---
title: "Three identical point charges, each with charge \\(+q\\), are initially fixed at the vertices of an equilateral triangle with side length \\(d\\). The distance from any vertex to the center of the triangle is \\(\\dfrac{d}{\\sqrt{3}}\\). An external agent slowly moves one of the charges from its vertex to the center of the triangle while the other two charges remain fixed at their original positions. How much work is done by the external agent to move the charge to the center?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/118048/"
date_modified: "2026-08-04T08:04:53+00:00"
---

# Three identical point charges, each with charge \(+q\), are initially fixed at the vertices of an equilateral triangle with side length \(d\). The distance from any vertex to the center of the triangle is \(\dfrac{d}{\sqrt{3}}\). An external agent slowly moves one of the charges from its vertex to the center of the triangle while the other two charges remain fixed at their original positions. How much work is done by the external agent to move the charge to the center?

Three identical point charges, each with charge \(+q\), are initially fixed at the vertices of an equilateral triangle with side length \(d\). The distance from any vertex to the center of the triangle is \(\dfrac{d}{\sqrt{3}}\). An external agent slowly moves one of the charges from its vertex to the center of the triangle while the other two charges remain fixed at their original positions. How much work is done by the external agent to move the charge to the center?

![An equilateral triangle with side length d is drawn with thin lines. Three filled black circles representing positive point charges, each labeled +q, are positioned at the three vertices. A dashed gray line with an arrowhead extends from the top vertex to the geometric center of the triangle, where a small hollow dotted circle indicates the new position. A dimension line labeled d marks the bottom side of the triangle. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830693-vBseyW.jpg)

- **A.** \(\dfrac{q^2}{4\pi\varepsilon_0 d} (\sqrt{3} - 1)\)
- **B.** \(\dfrac{q^2}{4\pi\varepsilon_0 d} (2\sqrt{3} - 2)\)
- **C.** \(\dfrac{q^2}{4\pi\varepsilon_0 d} (3\sqrt{3} - 3)\)
- **D.** \(\dfrac{q^2}{4\pi\varepsilon_0 d} (2\sqrt{3})\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118048/*