If the electric flux entering and leaving a closed surface are `6 xx 10^6` and `9 xx 10^6` SI units respectively, then the net charge inside the surface of permittivity of free space `epsilon_0` is

Options

A. `epsilon_0xx10^6`
B. `-epsilon_0xx10^6`
C. `-2epsilon_0 xx10^6`
D. `3epsilon_0xx10^6`

Explanation

To find the net charge inside a closed surface using the electric flux entering and leaving the surface, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Electric flux entering the surface, \( \Phi_{in} = 6 \times 10^6 \) SI units.
- Electric flux leaving the surface, \( \Phi_{out} = 9 \times 10^6 \) SI units.

2. **Determine the Net Electric Flux**:
- According to the convention used, we can take the entering flux as negative and the leaving flux as positive. Therefore, the net electric flux \( \Phi \) can be calculated as:
\[ \Phi = \Phi_{out} - \Phi_{in} \]
- Substituting the values:
\[ \Phi = 9 \times 10^6 - 6 \times 10^6 = 3 \times 10^6 \text{ SI units} \]

3. **Apply Gauss's Law**:
- Gauss's Law states that the net electric flux through a closed surface is equal to the charge enclosed \( Q \) divided by the permittivity of free space \( \epsilon_0 \):
\[ \Phi = \frac{Q}{\epsilon_0} \]
- Rearranging the formula to find the charge \( Q \):
\[ Q = \Phi \cdot \epsilon_0 \]

4. **Substitute the Net Electric Flux**:
- Now substitute the net flux we calculated:
\[ Q = (3 \times 10^6) \cdot \epsilon_0 \]

5. **Final Answer**:
- The net charge inside the surface is:
\[ Q = 3 \epsilon_0 \times 10^6 \text{ Coulombs} \]