---
title: "An insulating solid sphere of radius \\(R\\) has a non-uniform volume charge density given by \\(\\rho(r) = a r^2\\) for \\(0 \\le r \\le R\\), where \\(a\\) is a positive constant and \\(r\\) is the radial distance from the center. The total charge of the sphere is \\(Q\\). Which of the following expressions gives the magnitude of the electric field \\(E(r)\\) inside the sphere at a distance \\(r < R\\) from the center?"
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url: "https://nerd-notes.com/ubq/117963/"
date_modified: "2026-08-04T08:02:43+00:00"
---

# An insulating solid sphere of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = a r^2\) for \(0 \le r \le R\), where \(a\) is a positive constant and \(r\) is the radial distance from the center. The total charge of the sphere is \(Q\). Which of the following expressions gives the magnitude of the electric field \(E(r)\) inside the sphere at a distance \(r < R\) from the center?

An insulating solid sphere of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = a r^2\) for \(0 \le r \le R\), where \(a\) is a positive constant and \(r\) is the radial distance from the center. The total charge of the sphere is \(Q\). Which of the following expressions gives the magnitude of the electric field \(E(r)\) inside the sphere at a distance \(r < R\) from the center?

![A cross-sectional view of a solid sphere of radius R centered at the origin. The sphere is shaded with a radial gradient that is light at the center and becomes progressively darker toward the outer boundary at r = R, representing increasing volume charge density. A concentric dashed circle of radius r (where r < R) represents a spherical Gaussian surface inside the sphere. An arrow starts at the origin and extends to the dashed circle, labeled r. A second arrow starts at the origin and extends to the outer boundary of the sphere, labeled R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830562-mDCQtc.jpg)

- **A.** \(\dfrac{Q r}{4\pi \varepsilon_0 R^3}\)
- **B.** \(\dfrac{Q r^3}{4\pi \varepsilon_0 R^5}\)
- **C.** \(\dfrac{Q r^2}{4\pi \varepsilon_0 R^4}\)
- **D.** \(\dfrac{Q r^4}{4\pi \varepsilon_0 R^6}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117963/*