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title: "A thin, nonconducting circular disk of radius \\(R\\) lies in the \\(xy\\)-plane centered at the origin and carries a uniform positive surface charge density \\(\\sigma\\), corresponding to a total charge \\(Q = \\sigma \\pi R^2\\). The magnitude of the electric field at a point on the positive \\(z\\)-axis is given by the expression \\[E(z) = \\dfrac{\\sigma}{2\\varepsilon_0} \\left(1 – \\dfrac{z}{\\sqrt{z^2 + R^2}}\\right)\\] Which of the following statements correctly describes the behavior of the electric field in the limits \\(z \\ll R\\) and \\(z \\gg R\\)?"
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url: "https://nerd-notes.com/ubq/124741/"
date_modified: "2026-09-28T14:09:40+00:00"
---

# A thin, nonconducting circular disk of radius \(R\) lies in the \(xy\)-plane centered at the origin and carries a uniform positive surface charge density \(\sigma\), corresponding to a total charge \(Q = \sigma \pi R^2\). The magnitude of the electric field at a point on the positive \(z\)-axis is given by the expression
\[E(z) = \dfrac{\sigma}{2\varepsilon_0} \left(1 – \dfrac{z}{\sqrt{z^2 + R^2}}\right)\]
Which of the following statements correctly describes the behavior of the electric field in the limits \(z \ll R\) and \(z \gg R\)?

A thin, nonconducting circular disk of radius \(R\) lies in the \(xy\)-plane centered at the origin and carries a uniform positive surface charge density \(\sigma\), corresponding to a total charge \(Q = \sigma \pi R^2\). The magnitude of the electric field at a point on the positive \(z\)-axis is given by the expression
\[E(z) = \dfrac{\sigma}{2\varepsilon_0} \left(1 - \dfrac{z}{\sqrt{z^2 + R^2}}\right)\]
Which of the following statements correctly describes the behavior of the electric field in the limits \(z \ll R\) and \(z \gg R\)?

![A perspective view of a circular disk of radius \(R\) lying horizontally in the \(xy\)-plane, centered at the origin. The disk is drawn as a flat ellipse with light gray fill and a solid thin black boundary. A vertical dashed line passes through the center of the disk along the \(z\)-axis, extending upward and downward. An open origin point at the center of the disk has a horizontal radial arrow pointing toward the right edge of the disk labeled \(R\). A solid point on the positive \(z\)-axis above the disk is labeled \(P\), and a vertical double-headed dimension line extends from the origin to point \(P\), labeled \(z\). A straight solid arrow originating at point \(P\) points vertically upward along the \(z\)-axis, labeled \(\vec{E}\). The surface of the disk is labeled \(\sigma\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604580-LncUUJ.jpg)

- **A.** For \(z \ll R\), the electric field approaches \(\dfrac{\sigma}{\varepsilon_0}\), modeling an infinite conducting surface; for \(z \gg R\), the field decays as \(\dfrac{1}{z}\), modeling an infinitely long line of charge.
- **B.** For \(z \ll R\), the electric field is proportional to \(z\), increasing linearly from zero at the center; for \(z \gg R\), the leading-order nonzero term decays as \(\dfrac{1}{z^2}\), with magnitude \(\dfrac{Q}{4\pi\varepsilon_0 z^2}\).
- **C.** For \(z \ll R\), the electric field approaches \(\dfrac{\sigma}{2\varepsilon_0}\), modeling an infinite nonconducting sheet; for \(z \gg R\), the leading-order nonzero term decays as \(\dfrac{1}{z^3}\), modeling an electric dipole.
- **D.** For \(z \ll R\), the electric field approaches \(\dfrac{\sigma}{2\varepsilon_0}\), modeling an infinite nonconducting sheet; for \(z \gg R\), the leading-order nonzero term decays as \(\dfrac{1}{z^2}\), modeling a point charge with magnitude \(\dfrac{Q}{4\pi\varepsilon_0 z^2}\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124741/*