---
title: "A thin, nonconducting plastic rod is bent into a semicircle of radius \\(R\\) centered at the origin. The rod carries a non-uniform linear charge density given by \\(\\lambda(\\theta) = \\lambda_0 \\sin\\theta\\), where \\(\\lambda_0\\) is a positive constant and \\(\\theta\\) is the angle measured counterclockwise from the positive \\(x\\)-axis (\\(0 \\le \\theta \\le \\pi\\)). Assuming the electric potential is zero at infinity, what is the electric potential at the origin?"
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url: "https://nerd-notes.com/ubq/118157/"
date_modified: "2026-08-04T08:05:32+00:00"
---

# A thin, nonconducting plastic rod is bent into a semicircle of radius \(R\) centered at the origin. The rod carries a non-uniform linear charge density given by \(\lambda(\theta) = \lambda_0 \sin\theta\), where \(\lambda_0\) is a positive constant and \(\theta\) is the angle measured counterclockwise from the positive \(x\)-axis (\(0 \le \theta \le \pi\)). Assuming the electric potential is zero at infinity, what is the electric potential at the origin?

A thin, nonconducting plastic rod is bent into a semicircle of radius \(R\) centered at the origin. The rod carries a non-uniform linear charge density given by \(\lambda(\theta) = \lambda_0 \sin\theta\), where \(\lambda_0\) is a positive constant and \(\theta\) is the angle measured counterclockwise from the positive \(x\)-axis (\(0 \le \theta \le \pi\)). Assuming the electric potential is zero at infinity, what is the electric potential at the origin?

![A semicircular curved line of radius R is centered at the origin O in the xy-plane, lying entirely in the region y \ge 0. The arc starts at (R,0), passes through (0,R), and ends at (-R,0). A dashed straight line segment of length R extends from the origin O to a point on the arc at an angle \theta measured counterclockwise from the positive x-axis. A small segment of the arc at angle \theta is marked and labeled R\,d\theta. A curved arrow near the origin indicates the angle \theta from the positive x-axis to the dashed line. The origin is marked with a dot labeled O. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830732-FLZ7Qi.jpg)

- **A.** \(\dfrac{\lambda_0}{8\pi\varepsilon_0}\)
- **B.** \(\dfrac{\lambda_0}{4\pi\varepsilon_0}\)
- **C.** \(\dfrac{\lambda_0}{\pi\varepsilon_0}\)
- **D.** \(\dfrac{\lambda_0}{2\pi\varepsilon_0}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118157/*