---
title: "A solid non-conducting sphere of radius \\(R\\) has a non-uniform volume charge density given by \\(\\rho(r) = \\rho_0\\left(1 – \\dfrac{r}{R}\\right)\\) for \\(r \\le R\\) and \\(\\rho(r) = 0\\) for \\(r > R\\), where \\(\\rho_0\\) is a positive constant and \\(r\\) is the distance from the center of the sphere.  The electric potential is defined to be zero at an infinite distance from the sphere (\\(V(\\infty) = 0\\)).  Which of the following expressions represents the electric potential \\(V(r)\\) at a distance \\(r < R\\) from the center of the sphere?"
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url: "https://nerd-notes.com/ubq/121084/"
date_modified: "2026-08-23T04:57:45+00:00"
---

# A solid non-conducting sphere of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = \rho_0\left(1 – \dfrac{r}{R}\right)\) for \(r \le R\) and \(\rho(r) = 0\) for \(r > R\), where \(\rho_0\) is a positive constant and \(r\) is the distance from the center of the sphere.

The electric potential is defined to be zero at an infinite distance from the sphere (\(V(\infty) = 0\)).

Which of the following expressions represents the electric potential \(V(r)\) at a distance \(r < R\) from the center of the sphere?

A solid non-conducting sphere of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = \rho_0\left(1 - \dfrac{r}{R}\right)\) for \(r \le R\) and \(\rho(r) = 0\) for \(r > R\), where \(\rho_0\) is a positive constant and \(r\) is the distance from the center of the sphere.

The electric potential is defined to be zero at an infinite distance from the sphere (\(V(\infty) = 0\)).

Which of the following expressions represents the electric potential \(V(r)\) at a distance \(r < R\) from the center of the sphere?

![A single circle representing the spherical cross-section of radius \(R\). A horizontal dashed line segment extends from the central origin to the right boundary of the circle, with a single arrow pointing to the outer boundary labeled \(R\) above the line. A second dashed line segment extends from the center into the upper-right quadrant at an angle of 45 degrees, terminating at a solid black dot, with an arrow labeled \(r\) above it. The interior of the circle displays a smooth radial gradient that is dark gray at the center and fades to white at the outer boundary. The label \(\rho(r)\) appears in the upper-left interior of the circle. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461065-HJAd4a.jpg)

- **A.** \(V(r) = \dfrac{\rho_0 R^2}{12\varepsilon_0}\left(1 - 2\left(\dfrac{r}{R}\right)^2 + \left(\dfrac{r}{R}\right)^3\right)\)
- **B.** \(V(r) = \dfrac{\rho_0 R^2}{12\varepsilon_0}\left(1 + 2\left(\dfrac{r}{R}\right)^2 - \left(\dfrac{r}{R}\right)^3\right)\)
- **C.** \(V(r) = \dfrac{\rho_0 R^2}{6\varepsilon_0}\left(1 - 2\left(\dfrac{r}{R}\right)^2 + \left(\dfrac{r}{R}\right)^3\right)\)
- **D.** \(V(r) = \dfrac{\rho_0 R^2}{12\varepsilon_0}\left(2 - 2\left(\dfrac{r}{R}\right)^2 + \left(\dfrac{r}{R}\right)^3\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121084/*