---
title: "A particle with mass \\(m\\) and positive charge \\(q\\) is launched with speed \\(v\\) into a region of uniform magnetic field of magnitude \\(B\\). The velocity vector of the particle makes an angle \\(\\theta\\) with the direction of the magnetic field, where \\(0 < \\theta < \\dfrac{\\pi}{2}\\). The particle undergoes helical motion characterized by a trajectory radius \\(R\\) and a pitch \\(p\\), defined as the axial distance traveled parallel to the magnetic field during one complete revolution. Which of the following is a correct expression for the ratio \\(\\dfrac{p}{R}\\) of the pitch to the radius of the trajectory?"
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url: "https://nerd-notes.com/ubq/124848/"
date_modified: "2026-09-28T14:11:39+00:00"
---

# A particle with mass \(m\) and positive charge \(q\) is launched with speed \(v\) into a region of uniform magnetic field of magnitude \(B\). The velocity vector of the particle makes an angle \(\theta\) with the direction of the magnetic field, where \(0 < \theta < \dfrac{\pi}{2}\). The particle undergoes helical motion characterized by a trajectory radius \(R\) and a pitch \(p\), defined as the axial distance traveled parallel to the magnetic field during one complete revolution. Which of the following is a correct expression for the ratio \(\dfrac{p}{R}\) of the pitch to the radius of the trajectory?

A particle with mass \(m\) and positive charge \(q\) is launched with speed \(v\) into a region of uniform magnetic field of magnitude \(B\). The velocity vector of the particle makes an angle \(\theta\) with the direction of the magnetic field, where \(0 < \theta < \dfrac{\pi}{2}\). The particle undergoes helical motion characterized by a trajectory radius \(R\) and a pitch \(p\), defined as the axial distance traveled parallel to the magnetic field during one complete revolution. Which of the following is a correct expression for the ratio \(\dfrac{p}{R}\) of the pitch to the radius of the trajectory?

![A coordinate plane showing a uniform magnetic field represented by four parallel horizontal arrows pointing to the right, with a label \vec{B} near the top arrow. At the left side, a small solid circle represents a particle with a label +q immediately to its left. An arrow labeled \vec{v} originates at the center of the solid circle and extends upward and to the right into the magnetic field region. A horizontal dashed reference line extends to the right from the center of the particle, parallel to the magnetic field arrows. An arc with an angle label \theta is drawn between the horizontal dashed reference line and the arrow \vec{v}. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604699-zl6fiL.jpg)

- **A.** \(\dfrac{2\pi}{\tan\theta}\)
- **B.** \(2\pi\tan\theta\)
- **C.** \(2\pi\cos\theta\)
- **D.** \(\dfrac{2\pi}{\sin\theta}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124848/*