A

Step 1: According to Gauss's law, the total electric flux \(\Phi_E\) through a closed surface is given by the equation:
\[ \Phi_E = \frac{Q_{enc}}{\epsilon_0} \]
where \(Q_{enc}\) is the total charge enclosed within the surface and \(\epsilon_0\) is the permittivity of free space.

Step 2: Here, the surface is a hemispherical closed surface with charge \(q\) placed at the center. Because the charge is at the center of the hemisphere, it is essentially enclosed by this imaginary hemisphere if we consider an entire sphere.

Step 3: Since the charge \(q\) is the only charge enclosed by the hemisphere, we can write:
\[ Q_{enc} = q
\Phi_E = \frac{q}{\epsilon_0} \]
However, this equation gives the flux through a full closed surface (a sphere).

Step 4: The hemispherical surface represents half of a closed sphere, so the flux through the hemispherical surface will be half of the total flux:
\[ \Phi_{hemisphere} = \frac{1}{2} \left(\frac{q}{\epsilon_0}\right) = \frac{q}{2\epsilon_0} \]
Therefore, the electric flux through the hemispherical surface due to the charge \(q\) is given by \(\frac{q}{2\epsilon_0}\).
Final answer: Therefore, the electric flux through the given surface is \(\frac{q}{2\epsilon_0}\).