---
title: "An idealized ionic crystal lattice model consists of eight point charges, each of magnitude \\(q\\), located at the eight vertices of a cube of side length \\(s\\). The signs of the charges alternate along every edge of the cube, such that adjacent charges along any edge have opposite signs. What is the total electrostatic potential energy \\(U\\) of this eight-charge system?"
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url: "https://nerd-notes.com/ubq/118143/"
date_modified: "2026-08-04T08:05:30+00:00"
---

# An idealized ionic crystal lattice model consists of eight point charges, each of magnitude \(q\), located at the eight vertices of a cube of side length \(s\). The signs of the charges alternate along every edge of the cube, such that adjacent charges along any edge have opposite signs. What is the total electrostatic potential energy \(U\) of this eight-charge system?

An idealized ionic crystal lattice model consists of eight point charges, each of magnitude \(q\), located at the eight vertices of a cube of side length \(s\). The signs of the charges alternate along every edge of the cube, such that adjacent charges along any edge have opposite signs. What is the total electrostatic potential energy \(U\) of this eight-charge system?

![A 3D wireframe cube of side length s viewed at a three-quarter perspective. The eight vertices of the cube are marked with small filled circles labeled with charge signs. The vertex at the bottom-left-front is labeled +q, and moving along the three adjacent edges leads to three vertices labeled -q at the top-left-front, bottom-right-front, and bottom-left-back. Continuing along the remaining edges, the vertices alternate in sign such that every edge connects a +q charge to a -q charge. A double-headed arrow along one front bottom edge is labeled s. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830729-raqZ1A.jpg)

- **A.** \(\dfrac{q^2}{4\pi\varepsilon_0 s} \left( -24 + 12\sqrt{2} - \dfrac{8\sqrt{3}}{3} \right)\)
- **B.** \(\dfrac{q^2}{4\pi\varepsilon_0 s} \left( -12 - 6\sqrt{2} - \dfrac{4\sqrt{3}}{3} \right)\)
- **C.** \(\dfrac{q^2}{4\pi\varepsilon_0 s} \left( -12 + 3\sqrt{2} - \dfrac{4\sqrt{3}}{3} \right)\)
- **D.** \(\dfrac{q^2}{4\pi\varepsilon_0 s} \left( -12 + 6\sqrt{2} - \dfrac{4\sqrt{3}}{3} \right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118143/*