---
title: "A particle of mass \\(m\\) and positive charge \\(q\\) moves in a cyclotron accelerator within a uniform magnetic field of magnitude \\(B\\) directed perpendicular to its plane of motion. The particle accelerates twice per revolution across a gap between two semi-circular dees driven by an alternating potential difference of magnitude \\(\\Delta V\\). Assuming that the kinetic energy increases continuously over time, which of the following expressions represents the rate of change of the particle’s orbital radius with respect to time, \\(\\dfrac{dr}{dt}\\), as a function of its instantaneous orbital radius \\(r\\)?"
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url: "https://nerd-notes.com/ubq/118575/"
date_modified: "2026-08-04T08:11:24+00:00"
---

# A particle of mass \(m\) and positive charge \(q\) moves in a cyclotron accelerator within a uniform magnetic field of magnitude \(B\) directed perpendicular to its plane of motion. The particle accelerates twice per revolution across a gap between two semi-circular dees driven by an alternating potential difference of magnitude \(\Delta V\). Assuming that the kinetic energy increases continuously over time, which of the following expressions represents the rate of change of the particle’s orbital radius with respect to time, \(\dfrac{dr}{dt}\), as a function of its instantaneous orbital radius \(r\)?

A particle of mass \(m\) and positive charge \(q\) moves in a cyclotron accelerator within a uniform magnetic field of magnitude \(B\) directed perpendicular to its plane of motion. The particle accelerates twice per revolution across a gap between two semi-circular dees driven by an alternating potential difference of magnitude \(\Delta V\). Assuming that the kinetic energy increases continuously over time, which of the following expressions represents the rate of change of the particle's orbital radius with respect to time, \(\dfrac{dr}{dt}\), as a function of its instantaneous orbital radius \(r\)?

![A top-down view of a cyclotron accelerator consisting of two large semi-circular hollow metal regions labeled Dee 1 on the left and Dee 2 on the right, separated by a narrow vertical gap. In the center of the gap, a small solid circle represents a positively charged particle labeled q. A dashed line forms a continuous outward-spiraling path starting from the center dot and looping clockwise through both dees with increasing radius. At one point on the spiral path inside Dee 2, a solid arrow labeled v points tangentially along the path, and a radial dashed line extends from the center to that point labeled r. Four circle-dot symbols labeled \vec{B} are spaced evenly across the dees to indicate a uniform magnetic field directed out of the page. Across the narrow vertical gap between the dees, an AC voltage source symbol is connected at the bottom, labeled \Delta V. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831084-n4Y9Yt.jpg)

- **A.** \(\dfrac{\Delta V}{2 \pi B r}\)
- **B.** \(\dfrac{\Delta V}{\pi B r}\)
- **C.** \(\dfrac{2 \Delta V}{\pi B r}\)
- **D.** \(\dfrac{2 \pi \Delta V}{B r}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118575/*