---
title: "An electron of mass \\(m\\) and charge \\(-e\\) is injected at the origin \\((0,0)\\) into the \\(xy\\)-plane with an initial velocity \\(\\vec{v}_0 = v_0 \\hat{i}\\). In the region \\(x \\ge 0\\), a non-uniform magnetic field is directed out of the page, \\(\\vec{B} = (B_0 + \\beta x)\\hat{k}\\), where \\(B_0\\) and \\(\\beta\\) are positive constants. The dashed curves in the graphs represent a reference circular path of radius \\(R_0 = \\dfrac{m v_0}{e B_0}\\) corresponding to motion in a uniform magnetic field of magnitude \\(B_0\\). Which of the following graphs best represents the trajectory of the electron in the \\(xy\\)-plane?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/124930/"
date_modified: "2026-09-28T14:12:03+00:00"
---

# An electron of mass \(m\) and charge \(-e\) is injected at the origin \((0,0)\) into the \(xy\)-plane with an initial velocity \(\vec{v}_0 = v_0 \hat{i}\). In the region \(x \ge 0\), a non-uniform magnetic field is directed out of the page, \(\vec{B} = (B_0 + \beta x)\hat{k}\), where \(B_0\) and \(\beta\) are positive constants. The dashed curves in the graphs represent a reference circular path of radius \(R_0 = \dfrac{m v_0}{e B_0}\) corresponding to motion in a uniform magnetic field of magnitude \(B_0\). Which of the following graphs best represents the trajectory of the electron in the \(xy\)-plane?

An electron of mass \(m\) and charge \(-e\) is injected at the origin \((0,0)\) into the \(xy\)-plane with an initial velocity \(\vec{v}_0 = v_0 \hat{i}\). In the region \(x \ge 0\), a non-uniform magnetic field is directed out of the page, \(\vec{B} = (B_0 + \beta x)\hat{k}\), where \(B_0\) and \(\beta\) are positive constants. The dashed curves in the graphs represent a reference circular path of radius \(R_0 = \dfrac{m v_0}{e B_0}\) corresponding to motion in a uniform magnetic field of magnitude \(B_0\). Which of the following graphs best represents the trajectory of the electron in the \(xy\)-plane?

![A Cartesian coordinate plane with horizontal axis labeled x and vertical axis labeled y, intersecting at the origin (0,0). At the origin, a small filled circle represents the electron, from which a single horizontal arrow labeled \(\vec{v}_0\) points directly to the right along the positive x-axis. In the region x > 0, an array of exactly six circled dots is arranged in two vertical columns of three, indicating a magnetic field directed out of the page. A text label near the upper right reads \(\vec{B} = (B_0 + \beta x)\hat{k}\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604720-a35jim.jpg)

- **A.** Graph A
- **B.** Graph B
- **C.** Graph C
- **D.** Graph D

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124930/*