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title: "In a mass spectrometer, singly ionized carbon atoms (charge \\(q = +e\\)) are accelerated from rest through a constant potential difference \\(V_0\\) before entering a region of uniform magnetic field \\(B_0\\) directed perpendicular to their velocity. The ions travel in a semicircular path and strike a linear detector array at a distance \\(x\\) from the entrance slit. A beam containing two carbon isotopes with average mass \\(m_0 = 2.0 \\times 10^{-26} \\text{ kg}\\) strikes the detector, producing two distinct peaks centered at position \\(x_0 = 40 \\text{ cm}\\) with a small spatial separation of \\(\\Delta x = 1.0 \\text{ cm}\\). Based on the relationship between detector position and ion mass, what is the mass difference \\(\\Delta m\\) between the two isotopes?"
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url: "https://nerd-notes.com/ubq/118611/"
date_modified: "2026-08-04T08:11:48+00:00"
---

# In a mass spectrometer, singly ionized carbon atoms (charge \(q = +e\)) are accelerated from rest through a constant potential difference \(V_0\) before entering a region of uniform magnetic field \(B_0\) directed perpendicular to their velocity. The ions travel in a semicircular path and strike a linear detector array at a distance \(x\) from the entrance slit. A beam containing two carbon isotopes with average mass \(m_0 = 2.0 \times 10^{-26} \text{ kg}\) strikes the detector, producing two distinct peaks centered at position \(x_0 = 40 \text{ cm}\) with a small spatial separation of \(\Delta x = 1.0 \text{ cm}\). Based on the relationship between detector position and ion mass, what is the mass difference \(\Delta m\) between the two isotopes?

In a mass spectrometer, singly ionized carbon atoms (charge \(q = +e\)) are accelerated from rest through a constant potential difference \(V_0\) before entering a region of uniform magnetic field \(B_0\) directed perpendicular to their velocity. The ions travel in a semicircular path and strike a linear detector array at a distance \(x\) from the entrance slit. A beam containing two carbon isotopes with average mass \(m_0 = 2.0 \times 10^{-26} \text{ kg}\) strikes the detector, producing two distinct peaks centered at position \(x_0 = 40 \text{ cm}\) with a small spatial separation of \(\Delta x = 1.0 \text{ cm}\). Based on the relationship between detector position and ion mass, what is the mass difference \(\Delta m\) between the two isotopes?

![A horizontal rectangular diagram showing an ion trajectory. On the left, a small circular ion is labeled q with a horizontal arrow pointing to the right toward a vertical dashed line labeled V_0. To the right of the line is a large rectangular region containing a uniform grid of six small circles with central dots, labeled B_0 in the upper right. Inside this region, a curved dashed path begins at the left edge, forms a smooth semicircle opening upward and to the left, and ends at a horizontal solid line representing a detector array at the top. A double-headed horizontal arrow at the top measures the distance from the entrance point to the impact point and is labeled x. A small arrow points upward along the curve, labeled v. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831108-Hdjisq.jpg)

- **A.** \(5.0 \times 10^{-28} \text{ kg}\)
- **B.** \(1.0 \times 10^{-27} \text{ kg}\)
- **C.** \(2.0 \times 10^{-27} \text{ kg}\)
- **D.** \(4.0 \times 10^{-27} \text{ kg}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118611/*