A conducting hemispherical projection of radius $R$ is placed upon a ground plane of infinite extent. A point charge $q$ is placed a distance $d$ $(d > R)$ above the center of the hemisphere.

1 What is the force on $q$?

Hint: Try placing three image charges along the $z$ axis to make the plane and hemisphere have zero potential.

2 What is the total charge induced on the hemisphere at $r=R$ and on the ground plane $|y| > R$?

Hint:$$ \int \frac{{r} \,\mathrm d {r}}{\left[{r}^{2}+d^{2}\right]^{3 / 2}}=\frac{-1}{\sqrt{{r}^{2}+d^{2}}} $$