---
title: "Three solid insulating spheres, each of radius \\(R\\) and uniform mass density \\(\\rho_m\\), fall vertically through a viscous fluid in a region containing a uniform upward electric field of magnitude \\(E\\). The fluid exerts a resistive drag force \\(F_d = b v\\) on each sphere, where \\(b\\) is a constant. The volume charge densities \\(\\rho(r)\\) of the spheres as a function of radial distance \\(r\\) from their centers (\\(0 \\le r \\le R\\)) are given in the table below.  | Sphere | Volume Charge Density \\(\\rho(r)\\) | |—|—| | I | \\(\\rho_0 \\left(\\dfrac{r}{R}\\right)\\) | | II | \\(\\rho_0 \\left(\\dfrac{r}{R}\\right)^2\\) | | III | \\(\\dfrac{3}{5}\\rho_0\\) |  Assuming that the gravitational force on each sphere is greater than the electric force acting on it, which of the following correctly ranks the magnitudes of the downward terminal velocities \\(v_I\\), \\(v_{II}\\), and \\(v_{III}\\) of the three spheres?"
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url: "https://nerd-notes.com/ubq/118026/"
date_modified: "2026-08-04T08:03:01+00:00"
---

# Three solid insulating spheres, each of radius \(R\) and uniform mass density \(\rho_m\), fall vertically through a viscous fluid in a region containing a uniform upward electric field of magnitude \(E\). The fluid exerts a resistive drag force \(F_d = b v\) on each sphere, where \(b\) is a constant. The volume charge densities \(\rho(r)\) of the spheres as a function of radial distance \(r\) from their centers (\(0 \le r \le R\)) are given in the table below.

| Sphere | Volume Charge Density \(\rho(r)\) |
|—|—|
| I | \(\rho_0 \left(\dfrac{r}{R}\right)\) |
| II | \(\rho_0 \left(\dfrac{r}{R}\right)^2\) |
| III | \(\dfrac{3}{5}\rho_0\) |

Assuming that the gravitational force on each sphere is greater than the electric force acting on it, which of the following correctly ranks the magnitudes of the downward terminal velocities \(v_I\), \(v_{II}\), and \(v_{III}\) of the three spheres?

Three solid insulating spheres, each of radius \(R\) and uniform mass density \(\rho_m\), fall vertically through a viscous fluid in a region containing a uniform upward electric field of magnitude \(E\). The fluid exerts a resistive drag force \(F_d = b v\) on each sphere, where \(b\) is a constant. The volume charge densities \(\rho(r)\) of the spheres as a function of radial distance \(r\) from their centers (\(0 \le r \le R\)) are given in the table below.

| Sphere | Volume Charge Density \(\rho(r)\) |
|---|---|
| I | \(\rho_0 \left(\dfrac{r}{R}\right)\) |
| II | \(\rho_0 \left(\dfrac{r}{R}\right)^2\) |
| III | \(\dfrac{3}{5}\rho_0\) |

Assuming that the gravitational force on each sphere is greater than the electric force acting on it, which of the following correctly ranks the magnitudes of the downward terminal velocities \(v_I\), \(v_{II}\), and \(v_{III}\) of the three spheres?

![A vertical schematic showing a single sphere falling downward. A dashed vertical arrow labeled E points upward. From the center of the sphere, three distinct force vectors are drawn: a straight arrow pointing vertically downward labeled F_g, a straight arrow pointing vertically upward labeled F_E, and a second straight arrow pointing vertically upward labeled F_d. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830581-y5qYKw.jpg)

- **A.** \(v_{II} = v_{III} > v_I\)
- **B.** \(v_I > v_{II} = v_{III}\)
- **C.** \(v_{III} > v_I = v_{II}\)
- **D.** \(v_{II} > v_I > v_{III}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118026/*