---
title: "A particle with mass \\(m\\) and positive charge \\(q\\) enters a uniform magnetic field \\(\\vec{B}\\) with velocity \\(\\vec{v}\\). The velocity vector makes an angle \\(\\theta\\) (where \\(0 < \\theta < \\dfrac{\\pi}{2}\\)) relative to the direction of \\(\\vec{B}\\), causing the particle to execute a helical trajectory. Which of the following expressions correctly represents the radius \\(R\\) of the particle's helical path, and by what factor does the pitch \\(p\\) (the distance traveled parallel to \\(\\vec{B}\\) in one full revolution) change if the magnetic field magnitude is doubled?"
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url: "https://nerd-notes.com/ubq/118532/"
date_modified: "2026-08-04T08:11:14+00:00"
---

# A particle with mass \(m\) and positive charge \(q\) enters a uniform magnetic field \(\vec{B}\) with velocity \(\vec{v}\). The velocity vector makes an angle \(\theta\) (where \(0 < \theta < \dfrac{\pi}{2}\)) relative to the direction of \(\vec{B}\), causing the particle to execute a helical trajectory. Which of the following expressions correctly represents the radius \(R\) of the particle's helical path, and by what factor does the pitch \(p\) (the distance traveled parallel to \(\vec{B}\) in one full revolution) change if the magnetic field magnitude is doubled?

A particle with mass \(m\) and positive charge \(q\) enters a uniform magnetic field \(\vec{B}\) with velocity \(\vec{v}\). The velocity vector makes an angle \(\theta\) (where \(0 < \theta < \dfrac{\pi}{2}\)) relative to the direction of \(\vec{B}\), causing the particle to execute a helical trajectory. Which of the following expressions correctly represents the radius \(R\) of the particle's helical path, and by what factor does the pitch \(p\) (the distance traveled parallel to \(\vec{B}\) in one full revolution) change if the magnetic field magnitude is doubled?

![A three-dimensional spatial diagram showing a uniform magnetic field \(\vec{B}\) directed along the positive z-axis, represented by three vertical parallel blue arrows pointing straight upward, each labeled \(\vec{B}\). A particle with charge \(q\) follows a coiled helical trajectory spiraling upward around a vertical central dashed line along the z-axis. A green velocity vector arrow labeled \(\vec{v}\) originates from the particle at the start of the path, angled upward and to the right at an angle labeled \(\theta\) relative to the vertical z-axis. Horizontal dashed lines mark one complete turn of the helix, with a vertical double-headed arrow between two adjacent turns labeled pitch \(p\). A horizontal circular radius arrow from the vertical dashed axis to the helical path is labeled \(R\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831074-sRAkhP.jpg)

- **A.** Radius \(R = \dfrac{m v \sin\theta}{q B}\) ; Pitch Factor = \(\dfrac{1}{2}\)
- **B.** Radius \(R = \dfrac{m v \cos\theta}{q B}\) ; Pitch Factor = \(\dfrac{1}{2}\)
- **C.** Radius \(R = \dfrac{m v \sin\theta}{q B}\) ; Pitch Factor = \(2\)
- **D.** Radius \(R = \dfrac{m v}{q B}\) ; Pitch Factor = \(1\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118532/*