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title: "Two long, straight, parallel wires carry steady currents \\(I_1\\) and \\(I_2\\) perpendicular to the \\(xy\\)-plane. An arbitrary closed Amperian loop \\(C\\) in the \\(xy\\)-plane encloses wire 1 but does not enclose wire 2. A student calculates the line integral around loop \\(C\\) and finds \\(\\oint_C \\vec{B} \\cdot d\\vec{\\ell} = \\mu_0 I_1\\). Which of the following best explains why current \\(I_2\\) does not affect the value of this line integral, and how \\(I_2\\) relates to the total magnetic field \\(\\vec{B}\\) along path \\(C\\)?"
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url: "https://nerd-notes.com/ubq/118509/"
date_modified: "2026-08-04T08:11:11+00:00"
---

# Two long, straight, parallel wires carry steady currents \(I_1\) and \(I_2\) perpendicular to the \(xy\)-plane. An arbitrary closed Amperian loop \(C\) in the \(xy\)-plane encloses wire 1 but does not enclose wire 2. A student calculates the line integral around loop \(C\) and finds \(\oint_C \vec{B} \cdot d\vec{\ell} = \mu_0 I_1\). Which of the following best explains why current \(I_2\) does not affect the value of this line integral, and how \(I_2\) relates to the total magnetic field \(\vec{B}\) along path \(C\)?

Two long, straight, parallel wires carry steady currents \(I_1\) and \(I_2\) perpendicular to the \(xy\)-plane. An arbitrary closed Amperian loop \(C\) in the \(xy\)-plane encloses wire 1 but does not enclose wire 2. A student calculates the line integral around loop \(C\) and finds \(\oint_C \vec{B} \cdot d\vec{\ell} = \mu_0 I_1\). Which of the following best explains why current \(I_2\) does not affect the value of this line integral, and how \(I_2\) relates to the total magnetic field \(\vec{B}\) along path \(C\)?

![A two-dimensional schematic in the xy-plane showing two parallel long straight wires perpendicular to the page. Wire 1 is on the left, shown as a small circle with a central dot labeled I_1. Wire 2 is on the right, shown as a small circle with an X labeled I_2. An irregularly shaped closed loop labeled C surrounds wire 1, with a counterclockwise direction arrow on the loop. Wire 2 is completely outside loop C. At a representative point on loop C, a single vector arrow labeled B_total points tangentially along the loop. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831071-yLO6n8.jpg)

- **A.** Current \(I_2\) contributes to the magnetic field \(\vec{B}\) at individual points on path \(C\), but its line integral \(\oint_C \vec{B}_2 \cdot d\vec{\ell}\) equals zero because the total angular displacement around wire 2 along closed path \(C\) is zero.
- **B.** Current \(I_2\) does not contribute to the magnetic field \(\vec{B}\) at points on path \(C\) because magnetic fields produced by external currents cannot penetrate the enclosed surface of an Amperian loop.
- **C.** Current \(I_2\) contributes to both the local field \(\vec{B}\) and the line integral \(\oint_C \vec{B} \cdot d\vec{\ell}\), but its contribution to the integral cancels out only if path \(C\) possesses high spatial symmetry centered on wire 1.
- **D.** The magnetic field vector \(\vec{B}\) appearing in Ampère's law represents strictly the field produced by enclosed currents, so wire 2 affects neither \(\vec{B}\) nor the line integral \(\oint_C \vec{B} \cdot d\vec{\ell}\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118509/*