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title: "A point charge \\(+Q\\) is fixed at the origin. A physical electric dipole aligned along the \\(x\\)-axis consists of charge \\(-q\\) at \\(x = r\\) and charge \\(+q\\) at \\(x = r + s\\), where \\(s\\) is a fixed positive separation distance and \\(r > 0\\). The electrostatic potential energy of the dipole due to the charge \\(+Q\\) is given by \\(U(r) = kQq\\left(\\dfrac{1}{r + s} – \\dfrac{1}{r}\\right)\\). In the limit where the distance to the dipole is much greater than its size (\\(r \\gg s\\)), which of the following correctly gives the leading-order expression for \\(U(r)\\) including its first non-vanishing correction term in terms of the dipole moment \\(p = qs\\), and correctly compares \\(U(r)\\) to the ideal point-dipole potential energy \\(U_{\\text{ideal}} = -\\dfrac{kQp}{r^2}\\)?"
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url: "https://nerd-notes.com/ubq/124603/"
date_modified: "2026-09-28T14:08:48+00:00"
---

# A point charge \(+Q\) is fixed at the origin. A physical electric dipole aligned along the \(x\)-axis consists of charge \(-q\) at \(x = r\) and charge \(+q\) at \(x = r + s\), where \(s\) is a fixed positive separation distance and \(r > 0\). The electrostatic potential energy of the dipole due to the charge \(+Q\) is given by \(U(r) = kQq\left(\dfrac{1}{r + s} – \dfrac{1}{r}\right)\). In the limit where the distance to the dipole is much greater than its size (\(r \gg s\)), which of the following correctly gives the leading-order expression for \(U(r)\) including its first non-vanishing correction term in terms of the dipole moment \(p = qs\), and correctly compares \(U(r)\) to the ideal point-dipole potential energy \(U_{\text{ideal}} = -\dfrac{kQp}{r^2}\)?

A point charge \(+Q\) is fixed at the origin. A physical electric dipole aligned along the \(x\)-axis consists of charge \(-q\) at \(x = r\) and charge \(+q\) at \(x = r + s\), where \(s\) is a fixed positive separation distance and \(r > 0\). The electrostatic potential energy of the dipole due to the charge \(+Q\) is given by \(U(r) = kQq\left(\dfrac{1}{r + s} - \dfrac{1}{r}\right)\). In the limit where the distance to the dipole is much greater than its size (\(r \gg s\)), which of the following correctly gives the leading-order expression for \(U(r)\) including its first non-vanishing correction term in terms of the dipole moment \(p = qs\), and correctly compares \(U(r)\) to the ideal point-dipole potential energy \(U_{\text{ideal}} = -\dfrac{kQp}{r^2}\)?

![A horizontal axis labeled with an italic letter x extends to the right with an arrowhead. A solid black dot on the axis at the far left is labeled with a plus sign followed by Q, positioned above a vertical tick mark labeled 0. Further to the right along the horizontal axis, a filled circle is labeled with a minus sign followed by q, located above a tick mark labeled r. A double-headed horizontal arrow below the axis extends from the tick mark at 0 to the tick mark at r, with the label r centered beneath it. Further to the right of the minus charge, another filled circle is labeled with a plus sign followed by q, located above a tick mark labeled r + s. A smaller double-headed horizontal arrow spans the interval between the two charges with the label s centered above it. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604527-NWpgh9.jpg)

- **A.** \(U(r) \approx -\dfrac{kQp}{r^2} + \dfrac{kQps}{r^3}\), which indicates that \(U(r) > U_{\text{ideal}}\)
- **B.** \(U(r) \approx -\dfrac{kQp}{r^2} + \dfrac{kQps}{r^3}\), which indicates that \(U(r) < U_{\text{ideal}}\)
- **C.** \(U(r) \approx -\dfrac{kQp}{r^2} - \dfrac{kQps}{r^3}\), which indicates that \(U(r) < U_{\text{ideal}}\)
- **D.** \(U(r) \approx -\dfrac{kQp}{r^2} - \dfrac{kQps}{r^3}\), which indicates that \(U(r) > U_{\text{ideal}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124603/*