---
title: "A conducting bar of mass \\(m\\), length \\(\\ell\\), and negligible electrical resistance rests on two parallel, horizontal, frictionless conducting rails separated by a distance \\(\\ell\\). The rails are connected at one end by a resistor of resistance \\(R\\), and the rails themselves have negligible resistance. A uniform magnetic field of magnitude \\(B\\) is directed perpendicular to the plane of the rails, pointing out of the page. A constant horizontal pulling force of magnitude \\(F_0\\) is applied to the bar, causing it to accelerate to the right until it reaches a constant terminal speed \\(v_T\\). What is the rate at which thermal energy is dissipated in the resistor when the bar moves at speed \\(v_T\\)?"
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url: "https://nerd-notes.com/ubq/118727/"
date_modified: "2026-08-04T08:13:55+00:00"
---

# A conducting bar of mass \(m\), length \(\ell\), and negligible electrical resistance rests on two parallel, horizontal, frictionless conducting rails separated by a distance \(\ell\). The rails are connected at one end by a resistor of resistance \(R\), and the rails themselves have negligible resistance. A uniform magnetic field of magnitude \(B\) is directed perpendicular to the plane of the rails, pointing out of the page. A constant horizontal pulling force of magnitude \(F_0\) is applied to the bar, causing it to accelerate to the right until it reaches a constant terminal speed \(v_T\). What is the rate at which thermal energy is dissipated in the resistor when the bar moves at speed \(v_T\)?

A conducting bar of mass \(m\), length \(\ell\), and negligible electrical resistance rests on two parallel, horizontal, frictionless conducting rails separated by a distance \(\ell\). The rails are connected at one end by a resistor of resistance \(R\), and the rails themselves have negligible resistance. A uniform magnetic field of magnitude \(B\) is directed perpendicular to the plane of the rails, pointing out of the page. A constant horizontal pulling force of magnitude \(F_0\) is applied to the bar, causing it to accelerate to the right until it reaches a constant terminal speed \(v_T\). What is the rate at which thermal energy is dissipated in the resistor when the bar moves at speed \(v_T\)?

![A top-down view of two parallel horizontal rails separated by vertical distance \ell. The left end of the top rail connects to the top terminal of a resistor labeled R, and the left end of the bottom rail connects to the bottom terminal of R, forming a closed loop on the left. A vertical straight bar of length \ell rests perpendicularly across both rails. An arrow labeled F_0 originates at the center of the bar and points horizontally to the right. An arrow labeled v originates at the top of the bar and points horizontally to the right. Six small circles with central dots, arranged in a two-by-three grid around the bar, represent a uniform magnetic field directed out of the page, labeled B. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831235-7YcGYg.jpg)

- **A.** \(\dfrac{F_0^2 R}{4 B^2 \ell^2}\)
- **B.** \(\dfrac{F_0^2 R}{2 B^2 \ell^2}\)
- **C.** \(\dfrac{F_0^2 R}{B^2 \ell^2}\)
- **D.** \(\dfrac{2 F_0^2 R}{B^2 \ell^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118727/*