---
title: "A thin flat disk of radius \\(R\\) lies in the \\(xy\\)-plane centered at the origin. The disk has a non-uniform surface charge density given in polar coordinates by \\(\\sigma(r, \\theta) = \\sigma_0 \\left(\\dfrac{r}{R}\\right) \\cos\\theta\\), where \\(\\sigma_0\\) is a positive constant, \\(r\\) is the radial distance from the origin, and \\(\\theta\\) is measured counterclockwise from the positive \\(x\\)-axis. Which of the following expressions gives the magnitude of the electric field at the origin?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/118032/"
date_modified: "2026-08-04T08:03:07+00:00"
---

# A thin flat disk of radius \(R\) lies in the \(xy\)-plane centered at the origin. The disk has a non-uniform surface charge density given in polar coordinates by \(\sigma(r, \theta) = \sigma_0 \left(\dfrac{r}{R}\right) \cos\theta\), where \(\sigma_0\) is a positive constant, \(r\) is the radial distance from the origin, and \(\theta\) is measured counterclockwise from the positive \(x\)-axis. Which of the following expressions gives the magnitude of the electric field at the origin?

A thin flat disk of radius \(R\) lies in the \(xy\)-plane centered at the origin. The disk has a non-uniform surface charge density given in polar coordinates by \(\sigma(r, \theta) = \sigma_0 \left(\dfrac{r}{R}\right) \cos\theta\), where \(\sigma_0\) is a positive constant, \(r\) is the radial distance from the origin, and \(\theta\) is measured counterclockwise from the positive \(x\)-axis. Which of the following expressions gives the magnitude of the electric field at the origin?

![A circular disk of radius R centered at the origin in the xy-plane. The x-axis extends horizontally to the right and the y-axis extends vertically upward. In the first quadrant, a small shaded differential sector element is shown at radius r and angle theta relative to the positive x-axis. A dashed radial line segment labeled r extends from the origin to the element. A curved arrow labeled theta indicates the angle counterclockwise from the positive x-axis to the radial line. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830587-iUGf2h.jpg)

- **A.** \(\dfrac{\sigma_0}{2\varepsilon_0}\)
- **B.** \(\dfrac{\sigma_0}{\pi \varepsilon_0}\)
- **C.** \(\dfrac{\sigma_0}{4\varepsilon_0}\)
- **D.** \(\dfrac{\sigma_0}{8\varepsilon_0}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118032/*