---
title: "A solid nonconducting sphere of radius \\(R\\) carries a total positive charge \\(Q\\) distributed with a volume charge density \\(\\rho(r) = \\dfrac{\\beta}{r}\\) for \\(0 < r \\le R\\), where \\(\\beta\\) is a positive constant. A second nonconducting sphere of radius \\(R\\) carries the same total charge \\(Q\\) distributed uniformly throughout its volume. Which of the following statements correctly describes and explains the behavior of the electric field magnitude \\(E(r)\\) inside each sphere in the limit as \\(r \\to 0^+\\)?"
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url: "https://nerd-notes.com/ubq/124545/"
date_modified: "2026-09-28T14:08:35+00:00"
---

# A solid nonconducting sphere of radius \(R\) carries a total positive charge \(Q\) distributed with a volume charge density \(\rho(r) = \dfrac{\beta}{r}\) for \(0 < r \le R\), where \(\beta\) is a positive constant. A second nonconducting sphere of radius \(R\) carries the same total charge \(Q\) distributed uniformly throughout its volume. Which of the following statements correctly describes and explains the behavior of the electric field magnitude \(E(r)\) inside each sphere in the limit as \(r \to 0^+\)?

A solid nonconducting sphere of radius \(R\) carries a total positive charge \(Q\) distributed with a volume charge density \(\rho(r) = \dfrac{\beta}{r}\) for \(0 < r \le R\), where \(\beta\) is a positive constant. A second nonconducting sphere of radius \(R\) carries the same total charge \(Q\) distributed uniformly throughout its volume. Which of the following statements correctly describes and explains the behavior of the electric field magnitude \(E(r)\) inside each sphere in the limit as \(r \to 0^+\)?

![Two circular cross sections representing spheres of radius \(R\) sit side by side horizontally. The left circle represents the non-uniform sphere with radial charge density \(\rho(r) = \dfrac{\beta}{r}\), indicated by text above it. A solid arrow extends from the center of the left circle toward the upper right at an angle of 45 degrees to the outer boundary, labeled \(R\). Concentric with the left circle is a dashed circle of smaller radius, with an arrow from the center pointing horizontally to the right to the dashed boundary, labeled \(r\). The right circle represents the uniform sphere with text \(\rho = \rho_0\) above it. A solid arrow extends from the center of the right circle toward the upper right at 45 degrees to the outer boundary, labeled \(R\). Concentric with the right circle is an identical dashed circle of radius \(r\), with a horizontal arrow from the center to the dashed boundary, labeled \(r\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604515-j2oMD5.jpg)

- **A.** For the non-uniform sphere, \(E(r)\) approaches a finite non-zero value because the enclosed charge scales as \(r^2\), whereas for the uniform sphere, \(E(r) \to 0\) because the enclosed charge scales as \(r^3\).
- **B.** For the non-uniform sphere, \(E(r) \to \infty\) because the volume charge density diverges as \(r \to 0^+\), whereas for the uniform sphere, \(E(r) \to 0\) because the enclosed charge scales as \(r^3\).
- **C.** For both spheres, \(E(r) \to 0\) because spherical symmetry requires the net electric field to vanish at the center of any spherically symmetric charge distribution.
- **D.** For the non-uniform sphere, \(E(r) \to 0\) because the enclosed charge scales as \(r^3\), whereas for the uniform sphere, \(E(r)\) approaches a finite non-zero value because the charge density is constant throughout the volume.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124545/*