---
title: "Two point charges are fixed in place on the \\(x\\)-axis. A point charge of \\(+Q\\) is located at the origin (\\(x = 0\\)), and a point charge of \\(+4Q\\) is located at position \\(x = d\\)."
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/117737/"
date_modified: "2026-08-04T07:54:28+00:00"
---

# Two point charges are fixed in place on the \(x\)-axis. A point charge of \(+Q\) is located at the origin (\(x = 0\)), and a point charge of \(+4Q\) is located at position \(x = d\).

Two point charges are fixed in place on the \(x\)-axis. A point charge of \(+Q\) is located at the origin (\(x = 0\)), and a point charge of \(+4Q\) is located at position \(x = d\).

![A horizontal dashed line representing the x-axis. A small circle containing a plus sign is located at the left end of the axis and is labeled with text \(+Q\) above it and \(x = 0\) below it. A second small circle containing a plus sign is located further to the right on the axis and is labeled with text \(+4Q\) above it and \(x = d\) below it. No other lines, labels, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830067-r9vKAJ.jpg)

**Part a)** On the axes provided, **sketch** a graph of the electric potential \(V\) as a function of position \(x\) for the region between the two charges (\(0 < x < d\)). Your sketch should clearly show the general shape of the curve, its asymptotic behavior near the charges, and the approximate location of any minimum or maximum values relative to the midpoint \(x = d/2\).

**Part b)** **Derive** an expression for the position \(x\) between the two charges (\(0 < x < d\)) where the net electric field is zero. Express your answer in terms of \(d\) and fundamental constants.

**Part c)** Based on physical principles, **explain** how the shape of the graph you sketched in part (a) is consistent with your mathematical derivation in part (b).

**Part d)** Figure 3 shows a two-dimensional map of the equipotential isolines around the two charges. Point P is located on a circular isoline very close to the \(+Q\) charge. Point R is located on an isoline between the two charges. Answer the following questions about the electric field in this two-dimensional representation.


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117737/*