---
title: "A long, straight solid cylindrical wire of radius \\(R\\) carries a steady current \\(I_0\\) that is uniformly distributed across its cross section. A student considers a circular Amperian loop of radius \\(r\\) centered on the axis of the wire, where \\(r > R\\). If the radius of the loop is increased from \\(r = 2R\\) to \\(r = 4R\\), which of the following correctly describes what happens to the value of the line integral \\(\\oint \\vec{B} \\cdot d\\vec{\\ell}\\) around the loop and gives the correct justification?"
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url: "https://nerd-notes.com/ubq/121206/"
date_modified: "2026-08-23T04:58:51+00:00"
---

# A long, straight solid cylindrical wire of radius \(R\) carries a steady current \(I_0\) that is uniformly distributed across its cross section. A student considers a circular Amperian loop of radius \(r\) centered on the axis of the wire, where \(r > R\). If the radius of the loop is increased from \(r = 2R\) to \(r = 4R\), which of the following correctly describes what happens to the value of the line integral \(\oint \vec{B} \cdot d\vec{\ell}\) around the loop and gives the correct justification?

A long, straight solid cylindrical wire of radius \(R\) carries a steady current \(I_0\) that is uniformly distributed across its cross section. A student considers a circular Amperian loop of radius \(r\) centered on the axis of the wire, where \(r > R\). If the radius of the loop is increased from \(r = 2R\) to \(r = 4R\), which of the following correctly describes what happens to the value of the line integral \(\oint \vec{B} \cdot d\vec{\ell}\) around the loop and gives the correct justification?

![A circular cross section of a cylindrical wire centered at the origin, with a solid outline and light gray fill, labeled with a radius arrow from the center to the right edge labeled R. At the center of the circle, an encircled cross represents current flowing into the page, labeled I_0. A concentric dashed circular path of larger radius surrounds the wire, with a radius arrow extending from the center upward at an angle of 45 degrees to the dashed path, labeled r. An arrow on the dashed path indicates a clockwise orientation for the path element d\vec{\ell}. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461131-SlZrnV.jpg)

- **A.** The line integral is halved because the magnetic field magnitude \(B\) decreases inversely with distance \(r\).
- **B.** The line integral remains unchanged because the total current enclosed by the loop is equal to \(I_0\) for any \(r > R\).
- **C.** The line integral doubles because the circumference of the integration path is proportional to \(r\).
- **D.** The line integral quadruples because the cross-sectional area enclosed by the loop increases by a factor of \(4\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121206/*