---
title: "A conducting bar of length \\(\\ell\\) and resistance \\(R\\) slides horizontally with negligible friction along two parallel conducting rails of negligible resistance separated by a distance \\(\\ell\\). A uniform magnetic field \\(\\vec{B}\\) of magnitude \\(B\\) is directed perpendicularly into the page, and an ideal battery of emf \\(\\mathcal{E}\\) is connected across the left ends of the rails with its positive terminal at the top rail. An external agent pulls the bar to the right at a constant speed \\(v\\), where \\(v > \\dfrac{\\mathcal{E}}{B\\ell}\\). What is the mechanical power supplied by the external agent to maintain this motion?"
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url: "https://nerd-notes.com/ubq/121360/"
date_modified: "2026-08-23T04:59:58+00:00"
---

# A conducting bar of length \(\ell\) and resistance \(R\) slides horizontally with negligible friction along two parallel conducting rails of negligible resistance separated by a distance \(\ell\). A uniform magnetic field \(\vec{B}\) of magnitude \(B\) is directed perpendicularly into the page, and an ideal battery of emf \(\mathcal{E}\) is connected across the left ends of the rails with its positive terminal at the top rail. An external agent pulls the bar to the right at a constant speed \(v\), where \(v > \dfrac{\mathcal{E}}{B\ell}\). What is the mechanical power supplied by the external agent to maintain this motion?

A conducting bar of length \(\ell\) and resistance \(R\) slides horizontally with negligible friction along two parallel conducting rails of negligible resistance separated by a distance \(\ell\). A uniform magnetic field \(\vec{B}\) of magnitude \(B\) is directed perpendicularly into the page, and an ideal battery of emf \(\mathcal{E}\) is connected across the left ends of the rails with its positive terminal at the top rail. An external agent pulls the bar to the right at a constant speed \(v\), where \(v > \dfrac{\mathcal{E}}{B\ell}\). What is the mechanical power supplied by the external agent to maintain this motion?

![A schematic diagram showing two horizontal parallel straight rails connected at their left ends to form a circuit loop with a vertical line segment representing a conducting bar of length \(\ell\). On the left vertical boundary, an ideal battery symbol is drawn with the longer horizontal plate at the top and shorter plate at the bottom, labeled \(\mathcal{E}\). The vertical bar spans between the two horizontal rails and is positioned toward the right. A horizontal arrow labeled \(\vec{v}\) points to the right from the center of the vertical bar. Exactly eight small \(\times\) symbols representing a magnetic field directed into the page are arranged in two horizontal rows of four across the region, with a single label \(\vec{B}\) next to the top-right symbol. A vertical dimension line labeled \(\ell\) indicates the separation between the rails. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461198-5tgPLU.jpg)

- **A.** \(\dfrac{(B\ell v - \mathcal{E})^2}{R}\)
- **B.** \(\dfrac{(B\ell v)^2}{R}\)
- **C.** \(\dfrac{B\ell v(B\ell v + \mathcal{E})}{R}\)
- **D.** \(\dfrac{B\ell v(B\ell v - \mathcal{E})}{R}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121360/*