---
title: "Four long, straight parallel wires perpendicular to the page carry steady currents: Wire 1 carries current \\(3I_0\\) directed out of the page, Wire 2 carries current \\(I_0\\) directed into the page, Wire 3 carries current \\(2I_0\\) directed into the page, and Wire 4 carries current \\(2I_0\\) directed out of the page.  Three closed paths lie in the plane of the page and are each traversed counterclockwise: path \\(C_1\\) encloses only Wires 1 and 2, path \\(C_2\\) encloses only Wires 2, 3, and 4, and path \\(C_3\\) encloses only Wires 1, 2, and 3.  Which row in the table correctly identifies the value of the line integral \\(\\oint \\vec{B} \\cdot d\\vec{\\ell}\\) along each path?  | | Path \\(C_1\\) | Path \\(C_2\\) | Path \\(C_3\\) | | :— | :— | :— | :— | | Row A | \\(-2\\mu_0 I_0\\) | \\(+\\mu_0 I_0\\) | \\(0\\) | | Row B | \\(+2\\mu_0 I_0\\) | \\(-\\mu_0 I_0\\) | \\(0\\) | | Row C | \\(+4\\mu_0 I_0\\) | \\(+5\\mu_0 I_0\\) | \\(+6\\mu_0 I_0\\) | | Row D | \\(+2\\mu_0 I_0\\) | \\(-\\mu_0 I_0\\) | \\(+2\\mu_0 I_0\\) |"
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url: "https://nerd-notes.com/ubq/124982/"
date_modified: "2026-09-28T14:12:30+00:00"
---

# Four long, straight parallel wires perpendicular to the page carry steady currents: Wire 1 carries current \(3I_0\) directed out of the page, Wire 2 carries current \(I_0\) directed into the page, Wire 3 carries current \(2I_0\) directed into the page, and Wire 4 carries current \(2I_0\) directed out of the page.

Three closed paths lie in the plane of the page and are each traversed counterclockwise: path \(C_1\) encloses only Wires 1 and 2, path \(C_2\) encloses only Wires 2, 3, and 4, and path \(C_3\) encloses only Wires 1, 2, and 3.

Which row in the table correctly identifies the value of the line integral \(\oint \vec{B} \cdot d\vec{\ell}\) along each path?

| | Path \(C_1\) | Path \(C_2\) | Path \(C_3\) |
| :— | :— | :— | :— |
| Row A | \(-2\mu_0 I_0\) | \(+\mu_0 I_0\) | \(0\) |
| Row B | \(+2\mu_0 I_0\) | \(-\mu_0 I_0\) | \(0\) |
| Row C | \(+4\mu_0 I_0\) | \(+5\mu_0 I_0\) | \(+6\mu_0 I_0\) |
| Row D | \(+2\mu_0 I_0\) | \(-\mu_0 I_0\) | \(+2\mu_0 I_0\) |

Four long, straight parallel wires perpendicular to the page carry steady currents: Wire 1 carries current \(3I_0\) directed out of the page, Wire 2 carries current \(I_0\) directed into the page, Wire 3 carries current \(2I_0\) directed into the page, and Wire 4 carries current \(2I_0\) directed out of the page.

Three closed paths lie in the plane of the page and are each traversed counterclockwise: path \(C_1\) encloses only Wires 1 and 2, path \(C_2\) encloses only Wires 2, 3, and 4, and path \(C_3\) encloses only Wires 1, 2, and 3.

Which row in the table correctly identifies the value of the line integral \(\oint \vec{B} \cdot d\vec{\ell}\) along each path?

| | Path \(C_1\) | Path \(C_2\) | Path \(C_3\) |
| :--- | :--- | :--- | :--- |
| Row A | \(-2\mu_0 I_0\) | \(+\mu_0 I_0\) | \(0\) |
| Row B | \(+2\mu_0 I_0\) | \(-\mu_0 I_0\) | \(0\) |
| Row C | \(+4\mu_0 I_0\) | \(+5\mu_0 I_0\) | \(+6\mu_0 I_0\) |
| Row D | \(+2\mu_0 I_0\) | \(-\mu_0 I_0\) | \(+2\mu_0 I_0\) |

![A top-down view shows four circular wire cross-sections arranged horizontally in a plane. From left to right: the first circle contains a central black dot and is labeled \(3I_0\); the second circle contains an interior cross and is labeled \(I_0\); the third circle contains an interior cross and is labeled \(2I_0\); the fourth circle contains a central black dot and is labeled \(2I_0\). Three distinct closed loops surround subsets of these wires. A solid oval loop labeled \(C_1\) encloses the first and second circles, with a single counterclockwise arrowhead on its top boundary. A dashed oval loop labeled \(C_2\) encloses the second, third, and fourth circles, with a single counterclockwise arrowhead on its top boundary. A dotted oval loop labeled \(C_3\) encloses the first, second, and third circles, with a single counterclockwise arrowhead on its bottom boundary. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604750-rtzvnU.jpg)

- **A.** Row A
- **B.** Row B
- **C.** Row C
- **D.** Row D

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124982/*