---
title: "A thin, rigid insulating ring of radius \\(R\\) lies in the \\(xy\\)-plane and carries a total positive charge \\(Q\\) distributed uniformly along its circumference. A particle of mass \\(m\\) and negative charge \\(-q\\) is constrained to move along the central \\(z\\)-axis perpendicular to the plane of the ring. The particle is released from rest at a small displacement \\(z\\) from the center of the ring, where \\(z \\ll R\\). Which of the following expressions is the angular frequency \\(\\omega\\) of the particle’s small oscillations?"
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url: "https://nerd-notes.com/ubq/117986/"
date_modified: "2026-08-04T08:02:46+00:00"
---

# A thin, rigid insulating ring of radius \(R\) lies in the \(xy\)-plane and carries a total positive charge \(Q\) distributed uniformly along its circumference. A particle of mass \(m\) and negative charge \(-q\) is constrained to move along the central \(z\)-axis perpendicular to the plane of the ring. The particle is released from rest at a small displacement \(z\) from the center of the ring, where \(z \ll R\). Which of the following expressions is the angular frequency \(\omega\) of the particle’s small oscillations?

A thin, rigid insulating ring of radius \(R\) lies in the \(xy\)-plane and carries a total positive charge \(Q\) distributed uniformly along its circumference. A particle of mass \(m\) and negative charge \(-q\) is constrained to move along the central \(z\)-axis perpendicular to the plane of the ring. The particle is released from rest at a small displacement \(z\) from the center of the ring, where \(z \ll R\). Which of the following expressions is the angular frequency \(\omega\) of the particle's small oscillations?

![A thin circular ring of radius R lies horizontally, with positive charge symbols spaced uniformly around its circumference indicating a total charge Q. A vertical dashed z-axis passes through the center of the ring. A small particle labeled with charge -q and mass m is positioned on the z-axis at a distance z above the ring center. An arrow labeled z points from the center of the ring up to the particle. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830566-90lHXW.jpg)

- **A.** \(\sqrt{\dfrac{q Q}{2\pi\varepsilon_0 m R^3}}\)
- **B.** \(\sqrt{\dfrac{q Q}{8\pi\varepsilon_0 m R^3}}\)
- **C.** \(\sqrt{\dfrac{3 q Q}{4\pi\varepsilon_0 m R^3}}\)
- **D.** \(\sqrt{\dfrac{q Q}{4\pi\varepsilon_0 m R^3}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117986/*