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title: "A thin ring of radius \\(R\\) lies in the \\(xy\\)-plane and is centered at the origin. The ring carries a non-uniform linear charge density given by \\(\\lambda(\\theta) = \\lambda_0 \\cos(2\\theta)\\), where \\(\\theta\\) is the azimuthal angle measured counterclockwise from the positive \\(x\\)-axis and \\(\\lambda_0\\) is a positive constant. Which of the following expressions correctly gives the electric potential \\(V(z)\\) as a function of position \\(z\\) along the central \\(z\\)-axis, relative to \\(V = 0\\) at infinity?"
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url: "https://nerd-notes.com/ubq/118172/"
date_modified: "2026-08-04T08:05:42+00:00"
---

# A thin ring of radius \(R\) lies in the \(xy\)-plane and is centered at the origin. The ring carries a non-uniform linear charge density given by \(\lambda(\theta) = \lambda_0 \cos(2\theta)\), where \(\theta\) is the azimuthal angle measured counterclockwise from the positive \(x\)-axis and \(\lambda_0\) is a positive constant. Which of the following expressions correctly gives the electric potential \(V(z)\) as a function of position \(z\) along the central \(z\)-axis, relative to \(V = 0\) at infinity?

A thin ring of radius \(R\) lies in the \(xy\)-plane and is centered at the origin. The ring carries a non-uniform linear charge density given by \(\lambda(\theta) = \lambda_0 \cos(2\theta)\), where \(\theta\) is the azimuthal angle measured counterclockwise from the positive \(x\)-axis and \(\lambda_0\) is a positive constant. Which of the following expressions correctly gives the electric potential \(V(z)\) as a function of position \(z\) along the central \(z\)-axis, relative to \(V = 0\) at infinity?

![A three-dimensional Cartesian coordinate system showing x, y, and z axes meeting at an origin. A thin circular ring of radius R lies flat in the xy-plane, centered at the origin. An angle theta is indicated in the xy-plane measured counterclockwise from the positive x-axis to a point on the ring. A point labeled P lies on the positive z-axis at a height z above the origin. A straight dashed line segment connects a point on the ring at angle theta to point P on the z-axis, with the segment labeled r. A curved arrow along the ring shows the angle theta starting from the x-axis. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830741-ndWhig.jpg)

- **A.** \(V(z) = \dfrac{\lambda_0 R}{2\varepsilon_0 \sqrt{R^2 + z^2}}\)
- **B.** \(V(z) = \dfrac{\lambda_0 R}{2\pi\varepsilon_0 \sqrt{R^2 + z^2}}\)
- **C.** \(V(z) = 0\)
- **D.** \(V(z) = \dfrac{\lambda_0 R^2}{4\pi\varepsilon_0 (R^2 + z^2)}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118172/*