ext{m}^2$

Permittivity of free space, $\varepsilon_0 = 8.85 \times 10^{-12}~\text{C}^2/\text{Nm}^2$

Step 1: Formula for Electric Flux (Gauss’s Law)

Total electric flux, $\phi$, leaving the surface is:

$$ \phi = \frac{q}{\varepsilon_0} $$

where $q$ is the total charge on the sphere.

Step 2: Calculate Area of the Sphere

Radius, $r = \frac{d}{2} = \frac{3.5}{2}~\text{cm} = 1.75 \times 10^{-2}~\text{m}$

Area, $A$:

$$ A = 4\pi r^2 = 4\pi (1.75 \times 10^{-2})^2 $$

Calculate:

$$ A = 4 \times 3.14 \times (1.75 \times 10^{-2})^2 \\ = 12.56 \times (3.0625 \times 10^{-4}) \\ = 12.56 \times 3.0625 \times 10^{-4} \\ \approx 38.48 \times 10^{-4} \\ = 3.848 \times 10^{-3}~\text{m}^2 $$

Step 3: Find Total Charge

Total charge, $q = \sigma \times A$:

$$ q = 20 \times 10^{-6}~\text{C/m}^2 \times 3.848 \times 10^{-3}~\text{m}^2 \\ = 76.96 \times 10^{-9}~\text{C} \\ = 7.696 \times 10^{-8}~\text{C} $$

Step 4: Calculate the Total Electric Flux

$$ \phi = \frac{q}{\varepsilon_0} = \frac{7.696 \times 10^{-8}}{8.85 \times 10^{-12}} $$

Calculate denominator:

$$ \phi = \frac{7.696}{8.85} \times 10^{-8+12} \\ = 0.870 \times 10^{4} \\ = 8.7 \times 10^{3}~\text{Wb} $$

Step 5: Rounding off

Among the given options, the closest is:

$$ 8.7 \times 10^3~\text{Wb} \approx 87 \times 10^2~\text{Wb} $$

Final Answer:

Option C: $87 \times 10^2~\text{Wb}$

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