---
title: "A long wire carrying a steady current \\(I\\) is bent into a planar hairpin shape. The wire consists of two parallel, semi-infinite straight sections separated by a distance \\(2R\\) that are connected seamlessly by a semicircular arc of radius \\(R\\). What is the ratio of the magnitude of the magnetic field produced by the semicircular arc to the magnitude of the total magnetic field at the center of curvature of the arc, \\(B_{\\text{arc}} / B_{\\text{total}}\\)?"
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url: "https://nerd-notes.com/ubq/124795/"
date_modified: "2026-09-28T14:11:19+00:00"
---

# A long wire carrying a steady current \(I\) is bent into a planar hairpin shape. The wire consists of two parallel, semi-infinite straight sections separated by a distance \(2R\) that are connected seamlessly by a semicircular arc of radius \(R\). What is the ratio of the magnitude of the magnetic field produced by the semicircular arc to the magnitude of the total magnetic field at the center of curvature of the arc, \(B_{\text{arc}} / B_{\text{total}}\)?

A long wire carrying a steady current \(I\) is bent into a planar hairpin shape. The wire consists of two parallel, semi-infinite straight sections separated by a distance \(2R\) that are connected seamlessly by a semicircular arc of radius \(R\). What is the ratio of the magnitude of the magnetic field produced by the semicircular arc to the magnitude of the total magnetic field at the center of curvature of the arc, \(B_{\text{arc}} / B_{\text{total}}\)?

![A single thick solid line represents a wire in the plane of the page. At the top, a horizontal line segment extends from the left margin to the center-right. From this point, a semicircular arc of radius \(R\) curves downward and to the right, ending at a lower point directly below the start of the arc. From the bottom of the arc, a second horizontal line segment extends back toward the left margin, parallel to the top segment and separated from it by a vertical distance \(2R\). Exactly three small arrows along the wire indicate current direction: one pointing rightward on the top horizontal segment labeled \(I\), one pointing clockwise along the semicircular arc, and one pointing leftward on the bottom horizontal segment. A small solid dot labeled \(C\) marks the center of curvature of the semicircular arc. A vertical double-headed arrow between the two horizontal segments is labeled \(2R\). A dashed radial line segment from dot \(C\) to the arc is labeled \(R\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604678-7HxABy.jpg)

- **A.** \(\dfrac{2}{\pi + 2}\)
- **B.** \(\dfrac{\pi}{\pi + 4}\)
- **C.** \(\dfrac{\pi}{\pi + 1}\)
- **D.** \(\dfrac{\pi}{\pi + 2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124795/*