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title: "Two thin, parallel circular rings each of radius \\(R\\) carry uniform charges \\(+Q\\) and \\(-Q\\), respectively. Ring 1 is centered at \\((0, 0, a)\\) parallel to the \\(xy\\)-plane. Ring 2 is centered at \\((b, 0, -a)\\) parallel to the \\(xy\\)-plane, where \\(a > 0\\) and \\(b > 0\\). Which of the following expressions best represents the leading-order electric field vector \\(\\vec{E}\\) at a point \\((0, 0, z)\\) on the \\(z\\)-axis at a distance \\(z \\gg a, b, R\\)?"
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url: "https://nerd-notes.com/ubq/118028/"
date_modified: "2026-08-04T08:03:03+00:00"
---

# Two thin, parallel circular rings each of radius \(R\) carry uniform charges \(+Q\) and \(-Q\), respectively. Ring 1 is centered at \((0, 0, a)\) parallel to the \(xy\)-plane. Ring 2 is centered at \((b, 0, -a)\) parallel to the \(xy\)-plane, where \(a > 0\) and \(b > 0\). Which of the following expressions best represents the leading-order electric field vector \(\vec{E}\) at a point \((0, 0, z)\) on the \(z\)-axis at a distance \(z \gg a, b, R\)?

Two thin, parallel circular rings each of radius \(R\) carry uniform charges \(+Q\) and \(-Q\), respectively. Ring 1 is centered at \((0, 0, a)\) parallel to the \(xy\)-plane. Ring 2 is centered at \((b, 0, -a)\) parallel to the \(xy\)-plane, where \(a > 0\) and \(b > 0\). Which of the following expressions best represents the leading-order electric field vector \(\vec{E}\) at a point \((0, 0, z)\) on the \(z\)-axis at a distance \(z \gg a, b, R\)?

![A 3D Cartesian coordinate system with origin O. A vertical dashed axis labeled z extends upward and downward. A horizontal dashed axis pointing right is labeled x, and a dashed axis pointing obliquely out of the page is labeled y. Ring 1 is drawn as a horizontal ellipse centered at (0,0,a) on the z-axis, labeled with charge +Q and radius R. Ring 2 is drawn as an identical horizontal ellipse centered at (b,0,-a), shifted to the right along the x-axis, labeled with charge -Q and radius R. A point P is marked on the positive z-axis far above Ring 1 at position (0,0,z). Dotted dimension lines indicate height a above the xy-plane for Ring 1, depth -a below the xy-plane for Ring 2, and horizontal displacement b along the x-axis for Ring 2. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830583-5b7j1Z.jpg)

- **A.** \(\vec{E} \approx \dfrac{Q}{4\pi\varepsilon_0 z^3} \left( -b\,\hat{i} + 2a\,\hat{k} \right)\)
- **B.** \(\vec{E} \approx \dfrac{Q}{4\pi\varepsilon_0 z^3} \left( b\,\hat{i} + 2a\,\hat{k} \right)\)
- **C.** \(\vec{E} \approx \dfrac{Q}{4\pi\varepsilon_0 z^3} \left( b\,\hat{i} + 4a\,\hat{k} \right)\)
- **D.** \(\vec{E} \approx \dfrac{Q}{4\pi\varepsilon_0 z^3} \left( 2b\,\hat{i} + 4a\,\hat{k} \right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118028/*