---
title: "Two identical point charges, each with positive charge \\(+Q\\), are fixed along a vertical axis and separated by a distance \\(2d\\) in a region with a uniform downward gravitational acceleration of magnitude \\(g\\).  A small sphere of mass \\(m\\) and positive charge \\(q\\) is released from rest and remains in static equilibrium on the vertical axis at a position between the two fixed charges.  Which of the following statements correctly characterizes the equilibrium of the sphere with respect to small vertical displacements, and provides the correct physical justification?"
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url: "https://nerd-notes.com/ubq/124562/"
date_modified: "2026-09-28T14:08:38+00:00"
---

# Two identical point charges, each with positive charge \(+Q\), are fixed along a vertical axis and separated by a distance \(2d\) in a region with a uniform downward gravitational acceleration of magnitude \(g\).

A small sphere of mass \(m\) and positive charge \(q\) is released from rest and remains in static equilibrium on the vertical axis at a position between the two fixed charges.

Which of the following statements correctly characterizes the equilibrium of the sphere with respect to small vertical displacements, and provides the correct physical justification?

Two identical point charges, each with positive charge \(+Q\), are fixed along a vertical axis and separated by a distance \(2d\) in a region with a uniform downward gravitational acceleration of magnitude \(g\).

A small sphere of mass \(m\) and positive charge \(q\) is released from rest and remains in static equilibrium on the vertical axis at a position between the two fixed charges.

Which of the following statements correctly characterizes the equilibrium of the sphere with respect to small vertical displacements, and provides the correct physical justification?

![A vertical dashed line represents a vertical axis. At the top of the axis is a small solid circle labeled +Q. At the bottom of the axis is an identical small solid circle labeled +Q. The vertical separation between the two fixed charges is indicated by a vertical double-headed dimension arrow labeled 2d. Between the two charges, positioned along the vertical axis below the midpoint, is a small shaded circle labeled q, m. A single vertical downward arrow to the right of the axis is labeled g. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604518-nqa3k5.jpg)

- **A.** The equilibrium is unstable with respect to vertical displacements because the downward force of gravity is constant, so an upward displacement causes the upward electric force to exceed the downward forces.
- **B.** The equilibrium is unstable with respect to vertical displacements because electrostatic forces cannot produce a restoring force along any spatial axis without violating Gauss's law.
- **C.** The equilibrium is stable with respect to vertical displacements because an upward displacement increases the upward repulsive force from the bottom charge while decreasing the downward repulsive force from the top charge.
- **D.** The equilibrium is stable with respect to vertical displacements because an upward displacement decreases the upward repulsive force from the bottom charge and increases the downward repulsive force from the top charge, creating a net downward restoring force.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124562/*