---
title: "A parallel-plate capacitor with plates of area \\(A\\) is charged in a vacuum to a uniform surface charge density \\(+\\sigma\\) on one plate and \\(-\\sigma\\) on the other. The capacitor is then disconnected from the charging source so that the charge on the plates remains constant. In an experiment to determine \\(\\sigma\\), students measure the electric potential difference \\(\\Delta V\\) across the plates as the plate separation distance \\(d\\) is varied while keeping \\(d \\ll \\sqrt{A}\\).  Which of the following identifies the quantities that should be plotted on the vertical and horizontal axes to produce a linear graph, along with the correct expression relating \\(\\sigma\\) to the slope \\(m\\) of the resulting line of best fit?"
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url: "https://nerd-notes.com/ubq/121125/"
date_modified: "2026-08-23T04:58:12+00:00"
---

# A parallel-plate capacitor with plates of area \(A\) is charged in a vacuum to a uniform surface charge density \(+\sigma\) on one plate and \(-\sigma\) on the other. The capacitor is then disconnected from the charging source so that the charge on the plates remains constant. In an experiment to determine \(\sigma\), students measure the electric potential difference \(\Delta V\) across the plates as the plate separation distance \(d\) is varied while keeping \(d \ll \sqrt{A}\).

Which of the following identifies the quantities that should be plotted on the vertical and horizontal axes to produce a linear graph, along with the correct expression relating \(\sigma\) to the slope \(m\) of the resulting line of best fit?

A parallel-plate capacitor with plates of area \(A\) is charged in a vacuum to a uniform surface charge density \(+\sigma\) on one plate and \(-\sigma\) on the other. The capacitor is then disconnected from the charging source so that the charge on the plates remains constant. In an experiment to determine \(\sigma\), students measure the electric potential difference \(\Delta V\) across the plates as the plate separation distance \(d\) is varied while keeping \(d \ll \sqrt{A}\).

Which of the following identifies the quantities that should be plotted on the vertical and horizontal axes to produce a linear graph, along with the correct expression relating \(\sigma\) to the slope \(m\) of the resulting line of best fit?

![Two parallel horizontal flat rectangular plates of area \(A\) are positioned with a vertical separation distance \(d\). The top plate has a uniform charge density \(+\sigma\) indicated on its upper surface, and the bottom plate has a uniform charge density \(-\sigma\) indicated on its lower surface. Two lead wires connect the top and bottom plates to a circular voltmeter labeled \(\Delta V\). A vertical two-headed arrow labeled \(d\) extends between the inner faces of the plates. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461092-bCNrP1.jpg)

- **A.** Vertical axis: \(\Delta V\); Horizontal axis: \(\dfrac{1}{d}\); Relationship: \(\sigma = \dfrac{\varepsilon_0}{m}\)
- **B.** Vertical axis: \(\dfrac{1}{\Delta V}\); Horizontal axis: \(d\); Relationship: \(\sigma = \dfrac{1}{\varepsilon_0 m}\)
- **C.** Vertical axis: \(\Delta V\); Horizontal axis: \(d^2\); Relationship: \(\sigma = 2\varepsilon_0 m\)
- **D.** Vertical axis: \(\Delta V\); Horizontal axis: \(d\); Relationship: \(\sigma = \varepsilon_0 m\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121125/*