---
title: "A long, straight wire carrying a steady current \\(I_1\\) in the \\(+y\\)-direction lies along the \\(y\\)-axis in the \\(xy\\)-plane. In the region \\(x > 0\\), a thin semicircular wire of radius \\(R\\) is centered at the point \\((d, 0)\\), where \\(d > R\\), and carries a steady counterclockwise current \\(I_2\\). Points on the semicircle are parameterized by the angle \\(\\theta \\in [-\\pi/2, \\pi/2]\\) measured relative to the \\(+x\\)-axis, such that the position of an element on the wire is \\((d + R\\cos\\theta, R\\sin\\theta)\\). Which of the following integral expressions correctly represents the magnitude of the net magnetic force exerted on the semicircular wire by the straight wire?"
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/124804/"
date_modified: "2026-09-28T14:11:22+00:00"
---

# A long, straight wire carrying a steady current \(I_1\) in the \(+y\)-direction lies along the \(y\)-axis in the \(xy\)-plane. In the region \(x > 0\), a thin semicircular wire of radius \(R\) is centered at the point \((d, 0)\), where \(d > R\), and carries a steady counterclockwise current \(I_2\). Points on the semicircle are parameterized by the angle \(\theta \in [-\pi/2, \pi/2]\) measured relative to the \(+x\)-axis, such that the position of an element on the wire is \((d + R\cos\theta, R\sin\theta)\). Which of the following integral expressions correctly represents the magnitude of the net magnetic force exerted on the semicircular wire by the straight wire?

A long, straight wire carrying a steady current \(I_1\) in the \(+y\)-direction lies along the \(y\)-axis in the \(xy\)-plane. In the region \(x > 0\), a thin semicircular wire of radius \(R\) is centered at the point \((d, 0)\), where \(d > R\), and carries a steady counterclockwise current \(I_2\). Points on the semicircle are parameterized by the angle \(\theta \in [-\pi/2, \pi/2]\) measured relative to the \(+x\)-axis, such that the position of an element on the wire is \((d + R\cos\theta, R\sin\theta)\). Which of the following integral expressions correctly represents the magnitude of the net magnetic force exerted on the semicircular wire by the straight wire?

![The diagram shows an xy-coordinate plane with a vertical axis labeled y and a horizontal axis labeled x. A thick vertical solid line lies along the y-axis with a single upward arrow labeled \(I_1\). On the positive x-axis, a point is labeled \((d, 0)\). A semicircular arc of radius \(R\) is centered at \((d, 0)\) and curves to the right from \((d, -R)\) through \((d + R, 0)\) to \((d, R)\). A small curved arrow along the semicircle indicates the direction of current \(I_2\) flowing counterclockwise. A dashed straight line segment extends from \((d, 0)\) to an arbitrary point on the arc in the first quadrant, labeled with radius \(R\). A small curved angle marker between the positive x-axis and this dashed segment is labeled \(\theta\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604681-oC95G3.jpg)

- **A.** \(\dfrac{\mu_0 I_1 I_2 R}{2\pi} \int_{-\pi/2}^{\pi/2} \dfrac{1}{d + R\cos\theta}\,d\theta\)
- **B.** \(\dfrac{\mu_0 I_1 I_2 R}{2\pi} \int_{-\pi/2}^{\pi/2} \dfrac{\sin\theta}{d + R\cos\theta}\,d\theta\)
- **C.** \(\dfrac{\mu_0 I_1 I_2 R}{2\pi} \int_{-\pi/2}^{\pi/2} \dfrac{\cos\theta}{d + R\sin\theta}\,d\theta\)
- **D.** \(\dfrac{\mu_0 I_1 I_2 R}{2\pi} \int_{-\pi/2}^{\pi/2} \dfrac{\cos\theta}{d + R\cos\theta}\,d\theta\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124804/*