylinder, centered on and perpendicular to the central axis. Which of the following expressions correctly represents the orbital speed \(v\) of the particle in terms of \(r\), \(R\), \(\rho_0\), \(q\), \(m\), and fundamental constants?

An infinitely long non-conducting solid cylinder of radius \(R\) has a non-uniform 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. A particle of mass \(m\) and negative charge \(-q\) (where \(q > 0\)) moves in a stable circular orbit of radius \(r < R\) inside the cylinder, centered on and perpendicular to the central axis. Which of the following expressions correctly represents the orbital speed \(v\) of the particle in terms of \(r\), \(R\), \(\rho_0\), \(q\), \(m\), and fundamental constants?

![A cross-sectional view of a circular cylinder of radius R centered at the origin in the xy-plane. A shaded circular region represents the interior of the cylinder with radius R. A dashed concentric circle of radius r, where r < R, represents the circular orbit of a small particle. On the dashed circle, a small solid black circle labeled -q represents the particle. A curved vector arrow tangent to the orbit at -q shows the velocity vector v. A straight vector arrow pointing from -q toward the origin is labeled F_e. A radial line segment from the origin to the outer boundary is labeled R, and a radial line segment from the origin to the particle is labeled r. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830566-44cdVM.jpg)

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

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117982/*