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title: "Three identical point charges, each with charge \\(+q\\), are initially infinitely far apart from each other. An external agent brings the charges together and fixes them at the vertices of an equilateral triangle with side length \\(s\\), requiring a total work \\(W_1\\). The distance from each vertex to the geometric center of the triangle is \\(r = \\dfrac{s}{\\sqrt{3}}\\). A fourth identical point charge \\(+q\\) is then brought from infinity and placed at the center of the triangle, requiring an additional work \\(W_2\\) done by the external agent. What is the ratio \\(\\dfrac{W_2}{W_1}\\)?"
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url: "https://nerd-notes.com/ubq/121082/"
date_modified: "2026-08-23T04:57:44+00:00"
---

# Three identical point charges, each with charge \(+q\), are initially infinitely far apart from each other. An external agent brings the charges together and fixes them at the vertices of an equilateral triangle with side length \(s\), requiring a total work \(W_1\). The distance from each vertex to the geometric center of the triangle is \(r = \dfrac{s}{\sqrt{3}}\). A fourth identical point charge \(+q\) is then brought from infinity and placed at the center of the triangle, requiring an additional work \(W_2\) done by the external agent. What is the ratio \(\dfrac{W_2}{W_1}\)?

Three identical point charges, each with charge \(+q\), are initially infinitely far apart from each other. An external agent brings the charges together and fixes them at the vertices of an equilateral triangle with side length \(s\), requiring a total work \(W_1\). The distance from each vertex to the geometric center of the triangle is \(r = \dfrac{s}{\sqrt{3}}\). A fourth identical point charge \(+q\) is then brought from infinity and placed at the center of the triangle, requiring an additional work \(W_2\) done by the external agent. What is the ratio \(\dfrac{W_2}{W_1}\)?

![An equilateral triangle drawn with thin solid lines resting on a horizontal base. At each of the three vertices is a small filled circle labeled +q. The horizontal base of the triangle has a dimension line below it labeled s. At the geometric center of the triangle is an open circle labeled P. Three dashed line segments connect point P to each of the three vertices, and one of these dashed segments is labeled r. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461064-olOoe1.jpg)

- **A.** \(\sqrt{3}\)
- **B.** \(2\)
- **C.** \(3\)
- **D.** \(3\sqrt{3}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121082/*