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title: "Two long, straight, parallel conducting wires separated by a distance \\(d\\) carry steady currents of equal magnitude \\(I\\) in the same direction. In the absence of an external field, wire 1 exerts an attractive magnetic force per unit length \\(f_0\\) directly on wire 2. A uniform external magnetic field \\(\\vec{B}_{\\text{ext}}\\) is subsequently established throughout the entire region, oriented perpendicular to the plane containing the wires. Although the total magnetic force measured on each wire changes, the mutual force per unit length that wire 1 exerts on wire 2 remains equal to \\(f_0\\). Which of the following provides the correct physical explanation for this observation?"
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url: "https://nerd-notes.com/ubq/124812/"
date_modified: "2026-09-28T14:11:25+00:00"
---

# Two long, straight, parallel conducting wires separated by a distance \(d\) carry steady currents of equal magnitude \(I\) in the same direction. In the absence of an external field, wire 1 exerts an attractive magnetic force per unit length \(f_0\) directly on wire 2. A uniform external magnetic field \(\vec{B}_{\text{ext}}\) is subsequently established throughout the entire region, oriented perpendicular to the plane containing the wires. Although the total magnetic force measured on each wire changes, the mutual force per unit length that wire 1 exerts on wire 2 remains equal to \(f_0\). Which of the following provides the correct physical explanation for this observation?

Two long, straight, parallel conducting wires separated by a distance \(d\) carry steady currents of equal magnitude \(I\) in the same direction. In the absence of an external field, wire 1 exerts an attractive magnetic force per unit length \(f_0\) directly on wire 2. A uniform external magnetic field \(\vec{B}_{\text{ext}}\) is subsequently established throughout the entire region, oriented perpendicular to the plane containing the wires. Although the total magnetic force measured on each wire changes, the mutual force per unit length that wire 1 exerts on wire 2 remains equal to \(f_0\). Which of the following provides the correct physical explanation for this observation?

![Two vertical parallel lines representing two long wires, separated horizontally by a distance marked with a double-headed arrow labeled d. The left wire is labeled Wire 1 and has an upward vertical arrow alongside it labeled I. The right wire is labeled Wire 2 and has an upward vertical arrow alongside it labeled I. Between and around the two wires is a grid of six small encircled crosses arranged in two horizontal rows of three, labeled \(\vec{B}_{\text{ext}}\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604685-F02lQ6.jpg)

- **A.** Because magnetic fields obey the principle of linear superposition, the field generated by wire 1 at the location of wire 2 depends solely on wire 1's current and the separation distance; the external field contributes an independent Lorentz force to each conductor without modifying the field produced by either wire.
- **B.** According to Ampère's law, the line integral of the magnetic field around a closed path enclosing only one wire depends exclusively on the current inside that path; consequently, the external magnetic field cannot penetrate the space between the wires and leaves the local interaction field unchanged.
- **C.** Because both conductors carry identical currents in the same direction, the uniform external magnetic field exerts equal-magnitude forces in opposite directions on the two wires; these opposing external forces balance each other across the pair, preserving the net attractive interaction between them.
- **D.** Because magnetic forces act perpendicularly to charge velocity and do no work, the external magnetic field cannot transfer energy into the two-wire system; therefore, the spatial gradient of the system's magnetic potential energy that determines the mutual force remains strictly invariant.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124812/*