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title: "A thin, nonconducting ring of radius \\(R\\) lies in the \\(xy\\)-plane centered at the origin and carries a positive charge \\(Q\\) distributed uniformly along its circumference. The electric potential at a point on the positive \\(z\\)-axis is given by \\(V(z) = \\dfrac{Q}{4\\pi\\varepsilon_0\\sqrt{z^2 + R^2}}\\). Using the relationship between electric potential and electric field, which of the following gives the electric field vector \\(\\vec{E}(z)\\) on the positive \\(z\\)-axis and its correct first-order approximation in the limit \\(z \\gg R\\)?"
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url: "https://nerd-notes.com/ubq/124644/"
date_modified: "2026-09-28T14:08:55+00:00"
---

# A thin, nonconducting ring of radius \(R\) lies in the \(xy\)-plane centered at the origin and carries a positive charge \(Q\) distributed uniformly along its circumference. The electric potential at a point on the positive \(z\)-axis is given by \(V(z) = \dfrac{Q}{4\pi\varepsilon_0\sqrt{z^2 + R^2}}\). Using the relationship between electric potential and electric field, which of the following gives the electric field vector \(\vec{E}(z)\) on the positive \(z\)-axis and its correct first-order approximation in the limit \(z \gg R\)?

A thin, nonconducting ring of radius \(R\) lies in the \(xy\)-plane centered at the origin and carries a positive charge \(Q\) distributed uniformly along its circumference. The electric potential at a point on the positive \(z\)-axis is given by \(V(z) = \dfrac{Q}{4\pi\varepsilon_0\sqrt{z^2 + R^2}}\). Using the relationship between electric potential and electric field, which of the following gives the electric field vector \(\vec{E}(z)\) on the positive \(z\)-axis and its correct first-order approximation in the limit \(z \gg R\)?

![A horizontal flat ellipse centered at the origin represents a circular ring of radius \(R\) lying in the horizontal plane, labeled \(Q\). A vertical dashed coordinate axis labeled \(z\) extends upward through the center of the ring. A solid dot is marked on the vertical axis at a height labeled \(z\) above the center of the ring. A single straight arrow labeled \(R\) extends horizontally from the center of the ring to the perimeter of the ellipse. A unit vector arrow labeled \(\hat{k}\) points vertically upward along the \(z\)-axis. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604535-Z3L7In.jpg)

- **A.** \(\vec{E}(z) = \dfrac{Q}{4\pi\varepsilon_0 (z^2 + R^2)}\,\hat{k}\), which approximates to \(\dfrac{Q}{4\pi\varepsilon_0 z^2}\left(1 - \dfrac{R^2}{z^2}\right)\hat{k}\)
- **B.** \(\vec{E}(z) = \dfrac{Q z}{4\pi\varepsilon_0 (z^2 + R^2)^{3/2}}\,\hat{k}\), which approximates to \(\dfrac{Q}{4\pi\varepsilon_0 z^2}\left(1 - \dfrac{3R^2}{2z^2}\right)\hat{k}\)
- **C.** \(\vec{E}(z) = \dfrac{Q z}{4\pi\varepsilon_0 (z^2 + R^2)^{3/2}}\,\hat{k}\), which approximates to \(\dfrac{Q}{4\pi\varepsilon_0 z^2}\left(1 + \dfrac{3R^2}{2z^2}\right)\hat{k}\)
- **D.** \(\vec{E}(z) = -\dfrac{Q z}{4\pi\varepsilon_0 (z^2 + R^2)^{3/2}}\,\hat{k}\), which approximates to \(-\dfrac{Q}{4\pi\varepsilon_0 z^2}\left(1 - \dfrac{3R^2}{2z^2}\right)\hat{k}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124644/*