### Example Problem for Applying Gauss's Law#### InstructionsIn each figure, the dashed circle represents the cross section of a closed Gaussian surface.#### Diagram DescriptionIn 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 StatementThe 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?---### SolutionTo 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

Gauss law states that net electric flux through any closed surface is equal to the net charge…

In each figure, the dashed circle is the cross section of a closed Gaussian surface. +2.0 mC +2.0 mC q=? +3.0 mC -3.0 mC The net electric flux passing through the Gaussian surface is +3.9 × 108 N · m²/C. What is the value of the unknown charge?

In each figure, the dashed circle is the cross section of a closed Gaussian surface. +2.0 mC +2.0 mC q=? +3.0 mC -3.0 mC The net electric flux passing through the Gaussian surface is +3.9 × 108 N · m²/C. What is the value of the unknown charge?

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Question

### 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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