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title: "A thin, nonconducting ring of radius \\(R\\) lies in the \\(yz\\)-plane centered at the origin and carries a total positive charge \\(+Q\\) distributed uniformly along its circumference. A small electric dipole with fixed dipole moment of magnitude \\(p\\) is located on the \\(x\\)-axis at position \\(x > 0\\), oriented with its dipole moment vector \\(\\vec{p}\\) pointing in the \\(+x\\)-direction away from the ring. Assuming the electric potential energy \\(U\\) is defined to be zero at infinity, which of the following expressions correctly gives \\(U(x)\\) as a function of position \\(x\\)?"
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url: "https://nerd-notes.com/ubq/118134/"
date_modified: "2026-08-04T08:05:23+00:00"
---

# A thin, nonconducting ring of radius \(R\) lies in the \(yz\)-plane centered at the origin and carries a total positive charge \(+Q\) distributed uniformly along its circumference. A small electric dipole with fixed dipole moment of magnitude \(p\) is located on the \(x\)-axis at position \(x > 0\), oriented with its dipole moment vector \(\vec{p}\) pointing in the \(+x\)-direction away from the ring. Assuming the electric potential energy \(U\) is defined to be zero at infinity, which of the following expressions correctly gives \(U(x)\) as a function of position \(x\)?

A thin, nonconducting ring of radius \(R\) lies in the \(yz\)-plane centered at the origin and carries a total positive charge \(+Q\) distributed uniformly along its circumference. A small electric dipole with fixed dipole moment of magnitude \(p\) is located on the \(x\)-axis at position \(x > 0\), oriented with its dipole moment vector \(\vec{p}\) pointing in the \(+x\)-direction away from the ring. Assuming the electric potential energy \(U\) is defined to be zero at infinity, which of the following expressions correctly gives \(U(x)\) as a function of position \(x\)?

![A thin ring is drawn as a flat ellipse centered at the origin of a horizontal x-axis, representing a 3D perspective view. The ring has radius R, indicated by a solid line segment from the center to the top edge labeled R. The charge +Q is labeled near the ring. The horizontal x-axis extends to the right through the center of the ring. At a point on the x-axis labeled x, a small arrow representing the dipole moment vector points to the right and is labeled \vec{p}. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830723-d6QlqQ.jpg)

- **A.** \( U(x) = -\dfrac{1}{4\pi\varepsilon_0} \dfrac{Q p}{\sqrt{x^2 + R^2}} \)
- **B.** \( U(x) = -\dfrac{1}{4\pi\varepsilon_0} \dfrac{Q p}{x^2 + R^2} \)
- **C.** \( U(x) = -\dfrac{1}{4\pi\varepsilon_0} \dfrac{Q p x}{(x^2 + R^2)^{3/2}} \)
- **D.** \( U(x) = \dfrac{1}{4\pi\varepsilon_0} \dfrac{Q p x}{(x^2 + R^2)^{3/2}} \)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118134/*