---
title: "A thin, nonconducting disk of radius \\(R\\) lies in the \\(xy\\)-plane centered at the origin and carries a uniform positive surface charge density \\(\\sigma\\), giving a total charge \\(Q = \\sigma \\pi R^2\\).  The electric potential at a point on the positive \\(z\\)-axis is given by \\(V(z) = \\dfrac{\\sigma}{2\\varepsilon_0}\\left(\\sqrt{z^2 + R^2} – z\\right)\\).  Which of the following correctly describes the behavior of the potential \\(V(z)\\) and the axial electric field component \\(E_z(z) = -\\dfrac{dV}{dz}\\) in the asymptotic limits \\(z \\ll R\\) and \\(z \\gg R\\)?"
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url: "https://nerd-notes.com/ubq/124559/"
date_modified: "2026-09-28T14:08:38+00:00"
---

# A thin, nonconducting disk of radius \(R\) lies in the \(xy\)-plane centered at the origin and carries a uniform positive surface charge density \(\sigma\), giving a total charge \(Q = \sigma \pi R^2\).

The electric potential at a point on the positive \(z\)-axis is given by \(V(z) = \dfrac{\sigma}{2\varepsilon_0}\left(\sqrt{z^2 + R^2} – z\right)\).

Which of the following correctly describes the behavior of the potential \(V(z)\) and the axial electric field component \(E_z(z) = -\dfrac{dV}{dz}\) in the asymptotic limits \(z \ll R\) and \(z \gg R\)?

A thin, nonconducting disk of radius \(R\) lies in the \(xy\)-plane centered at the origin and carries a uniform positive surface charge density \(\sigma\), giving a total charge \(Q = \sigma \pi R^2\).

The electric potential at a point on the positive \(z\)-axis is given by \(V(z) = \dfrac{\sigma}{2\varepsilon_0}\left(\sqrt{z^2 + R^2} - z\right)\).

Which of the following correctly describes the behavior of the potential \(V(z)\) and the axial electric field component \(E_z(z) = -\dfrac{dV}{dz}\) in the asymptotic limits \(z \ll R\) and \(z \gg R\)?

![A flat ellipse viewed at a perspective tilt represents a circular disk centered at the origin in the horizontal plane. A shaded fill indicates uniform surface charge density, labeled with the symbol \(\sigma\) placed on the top surface of the disk. A straight solid line segment extends horizontally from the center of the disk to its right edge, labeled \(R\). A vertical solid line passes through the center of the disk, representing the \(z\)-axis, with an arrowhead at the top labeled \(z\). A solid dot is marked on the vertical axis above the center of the disk, labeled with a coordinate height \(z\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604517-eUi1Ak.jpg)

- **A.** For \(z \ll R\), \(V(z)\) is quadratic in \(z\), leading to \(E_z \to 0\) near the center; for \(z \gg R\), \(V(z)\) decreases as \(\dfrac{1}{z^2}\), behaving like an electric dipole.
- **B.** For \(z \ll R\), \(V(z)\) is linear in \(z\) with \(E_z \approx \dfrac{\sigma}{\varepsilon_0}\); for \(z \gg R\), \(V(z)\) decreases as \(\dfrac{1}{z}\), asymptotically matching the potential of a point charge of magnitude \(2Q\).
- **C.** For \(z \ll R\), \(V(z)\) is quadratic in \(z\) with \(E_z \approx \dfrac{\sigma z}{2\varepsilon_0 R}\); for \(z \gg R\), \(V(z)\) decreases as \(\dfrac{1}{z}\), asymptotically matching the potential of a point charge of magnitude \(Q\).
- **D.** For \(z \ll R\), \(V(z)\) is linear in \(z\) with a uniform field \(E_z \approx \dfrac{\sigma}{2\varepsilon_0}\); for \(z \gg R\), \(V(z)\) decreases as \(\dfrac{1}{z}\), asymptotically matching the potential of a point charge of magnitude \(Q\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124559/*