---
title: "Two point charges, each with positive charge \\(+Q\\), are held fixed on the y-axis at positions \\(y = +d\\) and \\(y = -d\\), as shown in Figure 1. A third point charge with positive charge \\(+q\\) is introduced to the system."
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url: "https://nerd-notes.com/ubq/117718/"
date_modified: "2026-08-04T07:53:48+00:00"
---

# Two point charges, each with positive charge \(+Q\), are held fixed on the y-axis at positions \(y = +d\) and \(y = -d\), as shown in Figure 1. A third point charge with positive charge \(+q\) is introduced to the system.

Two point charges, each with positive charge \(+Q\), are held fixed on the y-axis at positions \(y = +d\) and \(y = -d\), as shown in Figure 1. A third point charge with positive charge \(+q\) is introduced to the system.

![A standard Cartesian coordinate system with a horizontal x-axis and a vertical y-axis. Two filled circles representing point charges, each labeled '\(+Q\)', are located on the y-axis. One is at a tick mark labeled '\(d\)' above the origin, and the other is at a tick mark labeled '\(-d\)' below the origin. The origin is marked and labeled '\(O\)'. A point on the positive x-axis is marked with a small dot and labeled '\(P\)', located at a tick mark labeled '\(d\)'. No other charges, lines, or vectors are present.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830028-SrURkB.jpg)

**Part a)** The charge \(+q\) is placed at the origin \(O\). **Determine** the magnitude and direction (if any) of the net electric force exerted on the charge \(+q\). **Justify** your answer. *(2 points)*

**Part b)** The charge \(+q\) is then moved to point \(P\) at \(x = d\). **Derive** an expression for the electric potential \(V\) at point \(P\) due to the two fixed charges \(+Q\). Express your answer in terms of \(Q\), \(d\), and fundamental constants. *(2 points)*

**Part c)** **Derive** an expression for the magnitude of the net electric force exerted on the charge \(+q\) when it is located at point \(P\). Express your answer in terms of \(Q\), \(q\), \(d\), and fundamental constants. *(3 points)*

**Part d)** A student considers the work \(W\) required by an external force to move the charge \(+q\) at constant speed from infinity to three different positions on the positive x-axis: \(x = d\), \(x = 2d\), and \(x = 3d\). **Rank** the work required to move the charge to these three positions, from greatest to least. - [ ] \(W_d > W_{2d} > W_{3d}\) - [ ] \(W_{3d} > W_{2d} > W_d\) - [ ] \(W_d = W_{2d} = W_{3d}\) **Justify** your ranking using physical principles. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117718/*