---
title: "A closed right circular cylinder of radius \\(R\\) and length \\(L\\) (where \\(L > 2R\\)) is centered at the origin with its central axis aligned along the \\(z\\)-axis. An infinitely long, thin wire carrying a uniform linear charge density \\(\\lambda\\) is oriented parallel to the \\(x\\)-axis and lies in the \\(xy\\)-plane at a distance \\(d\\) (where \\(0 < d < R\\)) from the origin, passing through the cylinder. Which of the following expressions represents the net electric flux \\(\\Phi_E\\) through the entire closed surface of the cylinder?"
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url: "https://nerd-notes.com/ubq/118002/"
date_modified: "2026-08-04T08:02:50+00:00"
---

# A closed right circular cylinder of radius \(R\) and length \(L\) (where \(L > 2R\)) is centered at the origin with its central axis aligned along the \(z\)-axis. An infinitely long, thin wire carrying a uniform linear charge density \(\lambda\) is oriented parallel to the \(x\)-axis and lies in the \(xy\)-plane at a distance \(d\) (where \(0 < d < R\)) from the origin, passing through the cylinder. Which of the following expressions represents the net electric flux \(\Phi_E\) through the entire closed surface of the cylinder?

A closed right circular cylinder of radius \(R\) and length \(L\) (where \(L > 2R\)) is centered at the origin with its central axis aligned along the \(z\)-axis. An infinitely long, thin wire carrying a uniform linear charge density \(\lambda\) is oriented parallel to the \(x\)-axis and lies in the \(xy\)-plane at a distance \(d\) (where \(0 < d < R\)) from the origin, passing through the cylinder. Which of the following expressions represents the net electric flux \(\Phi_E\) through the entire closed surface of the cylinder?

![A 3D perspective diagram showing a closed right circular cylinder of radius \(R\) and length \(L\) centered at the origin of a three-dimensional Cartesian coordinate system with axes labeled \(x\), \(y\), and \(z\). The central axis of the cylinder lies along the vertical \(z\)-axis, extending from \(z = -L/2\) to \(z = L/2\). A straight, infinitely long thin line representing a wire lies in the horizontal \(xy\)-plane, oriented parallel to the \(x\)-axis. The wire is shifted along the positive \(y\)-axis by a distance \(d\) from the origin, where \(d < R\), so that it penetrates through the curved wall of the cylinder in two places. A dashed line segment from the origin along the \(y\)-axis to the wire is labeled \(d\). A radial line segment from the origin to the circular perimeter of the cylinder in the \(xy\)-plane is labeled \(R\). The wire is labeled with charge density \(\lambda\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830570-stIL99.jpg)

- **A.** \(\dfrac{2\lambda \sqrt{R^2 - d^2}}{\varepsilon_0}\)
- **B.** \(\dfrac{\lambda L}{\varepsilon_0}\)
- **C.** \(\dfrac{\lambda \sqrt{R^2 - d^2}}{\varepsilon_0}\)
- **D.** \(\dfrac{2\lambda (R - d)}{\varepsilon_0}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118002/*