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title: "A thin circular ring of radius \\(R\\) lies in the \\(xy\\)-plane, centered at the origin. The ring carries a non-uniform linear charge density given by \\(\\lambda(\\theta) = \\lambda_0 \\cos\\theta\\), where \\(\\lambda_0\\) is a positive constant and \\(\\theta\\) is the polar angle measured counterclockwise from the positive \\(x\\)-axis. A point \\(P\\) is located on the positive \\(z\\)-axis at a distance \\(z\\) from the origin. Which of the following expressions gives the magnitude of the net electric field at point \\(P\\)?"
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url: "https://nerd-notes.com/ubq/117999/"
date_modified: "2026-08-04T08:02:49+00:00"
---

# A thin circular ring of radius \(R\) lies in the \(xy\)-plane, centered at the origin. The ring carries a non-uniform linear charge density given by \(\lambda(\theta) = \lambda_0 \cos\theta\), where \(\lambda_0\) is a positive constant and \(\theta\) is the polar angle measured counterclockwise from the positive \(x\)-axis. A point \(P\) is located on the positive \(z\)-axis at a distance \(z\) from the origin. Which of the following expressions gives the magnitude of the net electric field at point \(P\)?

A thin circular ring of radius \(R\) lies in the \(xy\)-plane, centered at the origin. The ring carries a non-uniform linear charge density given by \(\lambda(\theta) = \lambda_0 \cos\theta\), where \(\lambda_0\) is a positive constant and \(\theta\) is the polar angle measured counterclockwise from the positive \(x\)-axis. A point \(P\) is located on the positive \(z\)-axis at a distance \(z\) from the origin. Which of the following expressions gives the magnitude of the net electric field at point \(P\)?

![A circular ring of radius \(R\) is rendered as a flat tilted ellipse centered at the origin of a three-dimensional Cartesian coordinate system. The horizontal axis extending to the right is labeled \(x\), the diagonal axis extending down and to the left is labeled \(y\), and the vertical axis extending straight up is labeled \(z\). The ring sits in the horizontal \(xy\)-plane. On the positive \(z\)-axis, a small solid black dot is plotted at a height \(z\) above the origin and is labeled \(P\). A dashed line segment of length \(R\) extends from the origin along the positive \(x\)-axis to the right edge of the ring, labeled \(R\). A curved arrow in the \(xy\)-plane indicates a polar angle \(\theta\) swept counterclockwise from the positive \(x\)-axis toward the positive \(y\)-axis. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830569-gToCTe.jpg)

- **A.** \(0\)
- **B.** \(\dfrac{\lambda_0 R^2}{4\varepsilon_0 (R^2 + z^2)^{3/2}}\)
- **C.** \(\dfrac{\lambda_0 R z}{4\varepsilon_0 (R^2 + z^2)^{3/2}}\)
- **D.** \(\dfrac{\lambda_0 R^2}{2\varepsilon_0 (R^2 + z^2)^{3/2}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117999/*