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title: "A thin, nonconducting semicircular arc of radius \\(R\\) is fixed in the \\(xy\\)-plane and centered at the origin. The arc extends from \\(\\theta = -\\dfrac{\\pi}{2}\\) to \\(\\theta = \\dfrac{\\pi}{2}\\), where \\(\\theta\\) is measured counterclockwise from the positive \\(x\\)-axis. The linear charge density along the arc varies according to \\(\\lambda(\\theta) = \\lambda_0 \\cos\\theta\\), where \\(\\lambda_0\\) is a positive constant. Assuming the electric potential is zero at infinity, which of the following expressions correctly represents the integral setup to determine the electric potential \\(V\\) at the origin?"
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url: "https://nerd-notes.com/ubq/121075/"
date_modified: "2026-08-23T04:57:41+00:00"
---

# A thin, nonconducting semicircular arc of radius \(R\) is fixed in the \(xy\)-plane and centered at the origin. The arc extends from \(\theta = -\dfrac{\pi}{2}\) to \(\theta = \dfrac{\pi}{2}\), where \(\theta\) is measured counterclockwise from the positive \(x\)-axis. The linear charge density along the arc varies according to \(\lambda(\theta) = \lambda_0 \cos\theta\), where \(\lambda_0\) is a positive constant. Assuming the electric potential is zero at infinity, which of the following expressions correctly represents the integral setup to determine the electric potential \(V\) at the origin?

A thin, nonconducting semicircular arc of radius \(R\) is fixed in the \(xy\)-plane and centered at the origin. The arc extends from \(\theta = -\dfrac{\pi}{2}\) to \(\theta = \dfrac{\pi}{2}\), where \(\theta\) is measured counterclockwise from the positive \(x\)-axis. The linear charge density along the arc varies according to \(\lambda(\theta) = \lambda_0 \cos\theta\), where \(\lambda_0\) is a positive constant. Assuming the electric potential is zero at infinity, which of the following expressions correctly represents the integral setup to determine the electric potential \(V\) at the origin?

![A Cartesian coordinate system in the xy-plane showing a horizontal x-axis and a vertical y-axis intersecting at the origin labeled O. A solid, curved semicircle of radius R lies entirely in the right half-plane, extending continuously from the point (0, -R) on the negative y-axis through (R, 0) on the positive x-axis to (0, R) on the positive y-axis. A thin dashed straight line connects the origin to a generic point on the semicircle in the first quadrant, with a single straight arrow along this dashed line labeled R pointing toward the upper-right at 45 degrees. A small curved angle arc labeled \theta extends counterclockwise from the positive x-axis to the dashed line. Near the apex of the semicircle, the label \lambda(\theta) = \lambda_0 \cos\theta appears. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461061-Captcp.jpg)

- **A.** \(V = \dfrac{\lambda_0}{4\pi\varepsilon_0} \int_{0}^{\pi} \cos\theta \, d\theta\)
- **B.** \(V = \dfrac{\lambda_0}{4\pi\varepsilon_0} \int_{-\pi/2}^{\pi/2} \cos\theta \, d\theta\)
- **C.** \(V = \dfrac{\lambda_0}{4\pi\varepsilon_0} \int_{-\pi/2}^{\pi/2} \cos^2\theta \, d\theta\)
- **D.** \(V = \dfrac{\lambda_0}{4\pi\varepsilon_0} \int_{0}^{\pi/2} \cos\theta \, d\theta\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121075/*