---
title: "Two point charges, \\(+q\\) and \\(-q\\), are held fixed on the \\(y\\)-axis at \\((0, a)\\) and \\((0, -a)\\), respectively, in vacuum.  Point \\(P\\) is located on the positive \\(x\\)-axis at a distance \\(x\\) from the origin.  Which of the following correctly gives the net electric field vector \\(\\vec{E}(x)\\) at point \\(P\\) and the power-law exponent \\(n\\) describing the asymptotic dependence of its magnitude, \\(E(x) \\propto x^n\\), in the limit \\(x \\gg a\\)?"
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url: "https://nerd-notes.com/ubq/124744/"
date_modified: "2026-09-28T14:09:49+00:00"
---

# Two point charges, \(+q\) and \(-q\), are held fixed on the \(y\)-axis at \((0, a)\) and \((0, -a)\), respectively, in vacuum.

Point \(P\) is located on the positive \(x\)-axis at a distance \(x\) from the origin.

Which of the following correctly gives the net electric field vector \(\vec{E}(x)\) at point \(P\) and the power-law exponent \(n\) describing the asymptotic dependence of its magnitude, \(E(x) \propto x^n\), in the limit \(x \gg a\)?

Two point charges, \(+q\) and \(-q\), are held fixed on the \(y\)-axis at \((0, a)\) and \((0, -a)\), respectively, in vacuum.

Point \(P\) is located on the positive \(x\)-axis at a distance \(x\) from the origin.

Which of the following correctly gives the net electric field vector \(\vec{E}(x)\) at point \(P\) and the power-law exponent \(n\) describing the asymptotic dependence of its magnitude, \(E(x) \propto x^n\), in the limit \(x \gg a\)?

![A Cartesian coordinate system with a horizontal x-axis and a vertical y-axis intersecting at an origin labeled O. On the positive y-axis, a solid black circle represents a charge labeled +q located at a distance a above the origin, indicated by a vertical dimension line labeled a. On the negative y-axis, an open circle with a black border represents a charge labeled -q located at a distance a below the origin, indicated by a vertical dimension line labeled a. On the positive x-axis, a black dot marks a point labeled P located at a distance x to the right of the origin, indicated by a horizontal dimension line along the axis labeled x. Straight dashed lines connect the charge +q to point P and the charge -q to point P. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604589-RX0VRY.jpg)

- **A.** \(\vec{E}(x) = \dfrac{1}{4\pi\varepsilon_0} \dfrac{2qx}{(x^2 + a^2)^{3/2}} \,\hat{i}\), with \(n = -2\)
- **B.** \(\vec{E}(x) = -\dfrac{1}{4\pi\varepsilon_0} \dfrac{2q}{x^2 + a^2} \,\hat{j}\), with \(n = -2\)
- **C.** \(\vec{E}(x) = -\dfrac{1}{4\pi\varepsilon_0} \dfrac{qa}{(x^2 + a^2)^{3/2}} \,\hat{j}\), with \(n = -3\)
- **D.** \(\vec{E}(x) = -\dfrac{1}{4\pi\varepsilon_0} \dfrac{2qa}{(x^2 + a^2)^{3/2}} \,\hat{j}\), with \(n = -3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124744/*