---
title: "A conducting rod of mass \\(m\\) and length \\(\\ell\\) moves without friction along two parallel horizontal conducting rails separated by a distance \\(\\ell\\) in a uniform, vertical magnetic field of magnitude \\(B\\). A resistor of resistance \\(R\\) connects the rails, and a constant external horizontal force of magnitude \\(F_0\\) pulls the rod away from the resistor, yielding the equation of motion \\(m \\dfrac{dv}{dt} = F_0 – \\dfrac{B^2 \\ell^2}{R}v\\). As the rod accelerates from rest, its speed approaches a terminal speed \\(v_T\\). Which of the following statements correctly describes the physical behavior of the system in the specified limit?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/124999/"
date_modified: "2026-09-28T14:13:04+00:00"
---

# A conducting rod of mass \(m\) and length \(\ell\) moves without friction along two parallel horizontal conducting rails separated by a distance \(\ell\) in a uniform, vertical magnetic field of magnitude \(B\). A resistor of resistance \(R\) connects the rails, and a constant external horizontal force of magnitude \(F_0\) pulls the rod away from the resistor, yielding the equation of motion \(m \dfrac{dv}{dt} = F_0 – \dfrac{B^2 \ell^2}{R}v\). As the rod accelerates from rest, its speed approaches a terminal speed \(v_T\). Which of the following statements correctly describes the physical behavior of the system in the specified limit?

A conducting rod of mass \(m\) and length \(\ell\) moves without friction along two parallel horizontal conducting rails separated by a distance \(\ell\) in a uniform, vertical magnetic field of magnitude \(B\). A resistor of resistance \(R\) connects the rails, and a constant external horizontal force of magnitude \(F_0\) pulls the rod away from the resistor, yielding the equation of motion \(m \dfrac{dv}{dt} = F_0 - \dfrac{B^2 \ell^2}{R}v\). As the rod accelerates from rest, its speed approaches a terminal speed \(v_T\). Which of the following statements correctly describes the physical behavior of the system in the specified limit?

![A top-down schematic of two horizontal parallel conducting lines separated by a vertical distance labeled \(\ell\). A vertical zigzag resistor labeled \(R\) connects the left ends of the two horizontal rails. A vertical solid rectangular bar labeled \(m\) rests across the rails to the right of the resistor. A single horizontal arrow originating from the center of the bar points to the right and is labeled \(F_0\). A regular grid of small 'x' symbols across the interior area indicates a uniform magnetic field labeled \(\vec{B}\) directed into the page. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604784-AQGa2x.jpg)

- **A.** As \(B \to 0\), the terminal speed \(v_T \to 0\) because the induced motional EMF decreases to zero, eliminating the current needed to sustain motion.
- **B.** As \(R \to 0\), the terminal speed \(v_T \to \infty\) because the loop resistance vanishes, allowing the induced current to accelerate the rod indefinitely.
- **C.** As \(R \to 0\), the terminal speed \(v_T \to 0\) because an infinitesimal speed induces the finite current necessary to produce a magnetic force that balances \(F_0\).
- **D.** As \(B \to 0\), the terminal speed \(v_T\) approaches a finite non-zero value because the rate of Joule heating balances the mechanical power input at a constant speed.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124999/*