---
title: "A solid, nonconducting sphere of radius \\(R\\) has a total negative charge \\(-Q\\) distributed uniformly throughout its volume. A narrow, frictionless radial tunnel extends from the center of the sphere to its surface. A small particle of mass \\(m\\) and positive charge \\(+q\\) is initially at the center of the sphere. In terms of \\(Q\\), \\(q\\), \\(m\\), \\(R\\), and fundamental constants, what is the minimum launch speed \\(v_0\\) required for the particle to escape to infinity?"
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url: "https://nerd-notes.com/ubq/121073/"
date_modified: "2026-08-23T04:57:41+00:00"
---

# A solid, nonconducting sphere of radius \(R\) has a total negative charge \(-Q\) distributed uniformly throughout its volume. A narrow, frictionless radial tunnel extends from the center of the sphere to its surface. A small particle of mass \(m\) and positive charge \(+q\) is initially at the center of the sphere. In terms of \(Q\), \(q\), \(m\), \(R\), and fundamental constants, what is the minimum launch speed \(v_0\) required for the particle to escape to infinity?

A solid, nonconducting sphere of radius \(R\) has a total negative charge \(-Q\) distributed uniformly throughout its volume. A narrow, frictionless radial tunnel extends from the center of the sphere to its surface. A small particle of mass \(m\) and positive charge \(+q\) is initially at the center of the sphere. In terms of \(Q\), \(q\), \(m\), \(R\), and fundamental constants, what is the minimum launch speed \(v_0\) required for the particle to escape to infinity?

![A cross-sectional schematic diagram of a sphere and radial tunnel. A large circle of radius \(R\) is filled with light gray shading. A solid radial line segment extends from the center of the circle to its rightmost edge, labeled \(R\) above the line. A label \(-Q\) is centered in the upper-left quadrant of the circle. Along the horizontal radius pointing to the right, a narrow white horizontal rectangular strip represents a tunnel from the center to the outer boundary. At the exact center of the circle, a small solid black dot represents a particle, labeled \(+q, m\) below the dot. A single horizontal arrow labeled \(v_0\) originates at the particle and points to the right along the tunnel. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461061-nyxHhR.jpg)

- **A.** \(\sqrt{\dfrac{qQ}{4\pi\varepsilon_0 m R}}\)
- **B.** \(\sqrt{\dfrac{qQ}{2\pi\varepsilon_0 m R}}\)
- **C.** \(\sqrt{\dfrac{3qQ}{4\pi\varepsilon_0 m R}}\)
- **D.** \(\sqrt{\dfrac{3qQ}{2\pi\varepsilon_0 m R}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121073/*