---
title: "A conducting bar of mass \\(m\\), length \\(\\ell\\), and electrical resistance \\(R\\) rests on two long, horizontal, frictionless conducting rails of negligible resistance located in a uniform, vertical magnetic field of magnitude \\(B\\). The rails are connected at one end by a wire of negligible resistance to form a closed circuit. As the bar moves, it experiences an ambient fluid drag force \\(\\vec{F}_{\\text{drag}} = -b\\vec{v}\\), where \\(b\\) is a positive constant and \\(\\vec{v}\\) is the velocity of the bar. At time \\(t = 0\\), the bar is launched along the rails with an initial speed \\(v_0\\). Which of the following expressions correctly represents the speed \\(v(t)\\) of the bar as a function of time \\(t\\)?"
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/118734/"
date_modified: "2026-08-04T08:14:01+00:00"
---

# A conducting bar of mass \(m\), length \(\ell\), and electrical resistance \(R\) rests on two long, horizontal, frictionless conducting rails of negligible resistance located in a uniform, vertical magnetic field of magnitude \(B\). The rails are connected at one end by a wire of negligible resistance to form a closed circuit. As the bar moves, it experiences an ambient fluid drag force \(\vec{F}_{\text{drag}} = -b\vec{v}\), where \(b\) is a positive constant and \(\vec{v}\) is the velocity of the bar. At time \(t = 0\), the bar is launched along the rails with an initial speed \(v_0\). Which of the following expressions correctly represents the speed \(v(t)\) of the bar as a function of time \(t\)?

A conducting bar of mass \(m\), length \(\ell\), and electrical resistance \(R\) rests on two long, horizontal, frictionless conducting rails of negligible resistance located in a uniform, vertical magnetic field of magnitude \(B\). The rails are connected at one end by a wire of negligible resistance to form a closed circuit. As the bar moves, it experiences an ambient fluid drag force \(\vec{F}_{\text{drag}} = -b\vec{v}\), where \(b\) is a positive constant and \(\vec{v}\) is the velocity of the bar. At time \(t = 0\), the bar is launched along the rails with an initial speed \(v_0\). Which of the following expressions correctly represents the speed \(v(t)\) of the bar as a function of time \(t\)?

![A top-down view of two parallel horizontal conducting rails oriented left-to-right, separated by a vertical distance \ell. A straight vertical wire connects the left ends of the two rails. A straight conducting bar of length \ell rests vertically across the rails, labeled 'Bar (mass m, resistance R)'. An arrow labeled v_0 originates from the bar and points to the right. A set of four magnetic field symbols, each drawn as an 'x' enclosed in a small circle and labeled \vec{B}, are arranged in a grid surrounding the rails to indicate a field directed into the page. An arrow labeled \vec{F}_{\text{drag}} originates from the bar and points to the left. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831241-8mhUWH.jpg)

- **A.** \(v(t) = v_0 \left(1 - \dfrac{b R + B^2 \ell^2}{m R} t\right)\)
- **B.** \(v(t) = \dfrac{v_0}{1 + \left(\dfrac{b R + B^2 \ell^2}{m R}\right) t}\)
- **C.** \(v(t) = v_0 \exp\left(-\dfrac{b R + B^2 \ell^2}{m R} t\right)\)
- **D.** \(v(t) = v_0 \exp\left(-\dfrac{2(b R + B^2 \ell^2)}{m R} t\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118734/*