Problem 19.23

19.23. a)
Gauss' Law states that the flux through a surface is given by $\Phi = \frac{Q_{inc}}{\epsilon_{0}}$
so, through half a sphere, the flux would be
$\Phi = \frac{+q}{2\epsilon_{0}}$

b)
There is no inclosed charge in the hemisphere since the point charge is a small distance above the plane. Therefore, the net flux must be zero. So, the flux flowing in through the flat surface must flow out of the hemisphere. Mathematically speaking,
$\Phi_{net} = \Phi_{plane} + \Phi_{hemisphere} = 0$
$\Phi_{plane} = \frac{-q}{2\epsilon_{0}}$