---
title: "A positive ion of mass \\(m\\) and charge \\(+q\\) is injected with an initial horizontal speed \\(v_0\\) into a region of space containing a uniform electric field of magnitude \\(E\\) and a uniform magnetic field of magnitude \\(B\\). The electric field is directed toward the bottom of the page, as shown in Figure 1. The ion passes through the region undeflected in a straight line."
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url: "https://nerd-notes.com/ubq/117816/"
date_modified: "2026-08-04T07:58:27+00:00"
---

# A positive ion of mass \(m\) and charge \(+q\) is injected with an initial horizontal speed \(v_0\) into a region of space containing a uniform electric field of magnitude \(E\) and a uniform magnetic field of magnitude \(B\). The electric field is directed toward the bottom of the page, as shown in Figure 1. The ion passes through the region undeflected in a straight line.

A positive ion of mass \(m\) and charge \(+q\) is injected with an initial horizontal speed \(v_0\) into a region of space containing a uniform electric field of magnitude \(E\) and a uniform magnetic field of magnitude \(B\). The electric field is directed toward the bottom of the page, as shown in Figure 1. The ion passes through the region undeflected in a straight line.

![A dashed rectangular box represents a region of uniform fields. On the left side of the box, a small circle with a '+' sign inside represents a positive ion. A horizontal arrow points from the ion to the right, entering the box, and is labeled \(v_0\). Inside the dashed box, three evenly spaced vertical arrows point downward, labeled \(\vec{E}\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830305-S2Jy51.jpg)

**Part a)** Determine the direction of the magnetic field and the forces exerted on the ion. *(3 points)*

**Part b)** **Derive** an expression for the speed \(v_0\) of the undeflected ion. Express your answer in terms of \(E\), \(B\), \(m\), \(q\), and physical constants, as appropriate. *(2 points)*

**Part c)** A second identical ion is injected into the same region with a speed \(v_1\) such that \(v_1 > v_0\). *(3 points)*

**Part d)** The electric field is now turned off, leaving only the uniform magnetic field. Ions of charge \(+q\) and mass \(m\) are injected into the region at various speeds \(v\). Each ion moves in a circular arc of radius \(R\) before exiting the region. On the axes provided, **sketch** a graph of the radius \(R\) as a function of the injection speed \(v\). *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117816/*