---
title: "A thin, flat plate shaped as a sector of a circle of radius \\(R\\) lies in the \\(xy\\)-plane with its vertex at the origin \\(O\\). The sector subtends an angle \\(\\theta_0\\) symmetric about the positive \\(x\\)-axis, spanning from \\(\\theta = -\\theta_0/2\\) to \\(\\theta = +\\theta_0/2\\). The plate carries a non-uniform surface charge density given by \\(\\sigma(r) = \\sigma_0 \\left(\\dfrac{r}{R}\\right)\\), where \\(\\sigma_0\\) is a positive constant and \\(r\\) is the radial distance from the origin. In terms of \\(\\sigma_0\\), \\(\\theta_0\\), and physical constants, what is the magnitude of the electric field at the origin \\(O\\)?"
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url: "https://nerd-notes.com/ubq/118000/"
date_modified: "2026-08-04T08:02:49+00:00"
---

# A thin, flat plate shaped as a sector of a circle of radius \(R\) lies in the \(xy\)-plane with its vertex at the origin \(O\). The sector subtends an angle \(\theta_0\) symmetric about the positive \(x\)-axis, spanning from \(\theta = -\theta_0/2\) to \(\theta = +\theta_0/2\). The plate carries a non-uniform surface charge density given by \(\sigma(r) = \sigma_0 \left(\dfrac{r}{R}\right)\), where \(\sigma_0\) is a positive constant and \(r\) is the radial distance from the origin. In terms of \(\sigma_0\), \(\theta_0\), and physical constants, what is the magnitude of the electric field at the origin \(O\)?

A thin, flat plate shaped as a sector of a circle of radius \(R\) lies in the \(xy\)-plane with its vertex at the origin \(O\). The sector subtends an angle \(\theta_0\) symmetric about the positive \(x\)-axis, spanning from \(\theta = -\theta_0/2\) to \(\theta = +\theta_0/2\). The plate carries a non-uniform surface charge density given by \(\sigma(r) = \sigma_0 \left(\dfrac{r}{R}\right)\), where \(\sigma_0\) is a positive constant and \(r\) is the radial distance from the origin. In terms of \(\sigma_0\), \(\theta_0\), and physical constants, what is the magnitude of the electric field at the origin \(O\)?

![A two-dimensional Cartesian xy-plane with origin labeled O. In the positive x region, a circular wedge sector of radius R extends symmetrically above and below the x-axis from angle -\theta_0/2 to +\theta_0/2. The curved outer boundary of the wedge is at radius R. A dashed angle arc indicates the total angular span \theta_0 centered on the x-axis. A small shaded differential area element dA is shown at radial position r and angle \theta relative to the x-axis. A straight dashed line segment of length r connects origin O to dA. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830569-oXetg4.jpg)

- **A.** \(\dfrac{\sigma_0 \theta_0}{4\pi \varepsilon_0}\)
- **B.** \(\dfrac{\sigma_0 \sin(\theta_0/2)}{2\pi \varepsilon_0}\)
- **C.** \(\dfrac{\sigma_0 \sin(\theta_0/2)}{\pi \varepsilon_0}\)
- **D.** \(\dfrac{\sigma_0 \sin\theta_0}{2\pi \varepsilon_0}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118000/*