---
title: "Two point charges, \\(+Q\\) and \\(-Q\\), are fixed on the x-axis at \\(x = -d\\) and \\(x = +d\\), respectively. The charges can be treated as ideal point charges. A mathematical Gaussian sphere of radius \\(R = d/2\\) is defined such that its center can be moved continuously along the x-axis."
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url: "https://nerd-notes.com/ubq/117851/"
date_modified: "2026-08-04T07:59:31+00:00"
---

# Two point charges, \(+Q\) and \(-Q\), are fixed on the x-axis at \(x = -d\) and \(x = +d\), respectively. The charges can be treated as ideal point charges. A mathematical Gaussian sphere of radius \(R = d/2\) is defined such that its center can be moved continuously along the x-axis.

Two point charges, \(+Q\) and \(-Q\), are fixed on the x-axis at \(x = -d\) and \(x = +d\), respectively. The charges can be treated as ideal point charges. A mathematical Gaussian sphere of radius \(R = d/2\) is defined such that its center can be moved continuously along the x-axis.

![A horizontal line representing the x-axis with a vertical y-axis crossing at the origin. A solid circle labeled +Q is placed on the x-axis at a tick mark labeled -d. A solid circle labeled -Q is placed on the x-axis at a tick mark labeled +d. A dashed circle representing a sphere is shown centered at an arbitrary point on the far left of the x-axis, with its radius indicated by an arrow labeled R = d/2. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830370-V3NqjY.jpg)

**Part a)** On the figure provided, **draw** a single arrow originating at the origin (\(x = 0, y = 0\)) to indicate the direction of the net electric field at that location. Then, **draw** a single arrow originating at the point (\(x = 0, y = d\)) on the y-axis to indicate the direction of the net electric field at that location.

**Part b)** **Derive** an expression for the magnitude of the net electric field at the origin. Express your answer in terms of \(Q\), \(d\), and physical constants, as appropriate.

**Part c)** The Gaussian sphere of radius \(R = d/2\) is initially centered at \(x = -3d\). The center of the sphere is slowly moved along the x-axis to \(x = +3d\). On the axes provided, **sketch** a graph of the total electric flux \(\Phi_E\) through the sphere as a function of the position \(x\) of its center.

**Part d)** **Justify** the shape of your graph from part (c), specifically referencing the application of Gauss's law and the intervals where the flux is positive, negative, and zero.

**Part e)** The point charge \(-Q\) is now replaced by a point charge \(+Q\). The Gaussian sphere is again moved from \(x = -3d\) to \(x = +3d\). **Indicate** how the graph sketched in part (c) would change for the region \(x > 0\). - [ ] The pulse would be identical to the original graph. - [ ] The pulse would be inverted (positive instead of negative). - [ ] The flux would be zero everywhere in the region. **Justify** your selection.


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117851/*