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title: "A thin, uniform conducting disk of radius \\(R\\) rotates in the \\(xy\\)-plane about a fixed vertical axis through its center with a constant angular speed \\(\\omega\\). A non-uniform magnetic field perpendicular to the plane of the disk is given by \\(\\vec{B}(r) = B_0 \\left(\\dfrac{r}{R}\\right)^2 \\hat{k}\\), where \\(r\\) is the radial distance from the axis of rotation and \\(B_0\\) is a positive constant. Which of the following expressions represents the magnitude of the induced electromotive force (EMF) between the center of the disk and its outer rim?"
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url: "https://nerd-notes.com/ubq/118760/"
date_modified: "2026-08-04T08:14:21+00:00"
---

# A thin, uniform conducting disk of radius \(R\) rotates in the \(xy\)-plane about a fixed vertical axis through its center with a constant angular speed \(\omega\). A non-uniform magnetic field perpendicular to the plane of the disk is given by \(\vec{B}(r) = B_0 \left(\dfrac{r}{R}\right)^2 \hat{k}\), where \(r\) is the radial distance from the axis of rotation and \(B_0\) is a positive constant. Which of the following expressions represents the magnitude of the induced electromotive force (EMF) between the center of the disk and its outer rim?

A thin, uniform conducting disk of radius \(R\) rotates in the \(xy\)-plane about a fixed vertical axis through its center with a constant angular speed \(\omega\). A non-uniform magnetic field perpendicular to the plane of the disk is given by \(\vec{B}(r) = B_0 \left(\dfrac{r}{R}\right)^2 \hat{k}\), where \(r\) is the radial distance from the axis of rotation and \(B_0\) is a positive constant. Which of the following expressions represents the magnitude of the induced electromotive force (EMF) between the center of the disk and its outer rim?

![A circular conducting disk of radius R lies flat in the xy-plane, centered at the origin. A vertical z-axis extends upward through the center of the disk, with a curved arrow around the z-axis indicating rotation at constant angular velocity \omega. A radial line segment extends from the origin to the outer rim of the disk, labeled R. Several upward-pointing magnetic field vectors \vec{B}(r) pass vertically through the disk, with shorter vectors near the center and longer vectors near the outer rim to represent the radial increase in field magnitude. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831260-PaJILI.jpg)

- **A.** \(\dfrac{1}{8} B_0 \omega R^2\)
- **B.** \(\dfrac{1}{6} B_0 \omega R^2\)
- **C.** \(\dfrac{1}{5} B_0 \omega R^2\)
- **D.** \(\dfrac{1}{4} B_0 \omega R^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118760/*