---
title: "A thin, nonconducting spherical shell of radius \\(R\\) is centered at the origin. The shell carries a non-uniform surface charge density given by \\(\\sigma(\\theta) = \\sigma_0 \\cos\\theta\\), where \\(\\sigma_0\\) is a positive constant and \\(\\theta\\) is the polar angle measured from the positive \\(z\\)-axis. Which of the following expressions gives the electric potential \\(V(z)\\) at a point on the positive \\(z\\)-axis at a distance \\(z > R\\) from the origin, relative to \\(V = 0\\) at infinity?"
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url: "https://nerd-notes.com/ubq/118122/"
date_modified: "2026-08-04T08:05:13+00:00"
---

# A thin, nonconducting spherical shell of radius \(R\) is centered at the origin. The shell carries a non-uniform surface charge density given by \(\sigma(\theta) = \sigma_0 \cos\theta\), where \(\sigma_0\) is a positive constant and \(\theta\) is the polar angle measured from the positive \(z\)-axis. Which of the following expressions gives the electric potential \(V(z)\) at a point on the positive \(z\)-axis at a distance \(z > R\) from the origin, relative to \(V = 0\) at infinity?

A thin, nonconducting spherical shell of radius \(R\) is centered at the origin. The shell carries a non-uniform surface charge density given by \(\sigma(\theta) = \sigma_0 \cos\theta\), where \(\sigma_0\) is a positive constant and \(\theta\) is the polar angle measured from the positive \(z\)-axis. Which of the following expressions gives the electric potential \(V(z)\) at a point on the positive \(z\)-axis at a distance \(z > R\) from the origin, relative to \(V = 0\) at infinity?

![A spherical shell of radius \(R\) centered at the origin of a 3D Cartesian axis system. A vertical line along the \(z\)-axis extends upward to a point labeled \(P\) at a distance \(z\) from the origin. A narrow circular ring on the upper hemisphere of the shell is highlighted at a polar angle \(\theta\) from the positive \(z\)-axis. An arrow labeled \(R\) extends from the origin to the outer edge of this ring element. Plus signs are distributed across the upper hemisphere and minus signs are distributed across the lower hemisphere. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830713-NCoe8I.jpg)

- **A.** \(\dfrac{\sigma_0 R^3}{3 \varepsilon_0 z^2}\)
- **B.** \(\dfrac{\sigma_0 R^2}{2 \varepsilon_0 z}\)
- **C.** \(\dfrac{\sigma_0 R^3}{2 \varepsilon_0 z^2}\)
- **D.** \(\dfrac{\sigma_0 R^3}{\varepsilon_0 z^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118122/*