College Physics
11th Edition
ISBN: 9781305952300
Author: Raymond A. Serway, Chris Vuille
Publisher: Cengage Learning
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![### Example Problem for Applying Gauss's Law
#### Instructions
In each figure, the dashed circle represents the cross section of a closed Gaussian surface.
#### Diagram Description
In the provided diagram:
- Several point charges are shown, both inside and outside the dashed circle (representing a Gaussian surface):
- Inside the circle:
- +2.0 mC (milliCoulombs)
- +3.0 mC
- + an unknown charge (labeled as \(q = ?\))
- Outside the circle:
- +2.0 mC
- -3.0 mC (Note: mC is milliCoulombs, a unit for measuring electric charge)
#### Problem Statement
The net electric flux passing through the Gaussian surface is given as \(+3.9 \times 10^8 \, \text{N} \cdot \text{m}^2/\text{C}\).
Question: What is the value of the unknown charge within the Gaussian surface?
---
### Solution
To find the unknown charge (\( q \)), we use Gauss's Law which states:
\[ \Phi_E = \frac{q_{\text{enc}}}{\epsilon_0} \]
where:
- \(\Phi_E\) is the electric flux through the surface.
- \(q_{\text{enc}}\) is the total charge enclosed within the Gaussian surface.
- \(\epsilon_0\) is the permittivity of free space, \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \).
Given data:
- \(\Phi_E = +3.9 \times 10^8 \, \text{N} \cdot \text{m}^2 / \text{C} \)
Calculate the total enclosed charge (\( q_{\text{enc}} \)):
\[ q_{\text{enc}} = \Phi_E \cdot \epsilon_0 \]
\[ q_{\text{enc}} = (3.9 \times 10^8 \, \text{N} \cdot \text{m}^2/\text{C}) \times (8.85 \times 10^{-12} \, \text{C}^2/\text](https://content.bartleby.com/qna-images/question/e4962ae7-1cb2-4276-8f6a-b9f851b67289/572c4209-239e-43c6-b405-a257f16668f2/w6ecv0q_processed.png)
Transcribed Image Text:### Example Problem for Applying Gauss's Law
#### Instructions
In each figure, the dashed circle represents the cross section of a closed Gaussian surface.
#### Diagram Description
In the provided diagram:
- Several point charges are shown, both inside and outside the dashed circle (representing a Gaussian surface):
- Inside the circle:
- +2.0 mC (milliCoulombs)
- +3.0 mC
- + an unknown charge (labeled as \(q = ?\))
- Outside the circle:
- +2.0 mC
- -3.0 mC (Note: mC is milliCoulombs, a unit for measuring electric charge)
#### Problem Statement
The net electric flux passing through the Gaussian surface is given as \(+3.9 \times 10^8 \, \text{N} \cdot \text{m}^2/\text{C}\).
Question: What is the value of the unknown charge within the Gaussian surface?
---
### Solution
To find the unknown charge (\( q \)), we use Gauss's Law which states:
\[ \Phi_E = \frac{q_{\text{enc}}}{\epsilon_0} \]
where:
- \(\Phi_E\) is the electric flux through the surface.
- \(q_{\text{enc}}\) is the total charge enclosed within the Gaussian surface.
- \(\epsilon_0\) is the permittivity of free space, \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \).
Given data:
- \(\Phi_E = +3.9 \times 10^8 \, \text{N} \cdot \text{m}^2 / \text{C} \)
Calculate the total enclosed charge (\( q_{\text{enc}} \)):
\[ q_{\text{enc}} = \Phi_E \cdot \epsilon_0 \]
\[ q_{\text{enc}} = (3.9 \times 10^8 \, \text{N} \cdot \text{m}^2/\text{C}) \times (8.85 \times 10^{-12} \, \text{C}^2/\text
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