### Problem StatementA uniform charge density of 65 nC/m³ is distributed throughout a spherical volume with a radius of 16.5 cm. Consider a cubical surface (with an edge length of 3.1 cm) completely inside the sphere. Determine the electric flux through this surface.**Instructions:** Round your answer to 2 decimal places.### Explanation**Diagram:**The provided diagram illustrates a sphere with a radius of 16.5 cm. Inside this sphere, there is a smaller cube with an edge length of 3.1 cm. The cube is entirely contained within the spherical volume.**Analysis:**The task requires determining the electric flux through the surface of the cube using the information on charge density.**Solution Steps:**1. **Volume of the Cube:** - Volume of the cube, $ V_{\text{cube}} $ = $ a^3 $ where $ a $ is the edge length of the cube. - Given $ a = 3.1 $ cm (or 0.031 m for consistency in SI units). \[ V_{\text{cube}} = (0.031)^3 = 2.9851 \times 10^{-5} \text{ m}^3 \]2. **Charge Enclosed in the Cube:** The charge density is uniform and given by $ \rho = 65 \text{ nC/m}^3 = 65 \times 10^{-9} \text{ C/m}^3 $. The total charge enclosed by the cube, $ Q_{\text{cube}} $: \[ Q_{\text{cube}} = \rho \times V_{\text{cube}} = 65 \times 10^{-9} \times 2.9851 \times 10^{-5} \text{ C} \] \[ Q_{\text{cube}} = 1.94 \times 10^{-12} \text{ C} \]3. **Electric Flux through the Cube’s Surface:** The electric flux, $ \Phi $, through any closed surface enclosing a charge $ Q $ is given by Gauss's Law: \[ \Phi = \frac{Q_{\text{cube}}}{\epsilon_0} \] where \( \epsilon_0 = 8.

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A uniform charge density of 65 nC/m3 is distributed throughout a spherical volume (radius = 16.5 cm). Consider a cubical (3.1 cm along the edge) surface completely inside the sphere. Determine the electric flux through this surface. %3D

A uniform charge density of 65 nC/m3 is distributed throughout a spherical volume (radius = 16.5 cm). Consider a cubical (3.1 cm along the edge) surface completely inside the sphere. Determine the electric flux through this surface. %3D

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Electric Charges and Fields

One of the fundamental properties is the electromagnetic property. Charge cannot be destroyed by any process and this contributes formally to the law of charge conservation.

Question

### Problem Statement

A uniform charge density of 65 nC/m³ is distributed throughout a spherical volume with a radius of 16.5 cm. Consider a cubical surface (with an edge length of 3.1 cm) completely inside the sphere. Determine the electric flux through this surface.

**Instructions:** Round your answer to 2 decimal places.

### Explanation

**Diagram:**
The provided diagram illustrates a sphere with a radius of 16.5 cm. Inside this sphere, there is a smaller cube with an edge length of 3.1 cm. The cube is entirely contained within the spherical volume.

**Analysis:**

The task requires determining the electric flux through the surface of the cube using the information on charge density.

**Solution Steps:**

1. **Volume of the Cube:**

- Volume of the cube, $ V_{\text{cube}} $ = $ a^3 $ where $ a $ is the edge length of the cube.
- Given $ a = 3.1 $ cm (or 0.031 m for consistency in SI units).

\[
V_{\text{cube}} = (0.031)^3 = 2.9851 \times 10^{-5} \text{ m}^3
\]

2. **Charge Enclosed in the Cube:**

The charge density is uniform and given by $ \rho = 65 \text{ nC/m}^3 = 65 \times 10^{-9} \text{ C/m}^3 $.

The total charge enclosed by the cube, $ Q_{\text{cube}} $:

\[
Q_{\text{cube}} = \rho \times V_{\text{cube}} = 65 \times 10^{-9} \times 2.9851 \times 10^{-5} \text{ C}
\]

\[
Q_{\text{cube}} = 1.94 \times 10^{-12} \text{ C}
\]

3. **Electric Flux through the Cube’s Surface:**

The electric flux, $ \Phi $, through any closed surface enclosing a charge $ Q $ is given by Gauss's Law:

\[
\Phi = \frac{Q_{\text{cube}}}{\epsilon_0}
\]

where \( \epsilon_0 = 8.

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