the net electric flux through each face of a die (singular of dice) has a magnitude in units of `10^(3)N.m^(2)//C` that is exactly equal to the number of sport N on the face ( 1 through 6) the flux is inward for N odd outword for N even. What is the net charge inside the die ?

Answer

To solve the problem, we need to calculate the net electric flux through the faces of a die and then use Gauss's Law to find the net charge inside the die.

### Step-by-Step Solution:

1. **Understanding the Electric Flux**:
The electric flux (Φ) through each face of the die is given in units of \(10^3 \, \text{N m}^2/\text{C}\) and is equal to the number of spots (N) on that face. For a die, the faces are numbered from 1 to 6.

2. **Identifying the Flux Direction**:
- For odd N (1, 3, 5), the flux is inward (negative).
- For even N (2, 4, 6), the flux is outward (positive).

3. **Calculating the Flux for Each Face**:
- Face 1: \(Φ_1 = -1 \times 10^3 \, \text{N m}^2/\text{C}\)
- Face 2: \(Φ_2 = +2 \times 10^3 \, \text{N m}^2/\text{C}\)
- Face 3: \(Φ_3 = -3 \times 10^3 \, \text{N m}^2/\text{C}\)
- Face 4: \(Φ_4 = +4 \times 10^3 \, \text{N m}^2/\text{C}\)
- Face 5: \(Φ_5 = -5 \times 10^3 \, \text{N m}^2/\text{C}\)
- Face 6: \(Φ_6 = +6 \times 10^3 \, \text{N m}^2/\text{C}\)

4. **Summing the Electric Flux**:
\[ \Phi_{\text{total}} = Φ_1 + Φ_2 + Φ_3 + Φ_4 + Φ_5 + Φ_6 \]
\[ = (-1 + 2 - 3 + 4 - 5 + 6) \times 10^3 \, \text{N m}^2/\text{C} \]
\[ = (3) \times 10^3 \, \text{N m}^2/\text{C} \]

5. **Using Gauss's Law**:
According to Gauss's Law, the total electric flux through a closed surface is equal to the charge enclosed (Q) divided by the permittivity of free space (\(ε_0\)):
\[ Φ_{\text{total}} = \frac{Q}{ε_0} \]
Rearranging gives us:
\[ Q = Φ_{\text{total}} \cdot ε_0 \]

6. **Calculating the Charge**:
The value of \(ε_0\) (permittivity of free space) is approximately \(8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2\).
\[ Q = (3 \times 10^3) \times (8.85 \times 10^{-12}) \]
\[ = 2.655 \times 10^{-8} \, \text{C} \]

### Final Answer:
The net charge inside the die is approximately \(2.66 \times 10^{-8} \, \text{C}\).