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title: "Two identical thin nonconducting rings, each of radius \\(R\\) and carrying a uniform positive charge \\(+Q\\), are fixed parallel to each other with their centers on the \\(x\\)-axis at \\(x = -d\\) and \\(x = +d\\), where \\(d < \\dfrac{R}{\\sqrt{2}}\\). The planes of both rings are perpendicular to the \\(x\\)-axis, and the origin is located midway between them. An electron of charge \\(-e\\) is constrained to move along the \\(x\\)-axis under the influence of the electrostatic force from the two rings. Which of the following best describes the qualitative graph of the electric force \\(F_x\\) exerted on the electron as a function of position \\(x\\) along the entire axis from \\(x = -\\infty\\) to \\(x = +\\infty\\)?"
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url: "https://nerd-notes.com/ubq/124665/"
date_modified: "2026-09-28T14:09:00+00:00"
---

# Two identical thin nonconducting rings, each of radius \(R\) and carrying a uniform positive charge \(+Q\), are fixed parallel to each other with their centers on the \(x\)-axis at \(x = -d\) and \(x = +d\), where \(d < \dfrac{R}{\sqrt{2}}\). The planes of both rings are perpendicular to the \(x\)-axis, and the origin is located midway between them. An electron of charge \(-e\) is constrained to move along the \(x\)-axis under the influence of the electrostatic force from the two rings. Which of the following best describes the qualitative graph of the electric force \(F_x\) exerted on the electron as a function of position \(x\) along the entire axis from \(x = -\infty\) to \(x = +\infty\)?

Two identical thin nonconducting rings, each of radius \(R\) and carrying a uniform positive charge \(+Q\), are fixed parallel to each other with their centers on the \(x\)-axis at \(x = -d\) and \(x = +d\), where \(d < \dfrac{R}{\sqrt{2}}\). The planes of both rings are perpendicular to the \(x\)-axis, and the origin is located midway between them. An electron of charge \(-e\) is constrained to move along the \(x\)-axis under the influence of the electrostatic force from the two rings. Which of the following best describes the qualitative graph of the electric force \(F_x\) exerted on the electron as a function of position \(x\) along the entire axis from \(x = -\infty\) to \(x = +\infty\)?

![A horizontal axis line labeled x passes horizontally through the center of the diagram, with an arrow pointing to the right. Two identical vertical ellipses representing thin circular rings of radius R are shown in perspective, centered on the horizontal axis at positions marked with short vertical tick marks labeled -d and +d. A central tick mark on the axis at the midpoint between the rings is labeled 0. Each ring has a charge label +Q positioned directly above it. A dashed radial line segment extends upward from the center of the left ring to its top rim, labeled R. A small filled circular dot representing an electron is located on the horizontal axis between the origin and the right ring, labeled -e. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604539-DzWzIT.jpg)

- **A.** A curve with rotational symmetry about the origin that diverges to vertical asymptotes at the ring positions \(x = \pm d\), with \(F_x \to +\infty\) as \(x \to d^-\) and \(F_x \to -\infty\) as \(x \to -d^+\).
- **B.** A continuous, smooth curve with rotational symmetry about the origin, having a negative slope at \(x = 0\), reaching a single local minimum at a position \(x > 0\), and asymptotically approaching zero as \(x \to \pm\infty\).
- **C.** A continuous, smooth curve with rotational symmetry about the origin, having a positive slope at \(x = 0\), reaching a single local maximum at a position \(x > 0\), and asymptotically approaching zero as \(x \to \pm\infty\).
- **D.** A continuous, smooth curve with reflectional symmetry across the vertical axis (\(F_x(-x) = F_x(x)\)), having a local minimum at \(x = 0\) and asymptotically approaching zero as \(x \to \pm\infty\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124665/*