---
title: "A point charge \\(+q\\) is fixed at the center of a hollow, thin-walled cylindrical shell of radius \\(R\\) and length \\(L\\). The flat circular ends of the cylinder are open, such that only the curved cylindrical surface is present. Let \\(\\Phi_{\\text{curved}}\\) represent the net electric flux passing outward through this curved surface. Which of the following statements correctly describes the limiting behavior of \\(\\Phi_{\\text{curved}}\\) as \\(L \\to 0\\) and as \\(L \\to \\infty\\)?"
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url: "https://nerd-notes.com/ubq/124528/"
date_modified: "2026-09-28T14:08:32+00:00"
---

# A point charge \(+q\) is fixed at the center of a hollow, thin-walled cylindrical shell of radius \(R\) and length \(L\). The flat circular ends of the cylinder are open, such that only the curved cylindrical surface is present. Let \(\Phi_{\text{curved}}\) represent the net electric flux passing outward through this curved surface. Which of the following statements correctly describes the limiting behavior of \(\Phi_{\text{curved}}\) as \(L \to 0\) and as \(L \to \infty\)?

A point charge \(+q\) is fixed at the center of a hollow, thin-walled cylindrical shell of radius \(R\) and length \(L\). The flat circular ends of the cylinder are open, such that only the curved cylindrical surface is present. Let \(\Phi_{\text{curved}}\) represent the net electric flux passing outward through this curved surface. Which of the following statements correctly describes the limiting behavior of \(\Phi_{\text{curved}}\) as \(L \to 0\) and as \(L \to \infty\)?

![A horizontal open cylindrical tube viewed from a slightly elevated angle. The cylinder has a circular left rim and a circular right rim, both drawn as open ellipses, with the interior visible to indicate the ends are uncapped. A horizontal dashed centerline extends along the cylindrical axis from the left rim to the right rim. At the exact midpoint of this centerline sits a small solid circle labeled \(+q\). A double-headed horizontal dimension arrow beneath the cylinder spans the full length from the left rim to the right rim, labeled \(L\). A single straight vertical arrow points upward from the centerline to the top edge of the right circular rim, labeled \(R\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604512-5eUEuG.jpg)

- **A.** As \(L \to 0\), \(\Phi_{\text{curved}} \to \dfrac{q}{\varepsilon_0}\), and as \(L \to \infty\), \(\Phi_{\text{curved}} \to \dfrac{q}{\varepsilon_0}\), because Gauss's law requires the net flux through any surface enclosing charge \(+q\) to equal \(\dfrac{q}{\varepsilon_0}\).
- **B.** As \(L \to 0\), \(\Phi_{\text{curved}} \to 0\), and as \(L \to \infty\), \(\Phi_{\text{curved}} \to \infty\), because the electric field at the curved surface remains directed outward while the surface area increases without bound.
- **C.** As \(L \to 0\), \(\Phi_{\text{curved}} \to \dfrac{q}{2\varepsilon_0}\), and as \(L \to \infty\), \(\Phi_{\text{curved}} \to \dfrac{q}{\varepsilon_0}\), because the flux is divided equally between the curved surface and the open ends when the length is small.
- **D.** As \(L \to 0\), \(\Phi_{\text{curved}} \to 0\), and as \(L \to \infty\), \(\Phi_{\text{curved}} \to \dfrac{q}{\varepsilon_0}\), because the total solid angle subtended by the two open ends approaches \(4\pi\) as \(L \to 0\) and approaches \(0\) as \(L \to \infty\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124528/*