---
title: "Three point charges are fixed at the vertices of an equilateral triangle of side length \\(d\\). Two of the vertices each hold a charge of \\(+q\\), and the third vertex holds a charge of \\(-q\\). A fourth particle with charge \\(+q\\) is brought from infinitely far away to point \\(P\\), which is located at the midpoint of the line segment connecting the two \\(+q\\) charges. Which of the following expressions represents the work done by an external agent to bring the fourth particle to point \\(P\\)?"
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url: "https://nerd-notes.com/ubq/117344/"
date_modified: "2026-08-04T06:51:52+00:00"
---

# Three point charges are fixed at the vertices of an equilateral triangle of side length \(d\). Two of the vertices each hold a charge of \(+q\), and the third vertex holds a charge of \(-q\). A fourth particle with charge \(+q\) is brought from infinitely far away to point \(P\), which is located at the midpoint of the line segment connecting the two \(+q\) charges. Which of the following expressions represents the work done by an external agent to bring the fourth particle to point \(P\)?

Three point charges are fixed at the vertices of an equilateral triangle of side length \(d\). Two of the vertices each hold a charge of \(+q\), and the third vertex holds a charge of \(-q\). A fourth particle with charge \(+q\) is brought from infinitely far away to point \(P\), which is located at the midpoint of the line segment connecting the two \(+q\) charges. Which of the following expressions represents the work done by an external agent to bring the fourth particle to point \(P\)?

![An equilateral triangle with side length labeled d. The bottom-left vertex has a circle labeled +q. The bottom-right vertex has a circle labeled +q. The top vertex has a circle labeled -q. The horizontal base connecting the two +q charges has a point labeled P located at its exact midpoint. A dashed vertical line segment extends from P up to the top vertex labeled -q. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785826312-dZNEVz.jpg)

- **A.** \(\dfrac{k q^2}{d} \left(4 - \dfrac{2\sqrt{3}}{3}\right)\)
- **B.** \(\dfrac{k q^2}{d} \left(4 + \dfrac{2\sqrt{3}}{3}\right)\)
- **C.** \(\dfrac{k q^2}{d} \left(2 - \dfrac{2\sqrt{3}}{3}\right)\)
- **D.** \(\dfrac{k q^2}{d} \left(4 - \sqrt{3}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117344/*