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title: "A thin insulating disk of radius \\(R\\) lying in the \\(xy\\)-plane is centered at the origin and carries a uniform positive charge \\(Q\\). The electric potential on the central axis at a distance \\(z > 0\\) from the center of the disk is given by the expression \\[ V(z) = \\dfrac{Q}{2\\pi \\varepsilon_0 R^2} \\left(\\sqrt{z^2 + R^2} – z\\right) \\] A point charge with the same total charge \\(Q\\) is fixed at the origin, producing an electric potential \\(V_{\\text{point}}(z) = \\dfrac{Q}{4\\pi\\varepsilon_0 z}\\) along the positive \\(z\\)-axis. Which of the following correctly compares \\(V(z)\\) to \\(V_{\\text{point}}(z)\\) for all finite \\(z > 0\\), and gives the leading dependence of the difference \\(\\Delta V(z) = V_{\\text{point}}(z) – V(z)\\) on \\(z\\) in the limit \\(z \\gg R\\)?"
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url: "https://nerd-notes.com/ubq/121099/"
date_modified: "2026-08-23T04:57:51+00:00"
---

# A thin insulating disk of radius \(R\) lying in the \(xy\)-plane is centered at the origin and carries a uniform positive charge \(Q\). The electric potential on the central axis at a distance \(z > 0\) from the center of the disk is given by the expression \[ V(z) = \dfrac{Q}{2\pi \varepsilon_0 R^2} \left(\sqrt{z^2 + R^2} – z\right) \] A point charge with the same total charge \(Q\) is fixed at the origin, producing an electric potential \(V_{\text{point}}(z) = \dfrac{Q}{4\pi\varepsilon_0 z}\) along the positive \(z\)-axis. Which of the following correctly compares \(V(z)\) to \(V_{\text{point}}(z)\) for all finite \(z > 0\), and gives the leading dependence of the difference \(\Delta V(z) = V_{\text{point}}(z) – V(z)\) on \(z\) in the limit \(z \gg R\)?

A thin insulating disk of radius \(R\) lying in the \(xy\)-plane is centered at the origin and carries a uniform positive charge \(Q\). The electric potential on the central axis at a distance \(z > 0\) from the center of the disk is given by the expression \[ V(z) = \dfrac{Q}{2\pi \varepsilon_0 R^2} \left(\sqrt{z^2 + R^2} - z\right) \] A point charge with the same total charge \(Q\) is fixed at the origin, producing an electric potential \(V_{\text{point}}(z) = \dfrac{Q}{4\pi\varepsilon_0 z}\) along the positive \(z\)-axis. Which of the following correctly compares \(V(z)\) to \(V_{\text{point}}(z)\) for all finite \(z > 0\), and gives the leading dependence of the difference \(\Delta V(z) = V_{\text{point}}(z) - V(z)\) on \(z\) in the limit \(z \gg R\)?

![A horizontal flat ellipse representing a thin circular disk of radius R is centered at the origin. A vertical dashed line extends upward from the center of the disk along the z-axis to a solid black dot labeled point P at a distance z above the origin. A straight solid line with an arrowhead extends from the center of the disk to its outer rim at a 30-degree angle, labeled R. Plus signs are evenly spaced across the top surface of the disk. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461070-dvxp0W.jpg)

- **A.** \(V(z) > V_{\text{point}}(z)\), and \(\Delta V(z)\) is proportional to \(\dfrac{1}{z^2}\)
- **B.** \(V(z) < V_{\text{point}}(z)\), and \(\Delta V(z)\) is proportional to \(\dfrac{1}{z^3}\)
- **C.** \(V(z) < V_{\text{point}}(z)\), and \(\Delta V(z)\) is proportional to \(\dfrac{1}{z^2}\)
- **D.** \(V(z) > V_{\text{point}}(z)\), and \(\Delta V(z)\) is proportional to \(\dfrac{1}{z^3}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121099/*