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title: "A thin, nonconducting wire carrying a uniform positive linear charge density \\(\\lambda_0\\) is bent into a planar spiral described in polar coordinates by \\(r(\\theta) = b\\theta\\) for \\(\\pi \\le \\theta \\le 2\\pi\\), where \\(b\\) is a positive constant. The spiral lies in the \\(xy\\)-plane with the origin \\((0,0)\\) located at \\(r = 0\\). Which of the following expressions correctly represents the integral set up to determine the \\(y\\)-component of the net electric field, \\(E_y\\), at the origin?"
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url: "https://nerd-notes.com/ubq/118009/"
date_modified: "2026-08-04T08:02:52+00:00"
---

# A thin, nonconducting wire carrying a uniform positive linear charge density \(\lambda_0\) is bent into a planar spiral described in polar coordinates by \(r(\theta) = b\theta\) for \(\pi \le \theta \le 2\pi\), where \(b\) is a positive constant. The spiral lies in the \(xy\)-plane with the origin \((0,0)\) located at \(r = 0\). Which of the following expressions correctly represents the integral set up to determine the \(y\)-component of the net electric field, \(E_y\), at the origin?

A thin, nonconducting wire carrying a uniform positive linear charge density \(\lambda_0\) is bent into a planar spiral described in polar coordinates by \(r(\theta) = b\theta\) for \(\pi \le \theta \le 2\pi\), where \(b\) is a positive constant. The spiral lies in the \(xy\)-plane with the origin \((0,0)\) located at \(r = 0\). Which of the following expressions correctly represents the integral set up to determine the \(y\)-component of the net electric field, \(E_y\), at the origin?

![A 2D schematic in the xy-plane showing a thin spiral wire starting at polar angle \theta = \pi on the negative x-axis at radius r = b\pi and winding counterclockwise to angle \theta = 2\pi on the positive x-axis at radius r = 2b\pi. The origin (0,0) is marked with a black dot labeled (0,0). The spiral wire is labeled with charge density \lambda_0. A small segment dq on the spiral at an arbitrary angle \theta in the third quadrant is highlighted, with a position vector \vec{r} pointing from the origin to dq labeled r = b\theta. An electric field vector d\vec{E} originates at (0,0) and points opposite to \vec{r}. Dashed coordinate axes x and y are shown. No other labels, lines, or text appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830572-ZIMh3P.jpg)

- **A.** \(-\dfrac{\lambda_0}{4\pi\varepsilon_0 b} \int_{\pi}^{2\pi} \dfrac{\sin\theta}{\theta} d\theta\)
- **B.** \(-\dfrac{\lambda_0 \sqrt{1+b^2}}{4\pi\varepsilon_0 b^2} \int_{\pi}^{2\pi} \dfrac{\sin\theta}{\theta^2} d\theta\)
- **C.** \(-\dfrac{\lambda_0}{4\pi\varepsilon_0 b} \int_{\pi}^{2\pi} \dfrac{\sqrt{1+\theta^2} \sin\theta}{\theta^2} d\theta\)
- **D.** \(-\dfrac{\lambda_0}{4\pi\varepsilon_0} \int_{\pi}^{2\pi} \dfrac{\sqrt{1+\theta^2} \sin\theta}{\theta} d\theta\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118009/*