To determine the electric flux through the sphere due to the dipoles, we can use Gauss's law, which states that the total electric flux \(\Phi_E\) through a closed surface is related to the enclosed charge \(Q_{enc}\) by the equation:
\[ \Phi_E = \frac{Q_{enc}}{\varepsilon_0} \]
where \(\varepsilon_0\) is the permittivity of free space, approximately \(8.85 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2\).
In this case, let's consider the characteristics of the dipoles:
- Each dipole consists of a +10 µC and a -10 µC charge.
- Since the dipole consists of equal and opposite charges, the total charge of each dipole is zero.
Now, when dealing with multiple dipoles:
- The total charge enclosed by the sphere (assuming it encloses all four dipoles) is:
\[ Q_{enc} = 4 \times (10 , \mu C - 10 , \mu C) = 4 \times 0 = 0 \]
As such, the enclosed charge within the sphere is zero. According to Gauss's Law:
\[ \Phi_E = \frac{Q_{enc}}{\varepsilon_0} = \frac{0}{\varepsilon_0} = 0 \]
Thus, the electric flux through the sphere due to these dipoles is:
\[ \Phi_E = 0 , \text{N} \cdot \text{m}^2/\text{C} \]
Therefore, the electric flux through the sphere is:
\[ \boxed{0.00 , \text{N} \cdot \text{m}^2/\text{C}} \]
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