---
title: "An alpha particle of mass \\(m\\) and charge \\(+2e\\) is projected directly toward a stationary gold nucleus of charge \\(+Ze\\). When the alpha particle is at a distance \\(d\\) from the nucleus, it has speed \\(v_0\\) directed radially inward toward the nucleus. Assuming the gold nucleus remains fixed in place and gravitational forces are negligible, which of the following expressions represents the distance of closest approach \\(r_{\\text{min}}\\) between the alpha particle and the nucleus?"
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url: "https://nerd-notes.com/ubq/124703/"
date_modified: "2026-09-28T14:09:10+00:00"
---

# An alpha particle of mass \(m\) and charge \(+2e\) is projected directly toward a stationary gold nucleus of charge \(+Ze\). When the alpha particle is at a distance \(d\) from the nucleus, it has speed \(v_0\) directed radially inward toward the nucleus. Assuming the gold nucleus remains fixed in place and gravitational forces are negligible, which of the following expressions represents the distance of closest approach \(r_{\text{min}}\) between the alpha particle and the nucleus?

An alpha particle of mass \(m\) and charge \(+2e\) is projected directly toward a stationary gold nucleus of charge \(+Ze\). When the alpha particle is at a distance \(d\) from the nucleus, it has speed \(v_0\) directed radially inward toward the nucleus. Assuming the gold nucleus remains fixed in place and gravitational forces are negligible, which of the following expressions represents the distance of closest approach \(r_{\text{min}}\) between the alpha particle and the nucleus?

![A horizontal dashed centerline extends across the frame. On the right side of the centerline sits a large solid circle labeled \(+Ze\) representing the target nucleus. On the left side of the centerline sits a smaller solid circle labeled \(+2e\) representing the incoming alpha particle. A straight horizontal arrow labeled \(v_0\) originates at the right edge of the alpha particle and points directly toward the nucleus. Below the centerline, a horizontal dimension line with arrowheads at both ends extends between vertical tick marks aligned with the centers of the two particles, labeled with the distance \(d\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604550-ZvJSIf.jpg)

- **A.** \(r_{\text{min}} = \dfrac{Ze^2 d}{Ze^2 + \pi\varepsilon_0 m v_0^2 d}\)
- **B.** \(r_{\text{min}} = \dfrac{Ze^2 d}{Ze^2 + 2\pi\varepsilon_0 m v_0^2 d}\)
- **C.** \(r_{\text{min}} = \dfrac{2Ze^2 d}{2Ze^2 + \pi\varepsilon_0 m v_0^2 d}\)
- **D.** \(r_{\text{min}} = \dfrac{Ze^2 d}{Ze^2 - \pi\varepsilon_0 m v_0^2 d}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124703/*