---
title: "A horizontal circular ring of radius \\(r\\), mass \\(m\\), and electrical resistance \\(R\\) falls vertically under the influence of gravity through a region with a non-uniform vertical magnetic field. The vertical component of the magnetic field varies linearly with height \\(z\\) according to \\(B_z(z) = B_0 + bz\\), where \\(B_0\\) and \\(b\\) are positive constants, and \\(z\\) is measured upward. Air resistance is negligible. Which of the following expressions correctly represents the magnitude of the ring’s terminal velocity \\(v_T\\)?"
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url: "https://nerd-notes.com/ubq/118762/"
date_modified: "2026-08-04T08:14:28+00:00"
---

# A horizontal circular ring of radius \(r\), mass \(m\), and electrical resistance \(R\) falls vertically under the influence of gravity through a region with a non-uniform vertical magnetic field. The vertical component of the magnetic field varies linearly with height \(z\) according to \(B_z(z) = B_0 + bz\), where \(B_0\) and \(b\) are positive constants, and \(z\) is measured upward. Air resistance is negligible. Which of the following expressions correctly represents the magnitude of the ring’s terminal velocity \(v_T\)?

A horizontal circular ring of radius \(r\), mass \(m\), and electrical resistance \(R\) falls vertically under the influence of gravity through a region with a non-uniform vertical magnetic field. The vertical component of the magnetic field varies linearly with height \(z\) according to \(B_z(z) = B_0 + bz\), where \(B_0\) and \(b\) are positive constants, and \(z\) is measured upward. Air resistance is negligible. Which of the following expressions correctly represents the magnitude of the ring's terminal velocity \(v_T\)?

![A physical setup diagram showing a horizontal circular ring of radius r falling vertically along a vertical z-axis. The vertical z-axis is drawn as a dashed line pointing upward with an arrowhead at the top labeled z. The ring is drawn as an ellipse viewed at a slight tilt, with radius r labeled from its center to its right edge. Below the ring, a downward vertical arrow is labeled v. Vertical magnetic field lines pass through the ring, with arrows pointing upward; the density of the magnetic field lines decreases towards the bottom to indicate a vertical magnetic field gradient B_z(z) = B_0 + bz. The ring is labeled with mass m and resistance R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831268-rxKJmX.jpg)

- **A.** \(\dfrac{4mg R}{\pi^2 r^4 b^2}\)
- **B.** \(\dfrac{2mg R}{\pi^2 r^4 b^2}\)
- **C.** \(\dfrac{mg R}{\pi^2 r^4 b^2}\)
- **D.** \(\dfrac{mg R}{2\pi^2 r^4 b^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118762/*