---
title: "A particle of mass \\(m\\) and positive charge \\(q\\) enters a uniform magnetic field \\(\\vec{B} = B_0 \\hat{k}\\) with an initial velocity \\(\\vec{v}_0\\) directed at an angle \\(\\theta_0\\) (where \\(0 < \\theta_0 < \\pi/2\\)) relative to the positive \\(z\\)-axis. In addition to the magnetic force, the particle experiences a linear drag force \\(\\vec{F}_D = -b \\vec{v}\\), where \\(b\\) is a positive constant. Which of the following pairs correctly describes the pitch angle \\(\\theta(t)\\) (the angle between \\(\\vec{v}\\) and \\(\\vec{B}\\)) and the radius \\(R(t)\\) of the particle's helical path as functions of time \\(t\\)?"
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url: "https://nerd-notes.com/ubq/118541/"
date_modified: "2026-08-04T08:11:15+00:00"
---

# A particle of mass \(m\) and positive charge \(q\) enters a uniform magnetic field \(\vec{B} = B_0 \hat{k}\) with an initial velocity \(\vec{v}_0\) directed at an angle \(\theta_0\) (where \(0 < \theta_0 < \pi/2\)) relative to the positive \(z\)-axis. In addition to the magnetic force, the particle experiences a linear drag force \(\vec{F}_D = -b \vec{v}\), where \(b\) is a positive constant. Which of the following pairs correctly describes the pitch angle \(\theta(t)\) (the angle between \(\vec{v}\) and \(\vec{B}\)) and the radius \(R(t)\) of the particle's helical path as functions of time \(t\)?

A particle of mass \(m\) and positive charge \(q\) enters a uniform magnetic field \(\vec{B} = B_0 \hat{k}\) with an initial velocity \(\vec{v}_0\) directed at an angle \(\theta_0\) (where \(0 < \theta_0 < \pi/2\)) relative to the positive \(z\)-axis. In addition to the magnetic force, the particle experiences a linear drag force \(\vec{F}_D = -b \vec{v}\), where \(b\) is a positive constant. Which of the following pairs correctly describes the pitch angle \(\theta(t)\) (the angle between \(\vec{v}\) and \(\vec{B}\)) and the radius \(R(t)\) of the particle's helical path as functions of time \(t\)?

![A 3D Cartesian coordinate system with x, y, and z axes labeled. A uniform magnetic field B is represented by three vertical parallel upward-pointing vector arrows aligned with the positive z-axis. At the origin, a small sphere representing a positively charged particle has an initial velocity vector v_0 directed in the y-z plane at an angle theta_0 relative to the z-axis. Originating from the particle, a tightening helical trajectory spirals upward around the z-axis, where both the radius of the coils and the vertical distance between consecutive turns visibly shrink as z increases. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831075-8zzfNZ.jpg)

- **A.** \(\theta(t) = \theta_0\) (constant), and \(R(t) = \dfrac{m v_0 \sin\theta_0}{q B_0} e^{-bt/m}\)
- **B.** \(\theta(t)\) decreases toward \(0\), and \(R(t) = \dfrac{m v_0 \sin\theta_0}{q B_0} e^{-bt/m}\)
- **C.** \(\theta(t)\) increases toward \(\pi/2\), and \(R(t) = \dfrac{m v_0 \sin\theta_0}{q B_0} \left(1 - \dfrac{bt}{m}\right)\)
- **D.** \(\theta(t) = \theta_0\) (constant), and \(R(t) = \dfrac{m v_0 \sin\theta_0}{q B_0} \left(1 - \dfrac{bt}{m}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118541/*