---
title: "A long, rigid horizontal wire carrying a steady current \\(I_1\\) is fixed in place. A second parallel, uniform horizontal wire of length \\(\\ell\\) and mass \\(m\\) carrying a steady current \\(I_2\\) is magnetically levitated in equilibrium at a height \\(h\\) directly above the fixed wire. The movable wire is constrained to move only along the vertical direction while remaining parallel to the fixed wire. Which of the following statements correctly characterizes the equilibrium of the movable wire with respect to small vertical displacements, and provides the correct justification?"
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url: "https://nerd-notes.com/ubq/124853/"
date_modified: "2026-09-28T14:11:41+00:00"
---

# A long, rigid horizontal wire carrying a steady current \(I_1\) is fixed in place. A second parallel, uniform horizontal wire of length \(\ell\) and mass \(m\) carrying a steady current \(I_2\) is magnetically levitated in equilibrium at a height \(h\) directly above the fixed wire. The movable wire is constrained to move only along the vertical direction while remaining parallel to the fixed wire. Which of the following statements correctly characterizes the equilibrium of the movable wire with respect to small vertical displacements, and provides the correct justification?

A long, rigid horizontal wire carrying a steady current \(I_1\) is fixed in place. A second parallel, uniform horizontal wire of length \(\ell\) and mass \(m\) carrying a steady current \(I_2\) is magnetically levitated in equilibrium at a height \(h\) directly above the fixed wire. The movable wire is constrained to move only along the vertical direction while remaining parallel to the fixed wire. Which of the following statements correctly characterizes the equilibrium of the movable wire with respect to small vertical displacements, and provides the correct justification?

![Two horizontal parallel line segments representing wires are shown in a vertical plane. The lower wire is drawn as a thick solid horizontal line fixed to two small hatched rectangular ground supports at its ends, with a horizontal rightward arrow along the wire labeled \(I_1\). Directly above the lower wire at a distance \(h\), the upper wire is drawn as a second thick solid horizontal line of the same length, with a horizontal leftward arrow along the wire labeled \(I_2\). A vertical two-headed dimension arrow between the centers of the two wires is labeled \(h\). A downward vertical arrow originating from the center of the upper wire is labeled \(mg\). A vertical dashed line passes through the centers of both wires, indicating the vertical axis of motion labeled \(y\). A downward arrow next to the lower right indicates the acceleration due to gravity, labeled \(g\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604700-fnYHcC.jpg)

- **A.** The equilibrium is unstable because an upward displacement decreases the upward magnetic force below the gravitational force, causing the resulting downward net force to accelerate the wire away from the equilibrium position.
- **B.** The equilibrium is unstable because the derivative of the net vertical force with respect to position is negative (\(\dfrac{dF_{\text{net}}}{dy} < 0\)), which indicates that any displacement produces a net force in the same direction as the displacement.
- **C.** The equilibrium is stable because the two wires carry parallel currents that repel each other, and an upward displacement decreases the upward magnetic force so that the constant downward gravitational force produces a net restoring force.
- **D.** The equilibrium is stable because the two wires carry antiparallel currents that repel each other, and an upward displacement decreases the upward magnetic force so that the constant downward gravitational force produces a net restoring force.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124853/*