---
title: "A long, straight wire carries a steady current directed out of the page. Two closed circular Amperian paths, labeled \\(C_1\\) and \\(C_2\\), are situated in a plane perpendicular to the wire. Path \\(C_1\\) is centered on the wire. Path \\(C_2\\) has the same radius as path \\(C_1\\) but is off-center, though it still encloses the wire completely. Which of the following correctly compares the line integral \\(\\oint \\vec{B} \\cdot d\\vec{\\ell}\\) along each path and describes the practical use of Ampère’s law to calculate the magnitude of the magnetic field \\(B\\) along path \\(C_2\\)?"
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url: "https://nerd-notes.com/ubq/118526/"
date_modified: "2026-08-04T08:11:14+00:00"
---

# A long, straight wire carries a steady current directed out of the page. Two closed circular Amperian paths, labeled \(C_1\) and \(C_2\), are situated in a plane perpendicular to the wire. Path \(C_1\) is centered on the wire. Path \(C_2\) has the same radius as path \(C_1\) but is off-center, though it still encloses the wire completely. Which of the following correctly compares the line integral \(\oint \vec{B} \cdot d\vec{\ell}\) along each path and describes the practical use of Ampère’s law to calculate the magnitude of the magnetic field \(B\) along path \(C_2\)?

A long, straight wire carries a steady current directed out of the page. Two closed circular Amperian paths, labeled \(C_1\) and \(C_2\), are situated in a plane perpendicular to the wire. Path \(C_1\) is centered on the wire. Path \(C_2\) has the same radius as path \(C_1\) but is off-center, though it still encloses the wire completely. Which of the following correctly compares the line integral \(\oint \vec{B} \cdot d\vec{\ell}\) along each path and describes the practical use of Ampère's law to calculate the magnitude of the magnetic field \(B\) along path \(C_2\)?

![A long straight wire perpendicular to the page is represented by a small circle containing a dot, indicating current out of the page. Two closed circular paths, labeled C1 and C2, are drawn in the plane of the page. Path C1 is a circle centered exactly on the wire. Path C2 is a circle of equal radius to C1, but positioned so that the wire is off-center inside C2, closer to the left side of C2. Counterclockwise integration arrows are marked on both circular paths.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/fig1-1785831073-1ieYqA.jpg)

- **A.** The value of \(\oint \vec{B} \cdot d\vec{\ell}\) is greater for path \(C_1\) than for path \(C_2\), because path \(C_1\) is aligned with circular magnetic field lines while path \(C_2\) intersects field lines at non-perpendicular angles.
- **B.** The value of \(\oint \vec{B} \cdot d\vec{\ell}\) is greater for path \(C_2\) than for path \(C_1\), because the wire is closer to part of path \(C_2\), creating a stronger magnetic field along that segment.
- **C.** The value of \(\oint \vec{B} \cdot d\vec{\ell}\) is the same for both paths, but Ampère's law cannot easily be used to determine the magnitude of \(\vec{B}\) along path \(C_2\) because \(B\) is not uniform in magnitude along that path.
- **D.** The value of \(\oint \vec{B} \cdot d\vec{\ell}\) is the same for both paths, and Ampère's law can easily be used to determine the magnitude of \(\vec{B}\) at any point along path \(C_2\) because both paths enclose the exact same current.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118526/*