---
title: "A thin, flat plate in the shape of a circular sector of radius \\(R\\) and subtended angle \\(\\theta_0\\) lies in the plane of the page. The surface charge density on the sector varies with radial distance \\(r\\) from the vertex according to \\(\\sigma(r) = \\sigma_0 \\dfrac{r}{R}\\), where \\(\\sigma_0\\) is a positive constant. The total electric charge on the sector is \\(Q\\). In terms of \\(Q\\), \\(R\\), and fundamental constants, what is the electric potential at the vertex of the sector, relative to zero at infinity?"
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url: "https://nerd-notes.com/ubq/118164/"
date_modified: "2026-08-04T08:05:35+00:00"
---

# A thin, flat plate in the shape of a circular sector of radius \(R\) and subtended angle \(\theta_0\) lies in the plane of the page. The surface charge density on the sector varies with radial distance \(r\) from the vertex according to \(\sigma(r) = \sigma_0 \dfrac{r}{R}\), where \(\sigma_0\) is a positive constant. The total electric charge on the sector is \(Q\). In terms of \(Q\), \(R\), and fundamental constants, what is the electric potential at the vertex of the sector, relative to zero at infinity?

A thin, flat plate in the shape of a circular sector of radius \(R\) and subtended angle \(\theta_0\) lies in the plane of the page. The surface charge density on the sector varies with radial distance \(r\) from the vertex according to \(\sigma(r) = \sigma_0 \dfrac{r}{R}\), where \(\sigma_0\) is a positive constant. The total electric charge on the sector is \(Q\). In terms of \(Q\), \(R\), and fundamental constants, what is the electric potential at the vertex of the sector, relative to zero at infinity?

![A flat circular sector of radius R and subtended angle \theta_0 lies in the xy-plane with its vertex at the origin labeled O. The sector extends symmetrically from angle -\theta_0/2 to +\theta_0/2 relative to the positive x-axis. A small shaded differential area element is located at distance r from the origin and angle \theta from the x-axis, with radial width dr and angular width d\theta. A straight dashed line segment labeled R extends from the origin along the upper straight edge of the sector to its outer circular arc. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830734-buj5d3.jpg)

- **A.** \(\dfrac{Q}{4\pi\varepsilon_0 R}\)
- **B.** \(\dfrac{3Q}{8\pi\varepsilon_0 R}\)
- **C.** \(\dfrac{Q}{2\pi\varepsilon_0 R}\)
- **D.** \(\dfrac{3Q}{4\pi\varepsilon_0 R}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118164/*