---
title: "Two isotopes, Ion 1 and Ion 2, enter a mass spectrometer deflection chamber containing a uniform magnetic field \\(\\vec{B}\\) directed perpendicular to their velocity vectors. Ion 1 has mass \\(m_1 = m\\) and charge \\(q_1 = +2e\\), while Ion 2 has mass \\(m_2 = 3m\\) and charge \\(q_2 = +e\\). If both ions enter the magnetic field region with the same kinetic energy \\(K\\), what is the ratio \\(r_2 / r_1\\) of the radius of the circular path of Ion 2 to that of Ion 1?"
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url: "https://nerd-notes.com/ubq/117440/"
date_modified: "2026-08-04T06:52:10+00:00"
---

# Two isotopes, Ion 1 and Ion 2, enter a mass spectrometer deflection chamber containing a uniform magnetic field \(\vec{B}\) directed perpendicular to their velocity vectors. Ion 1 has mass \(m_1 = m\) and charge \(q_1 = +2e\), while Ion 2 has mass \(m_2 = 3m\) and charge \(q_2 = +e\). If both ions enter the magnetic field region with the same kinetic energy \(K\), what is the ratio \(r_2 / r_1\) of the radius of the circular path of Ion 2 to that of Ion 1?

Two isotopes, Ion 1 and Ion 2, enter a mass spectrometer deflection chamber containing a uniform magnetic field \(\vec{B}\) directed perpendicular to their velocity vectors. Ion 1 has mass \(m_1 = m\) and charge \(q_1 = +2e\), while Ion 2 has mass \(m_2 = 3m\) and charge \(q_2 = +e\). If both ions enter the magnetic field region with the same kinetic energy \(K\), what is the ratio \(r_2 / r_1\) of the radius of the circular path of Ion 2 to that of Ion 1?

![A schematic diagram of a mass spectrometer chamber. On the left vertical wall, a narrow entrance slit opens into a rectangular deflection region containing a uniform magnetic field indicated by black dots labeled B pointing out of the page. Two positive ions enter horizontally to the right through the slit at the same entry point. Inside the field, two circular arc trajectories bend downward toward a detector plate along the bottom wall. Path 1 forms a smaller semicircular arc of radius r1, and Path 2 forms a larger semicircular arc of radius r2. Labels r1 and r2 indicate the radii of the two arcs from their respective centers along the entry line. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785826330-QxCk5x.jpg)

- **A.** \(\dfrac{\sqrt{3}}{2}\)
- **B.** \(\sqrt{3}\)
- **C.** \(\sqrt{6}\)
- **D.** \(2\sqrt{3}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117440/*