---
title: "A thin, horizontal ring of radius \\(R\\) carries a uniform positive charge \\(Q\\). A small particle of mass \\(m\\) and positive charge \\(q\\) is constrained to move along the vertical \\(z\\)-axis passing through the center of the ring, where \\(z > 0\\) is measured upward from the center. A uniform downward gravitational field \\(\\vec{g}\\) acts on the particle, producing an equilibrium height \\(z_0\\) where the electrostatic force balances the gravitational force. Which of the following correctly pairs the height \\(z_{\\text{max}}\\) at which the upward electrostatic force on the particle is maximized with the range of equilibrium heights \\(z_0\\) for which the particle is in stable equilibrium against small vertical displacements?"
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url: "https://nerd-notes.com/ubq/118010/"
date_modified: "2026-08-04T08:02:52+00:00"
---

# A thin, horizontal ring of radius \(R\) carries a uniform positive charge \(Q\). A small particle of mass \(m\) and positive charge \(q\) is constrained to move along the vertical \(z\)-axis passing through the center of the ring, where \(z > 0\) is measured upward from the center. A uniform downward gravitational field \(\vec{g}\) acts on the particle, producing an equilibrium height \(z_0\) where the electrostatic force balances the gravitational force. Which of the following correctly pairs the height \(z_{\text{max}}\) at which the upward electrostatic force on the particle is maximized with the range of equilibrium heights \(z_0\) for which the particle is in stable equilibrium against small vertical displacements?

A thin, horizontal ring of radius \(R\) carries a uniform positive charge \(Q\). A small particle of mass \(m\) and positive charge \(q\) is constrained to move along the vertical \(z\)-axis passing through the center of the ring, where \(z > 0\) is measured upward from the center. A uniform downward gravitational field \(\vec{g}\) acts on the particle, producing an equilibrium height \(z_0\) where the electrostatic force balances the gravitational force. Which of the following correctly pairs the height \(z_{\text{max}}\) at which the upward electrostatic force on the particle is maximized with the range of equilibrium heights \(z_0\) for which the particle is in stable equilibrium against small vertical displacements?

![A thin horizontal ring of radius R is centered at the origin in the horizontal plane and labeled with charge +Q. A vertical z-axis extends upward through the center of the ring. A small particle of mass m and positive charge +q is located on the z-axis at a height z above the plane of the ring. A downward vector arrow labeled \vec{g} is placed near the particle. An upward force vector arrow labeled \vec{F}_e starts at the particle and points upward. A downward force vector arrow labeled \vec{F}_g starts at the particle and points downward. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830572-ZDIGBO.jpg)

- **A.** \(z_{\text{max}} = \dfrac{R}{\sqrt{2}}\); Stable for \(z_0 > \dfrac{R}{\sqrt{2}}\)
- **B.** \(z_{\text{max}} = \dfrac{R}{\sqrt{2}}\); Stable for \(0 < z_0 < \dfrac{R}{\sqrt{2}}\)
- **C.** \(z_{\text{max}} = R\sqrt{2}\); Stable for \(z_0 > R\sqrt{2}\)
- **D.** \(z_{\text{max}} = \dfrac{R}{2}\); Stable for \(0 < z_0 < \dfrac{R}{2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118010/*