---
title: "A solid nonconducting sphere of radius \\(R\\) has a non-uniform volume charge density given by \\(\\rho(r) = \\rho_0 \\dfrac{r}{R}\\) for \\(r \\le R\\), where \\(\\rho_0\\) is a constant, and a total net charge \\(Q\\). The sphere is assembled by bringing infinitesimal charge elements from infinity and depositing them layer by layer in concentric spherical shells.  Which of the following expressions represents the total work required by an external agent to assemble this sphere?"
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url: "https://nerd-notes.com/ubq/121101/"
date_modified: "2026-08-23T04:57:54+00:00"
---

# A solid nonconducting sphere of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = \rho_0 \dfrac{r}{R}\) for \(r \le R\), where \(\rho_0\) is a constant, and a total net charge \(Q\). The sphere is assembled by bringing infinitesimal charge elements from infinity and depositing them layer by layer in concentric spherical shells.

Which of the following expressions represents the total work required by an external agent to assemble this sphere?

A solid nonconducting sphere of radius \(R\) has a non-uniform volume charge density given by \(\rho(r) = \rho_0 \dfrac{r}{R}\) for \(r \le R\), where \(\rho_0\) is a constant, and a total net charge \(Q\). The sphere is assembled by bringing infinitesimal charge elements from infinity and depositing them layer by layer in concentric spherical shells.

Which of the following expressions represents the total work required by an external agent to assemble this sphere?

![A large circle of radius R representing a solid sphere with a central origin point. Shading within the circle gets progressively darker from the center outward to represent non-uniform volume charge density. A dashed thin circular ring of radius r and thickness dr is shown within the sphere. A straight radial line from the center to the dashed ring is labeled r, and a straight radial line from the center to the outer perimeter is labeled R. An arrow pointing inward from outside the sphere toward the dashed ring indicates an infinitesimal charge element dq being brought in from infinity. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461074-Nmc8aW.jpg)

- **A.** \(\dfrac{Q^2}{7\pi\varepsilon_0 R}\)
- **B.** \(\dfrac{Q^2}{8\pi\varepsilon_0 R}\)
- **C.** \(\dfrac{3Q^2}{20\pi\varepsilon_0 R}\)
- **D.** \(\dfrac{Q^2}{5\pi\varepsilon_0 R}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121101/*