---
title: "A very long, solid nonconducting cylinder of radius \\(R\\) has a uniform positive volume charge density and a total linear charge density \\(+\\lambda\\). A particle of mass \\(m\\) and positive charge \\(q\\) is released from rest at a radial distance \\(r_1\\) from the cylinder’s central axis, where \\(r_1  R\\). Which of the following expressions correctly sets up the integral for the work \\(W\\) done on the particle by the electric field during this displacement?"
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url: "https://nerd-notes.com/ubq/124648/"
date_modified: "2026-09-28T14:08:56+00:00"
---

# A very long, solid nonconducting cylinder of radius \(R\) has a uniform positive volume charge density and a total linear charge density \(+\lambda\). A particle of mass \(m\) and positive charge \(q\) is released from rest at a radial distance \(r_1\) from the cylinder’s central axis, where \(r_1  R\). Which of the following expressions correctly sets up the integral for the work \(W\) done on the particle by the electric field during this displacement?

A very long, solid nonconducting cylinder of radius \(R\) has a uniform positive volume charge density and a total linear charge density \(+\lambda\). A particle of mass \(m\) and positive charge \(q\) is released from rest at a radial distance \(r_1\) from the cylinder's central axis, where \(r_1 < R\). The particle travels radially outward under the influence of the electric field to a final radial distance \(r_2\), where \(r_2 > R\). Which of the following expressions correctly sets up the integral for the work \(W\) done on the particle by the electric field during this displacement?

![A cross-sectional view of a long cylinder and the path of a charged particle. A large circle centered at the origin represents the cylinder of radius \(R\), filled with light gray shading. A horizontal dashed reference line extends from the cylinder's center to the right beyond the circle. Along this horizontal line, three positions are indicated: a radial distance \(r_1\) inside the circle, the cylinder radius \(R\) at the circle boundary, and a radial distance \(r_2\) outside the circle. At position \(r_1\), a small solid black circle represents the particle, labeled \(+q\), with a short horizontal arrow pointing to the right indicating its direction of motion toward \(r_2\). A dimension arrow from the center to the cylinder boundary is labeled \(R\). The region inside the circle is labeled \(+\lambda\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604535-stFLx3.jpg)

- **A.** \(W = \dfrac{q\lambda}{2\pi\varepsilon_0} \int_{r_1}^{r_2} \dfrac{1}{r}\,dr\)
- **B.** \(W = \dfrac{q\lambda}{2\pi\varepsilon_0} \left( \int_{r_1}^{R} \dfrac{1}{r}\,dr + \int_{R}^{r_2} \dfrac{r}{R^2}\,dr \right)\)
- **C.** \(W = \dfrac{q\lambda}{2\pi\varepsilon_0} \left( \int_{r_1}^{R} \dfrac{r}{R^2}\,dr + \int_{R}^{r_2} \dfrac{1}{r}\,dr \right)\)
- **D.** \(W = -\dfrac{q\lambda}{2\pi\varepsilon_0} \left( \int_{r_1}^{R} \dfrac{r}{R^2}\,dr + \int_{R}^{r_2} \dfrac{1}{r}\,dr \right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124648/*