Five thousand lines of force enter a certain volume of space and three thousand lines emerge from it. What is the total charge in coulomb within this volume ?

Explanation

To solve the problem, we will use Gauss's Law, which relates the electric flux through a closed surface to the charge enclosed within that surface.

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Number of lines of force entering the volume: \( N_{\text{in}} = 5000 \)
- Number of lines of force emerging from the volume: \( N_{\text{out}} = 3000 \)

2. **Calculate the Net Electric Flux**:
- The net electric flux (\( \Phi \)) through the closed surface is given by the difference between the lines entering and the lines exiting:
\[ \Phi = N_{\text{out}} - N_{\text{in}} = 3000 - 5000 = -2000 \]

3. **Apply Gauss's Law**:
- According to Gauss's Law, the electric flux (\( \Phi \)) is related to the charge (\( q \)) enclosed within the volume by the equation:
\[ \Phi = \frac{q}{\epsilon_0} \]
where \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \).

4. **Rearranging the Equation to Find Charge**:
- Rearranging the equation gives:
\[ q = \Phi \cdot \epsilon_0 \]

5. **Substituting the Values**:
- Substitute \( \Phi = -2000 \) and \( \epsilon_0 = 8.85 \times 10^{-12} \):
\[ q = -2000 \cdot (8.85 \times 10^{-12}) = -1.77 \times 10^{-8} \, \text{C} \]

6. **Final Result**:
- The total charge enclosed within the volume is:
\[ q = -1.77 \times 10^{-8} \, \text{C} \]

### Summary:
The total charge in coulombs within the volume is \( -1.77 \times 10^{-8} \, \text{C} \).