---
title: "A particle with rest mass \\(m\\) and positive charge \\(q\\) travels at constant speed \\(v\\) in a circular orbit perpendicular to a uniform magnetic field of magnitude \\(B\\).  The magnitude of the particle’s relativistic momentum is given by \\(p = \\gamma m v\\), where \\(\\gamma = \\dfrac{1}{\\sqrt{1 – \\dfrac{v^2}{c^2}}}\\) and \\(c\\) is the speed of light, resulting in a centripetal force of magnitude \\(F_c = \\dfrac{\\gamma m v^2}{R}\\), where \\(R\\) is the orbital radius.  Which of the following is a correct expression for the orbital period \\(T\\) of the particle?"
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url: "https://nerd-notes.com/ubq/121254/"
date_modified: "2026-08-23T04:59:02+00:00"
---

# A particle with rest mass \(m\) and positive charge \(q\) travels at constant speed \(v\) in a circular orbit perpendicular to a uniform magnetic field of magnitude \(B\).

The magnitude of the particle’s relativistic momentum is given by \(p = \gamma m v\), where \(\gamma = \dfrac{1}{\sqrt{1 – \dfrac{v^2}{c^2}}}\) and \(c\) is the speed of light, resulting in a centripetal force of magnitude \(F_c = \dfrac{\gamma m v^2}{R}\), where \(R\) is the orbital radius.

Which of the following is a correct expression for the orbital period \(T\) of the particle?

A particle with rest mass \(m\) and positive charge \(q\) travels at constant speed \(v\) in a circular orbit perpendicular to a uniform magnetic field of magnitude \(B\).

The magnitude of the particle's relativistic momentum is given by \(p = \gamma m v\), where \(\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}}\) and \(c\) is the speed of light, resulting in a centripetal force of magnitude \(F_c = \dfrac{\gamma m v^2}{R}\), where \(R\) is the orbital radius.

Which of the following is a correct expression for the orbital period \(T\) of the particle?

![A circular dashed trajectory of radius \(R\) lies in the plane of the page. At the top of the circular path, a small solid circle representing a particle labeled \(+q\) has a straight horizontal velocity vector arrow pointing to the right, labeled \(v\). Exactly four cross symbols indicating a uniform magnetic field \(\vec{B}\) directed into the page are arranged symmetrically around the orbit. A single straight arrow extends from the center of the circle to the orbit path toward the upper-right at 45 degrees, labeled \(R\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461141-4smb1M.jpg)

- **A.** \(\dfrac{\pi m}{q B \sqrt{1 - \dfrac{v^2}{c^2}}}\)
- **B.** \(\dfrac{2\pi m \sqrt{1 - \dfrac{v^2}{c^2}}}{q B}\)
- **C.** \(\dfrac{2\pi m}{q B \left(1 - \dfrac{v^2}{c^2}\right)}\)
- **D.** \(\dfrac{2\pi m}{q B \sqrt{1 - \dfrac{v^2}{c^2}}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121254/*