How do I prove that Gauss's Law works and is valid for a sphere with a circular disk on top?
I'm given the radius of the sphere R$(\theta_0,\pi)$, radius of the disc $(R \ sin \ \theta_0)$, the center of the circular disc:
$$z=z_{0}=R \ sin \ \theta _0$$
and the differential area element for both the sphere ($e_{r}R \ sin \ \theta _{0}$) and the circular disc ($e_{z} s \ ds d\phi$).
Where:
$$s = \sqrt{r^{2}-z^{2}} \ \ \ \ \ for \ \ \ \ s(0, e_{r}R \ sin \ \theta _{0})$$
With $\phi(0,2\pi)$ and $e_{r}e_{z}=cos \ \theta$.
So far, I have only placed the integral equations regarding Gauss's Law:
I also consulted other resources and they said that the equation for the sphere should be utilized, so that everything is simplified. I also saw sources using the steradian method, however, my professor strictly wants me to use the differentials provided. I'm at a dead end, because most sources together with my lectures only cover examples using symmetric figure.
