A Flux Divided

Consider a sphere $S_1$ of radius $R=\frac{\sqrt{5}}{2}$ centered at the charge; $S_1$ intersects the original sphere in the circle $x^2+y^2=1, z=0$ .

By Gauss' Law, the flux through the upper half of the original sphere is equal to the flux through the cap $C$ of $S_1$ with height $h=R+\frac{1}{2}$ .

The fraction we seek is $\frac{\text{area of}\ C}{\text{area of}\ S_1}=\frac{2\pi Rh}{4\pi R^2}=\frac{1}{2}+\frac{1}{2\sqrt{5}} \approx \boxed{0.7236}$ .

The flux is proportional to the solid angle covered by the surface.

The solid angle not covered by the surface corresponds to a cone of apex angle $2\theta$ and yielding a solid angle of $2 \pi (1-\cos\theta)$ , where $\theta$ is defined by $\tan\theta =1/(1/2)=2$ .

The solid angle covered by the half-sphere is $4\pi - 2\pi(1-\cos\theta)=2\pi(1+\cos\theta)$ and the fraction is $\eta= (1+\cos\theta)/2= (1+\frac{1}{\sqrt{1+\tan^2\theta}})/2 = (1+1/\sqrt{5})/2=0.7236$ .