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title: "A thin, flat, nonconducting annular disk lies in the xy-plane centered at the origin, with inner radius a and outer radius b, where 0 < a < b. The disk carries a uniform positive surface charge density \\(\\sigma\\). The electric field magnitude \\(E(z)\\) is evaluated at points along the positive z-axis. Which of the following pairs correctly identifies the leading functional dependence of \\(E(z)\\) on distance z in the near-center regime (\\(0 < z \\ll a\\)) and in the far-field regime (\\(z \\gg b\\))?"
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url: "https://nerd-notes.com/ubq/124546/"
date_modified: "2026-09-28T14:08:35+00:00"
---

# A thin, flat, nonconducting annular disk lies in the xy-plane centered at the origin, with inner radius a and outer radius b, where 0 < a < b. The disk carries a uniform positive surface charge density \(\sigma\). The electric field magnitude \(E(z)\) is evaluated at points along the positive z-axis. Which of the following pairs correctly identifies the leading functional dependence of \(E(z)\) on distance z in the near-center regime (\(0 < z \ll a\)) and in the far-field regime (\(z \gg b\))?

A thin, flat, nonconducting annular disk lies in the xy-plane centered at the origin, with inner radius a and outer radius b, where 0 < a < b. The disk carries a uniform positive surface charge density \(\sigma\). The electric field magnitude \(E(z)\) is evaluated at points along the positive z-axis. Which of the following pairs correctly identifies the leading functional dependence of \(E(z)\) on distance z in the near-center regime (\(0 < z \ll a\)) and in the far-field regime (\(z \gg b\))?

![A perspective view of a flat horizontal annulus centered at the origin of a three-dimensional coordinate system. The annulus is bounded by an inner ellipse of semi-major axis a and an outer ellipse of semi-major axis b, with shaded cross-hatching between the two concentric boundaries representing uniform surface charge density \(\sigma\). A vertical solid axis extends upward perpendicular to the plane through the origin, labeled z at its top arrow. A single point is marked on the positive z-axis at height z. A horizontal arrow points from the origin to the inner boundary labeled a, and a second horizontal arrow points from the origin to the outer boundary labeled b. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604515-JohPen.jpg)

- **A.** Near-center: Proportional to \(z\) | Far-field: Proportional to \(z^{-2}\)
- **B.** Near-center: Independent of \(z\) | Far-field: Proportional to \(z^{-2}\)
- **C.** Near-center: Proportional to \(z\) | Far-field: Proportional to \(z^{-3}\)
- **D.** Near-center: Independent of \(z\) | Far-field: Proportional to \(z^{-3}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124546/*