---
title: "A thin, flat circular disk of radius \\(R\\) lies in the \\(xy\\)-plane, centered at the origin. The disk carries a non-uniform surface charge density given by \\(\\sigma(r) = \\sigma_0 \\left(\\dfrac{r}{R}\\right)^2\\), where \\(r\\) is the distance from the origin and \\(\\sigma_0\\) is a positive constant. What is the ratio of the electric potential \\(V(R)\\) at a point on the central axis at \\(z = R\\) to the electric potential \\(V(0)\\) at the center of the disk?"
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url: "https://nerd-notes.com/ubq/118174/"
date_modified: "2026-08-04T08:05:48+00:00"
---

# A thin, flat circular disk of radius \(R\) lies in the \(xy\)-plane, centered at the origin. The disk carries a non-uniform surface charge density given by \(\sigma(r) = \sigma_0 \left(\dfrac{r}{R}\right)^2\), where \(r\) is the distance from the origin and \(\sigma_0\) is a positive constant. What is the ratio of the electric potential \(V(R)\) at a point on the central axis at \(z = R\) to the electric potential \(V(0)\) at the center of the disk?

A thin, flat circular disk of radius \(R\) lies in the \(xy\)-plane, centered at the origin. The disk carries a non-uniform surface charge density given by \(\sigma(r) = \sigma_0 \left(\dfrac{r}{R}\right)^2\), where \(r\) is the distance from the origin and \(\sigma_0\) is a positive constant. What is the ratio of the electric potential \(V(R)\) at a point on the central axis at \(z = R\) to the electric potential \(V(0)\) at the center of the disk?

![A thin circular disk rendered in gray shading is shown tilted in perspective in the xy-plane. A vertical z-axis extends upward from the origin at the center of the disk. A point labeled (0,0,R) is marked with a filled dot on the z-axis at height R above the center. A dashed circle of radius r concentric with the disk is drawn on the disk surface, representing a thin ring element of width dr. A straight line segment connects a point on the dashed ring to the dot at (0,0,R), forming a right triangle with a vertical dashed line of height R along the z-axis and a horizontal dashed line of length r along the disk surface. The radius of the full disk is labeled R at the outer edge along the surface. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830748-smXHXb.jpg)

- **A.** \(\dfrac{2 - \sqrt{2}}{2}\)
- **B.** \(\sqrt{2} - 1\)
- **C.** \(2 - \sqrt{2}\)
- **D.** \(2\sqrt{2} - 2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118174/*