---
title: "A very long, nonconducting cylindrical shell of inner radius \\(R\\) and outer radius \\(2R\\) has a non-uniform volume charge density given by \\(\\rho(r) = \\beta r\\) for \\(R \\le r \\le 2R\\), where \\(\\beta\\) is a positive constant and \\(r\\) is the radial distance from the central axis. The regions \\(r  2R\\) are empty space. The cylinder is sufficiently long that edge effects can be ignored."
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url: "https://nerd-notes.com/ubq/117837/"
date_modified: "2026-08-04T07:59:06+00:00"
---

# A very long, nonconducting cylindrical shell of inner radius \(R\) and outer radius \(2R\) has a non-uniform volume charge density given by \(\rho(r) = \beta r\) for \(R \le r \le 2R\), where \(\beta\) is a positive constant and \(r\) is the radial distance from the central axis. The regions \(r  2R\) are empty space. The cylinder is sufficiently long that edge effects can be ignored.

A very long, nonconducting cylindrical shell of inner radius \(R\) and outer radius \(2R\) has a non-uniform volume charge density given by \(\rho(r) = \beta r\) for \(R \le r \le 2R\), where \(\beta\) is a positive constant and \(r\) is the radial distance from the central axis. The regions \(r < R\) and \(r > 2R\) are empty space. The cylinder is sufficiently long that edge effects can be ignored.

![A 2D cross-sectional view of a very long cylindrical shell. Two concentric circles are centered on a dot representing the central axis. The inner circle has radius R and the outer circle has radius 2R. The annular region between the inner and outer circles is shaded gray to indicate the insulating material. An arrow pointing straight right from the center to the inner circle is labeled R. An arrow pointing toward the upper-right at a 45 degree angle from the center to the outer circle is labeled 2R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830345-V5u80j.jpg)

**Part a)** **Derive** an expression for the total charge per unit length \(\lambda\) of the cylindrical shell. Express your answer in terms of \(\beta\), \(R\), and physical constants, as appropriate. *(2 points)*

**Part b)** **Derive** expressions for the magnitude of the electric field \(E\) in each of the following regions. Express your answers in terms of \(\beta\), \(R\), \(r\), \(\varepsilon_0\), and physical constants, as appropriate. *(5 points)*

**Part c)** On the axes provided, **sketch** a graph of the magnitude of the electric field \(E\) as a function of the radial distance \(r\) from the central axis, from \(r = 0\) to \(r = 4R\). *(3 points)*

**Part d)** A particle of charge \(-q\) and mass \(m\) is in a stable circular orbit around the cylindrical shell at a radial distance \(r = 3R\). **Derive** an expression for the orbital speed \(v\) of the particle. Express your answer in terms of \(\beta\), \(R\), \(q\), \(m\), \(\varepsilon_0\), and physical constants, as appropriate. *(3 points)*

**Part e)** Suppose the nonconducting cylindrical shell is replaced with a solid conducting cylindrical shell that has the same inner radius \(R\), outer radius \(2R\), and the same total charge per unit length \(\lambda\) as derived in part (a). *(4 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117837/*