---
title: "A thin, straight conducting rod of length \\(L\\) is pivoted at one end and rotates with a constant angular speed \\(\\omega\\) in a plane perpendicular to a magnetic field. The magnetic field is directed into the page and has a non-uniform magnitude that varies with the distance \\(r\\) from the pivot according to \\(B(r) = B_0 \\left(1 + \\dfrac{r}{L}\\right)\\), where \\(B_0\\) is a positive constant. Which of the following expressions represents the magnitude of the induced electromotive force \\(\\mathcal{E}\\) between the two ends of the rod?"
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url: "https://nerd-notes.com/ubq/121329/"
date_modified: "2026-08-23T04:59:44+00:00"
---

# A thin, straight conducting rod of length \(L\) is pivoted at one end and rotates with a constant angular speed \(\omega\) in a plane perpendicular to a magnetic field. The magnetic field is directed into the page and has a non-uniform magnitude that varies with the distance \(r\) from the pivot according to \(B(r) = B_0 \left(1 + \dfrac{r}{L}\right)\), where \(B_0\) is a positive constant. Which of the following expressions represents the magnitude of the induced electromotive force \(\mathcal{E}\) between the two ends of the rod?

A thin, straight conducting rod of length \(L\) is pivoted at one end and rotates with a constant angular speed \(\omega\) in a plane perpendicular to a magnetic field. The magnetic field is directed into the page and has a non-uniform magnitude that varies with the distance \(r\) from the pivot according to \(B(r) = B_0 \left(1 + \dfrac{r}{L}\right)\), where \(B_0\) is a positive constant. Which of the following expressions represents the magnitude of the induced electromotive force \(\mathcal{E}\) between the two ends of the rod?

![A horizontal circular path is indicated by a dashed arc in the plane of the page. A thin straight line representing a conducting rod of length \(L\) extends from a small solid circular pivot at the center of the arc toward the upper-right at an angle. A curved arrow near the pivot indicates rotation with angular speed \(\omega\) in the counterclockwise direction. An array of eight \(\times\) symbols distributed across the region indicates a magnetic field directed into the page, with the symbols spaced more densely at greater distances from the pivot. A label \(L\) with a two-headed dimension arrow indicates the total length of the rod from the pivot to the outer tip. The magnetic field is labeled \(B(r)\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461184-fNryai.jpg)

- **A.** \(\dfrac{1}{2} B_0 \omega L^2\)
- **B.** \(\dfrac{2}{3} B_0 \omega L^2\)
- **C.** \(\dfrac{3}{4} B_0 \omega L^2\)
- **D.** \(\dfrac{5}{6} B_0 \omega L^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121329/*