(Solved): Consider the sphere with radius a shown below, carrying a uniform surface charge density S. a. ...

Consider the sphere with radius $a$ shown below, carrying a uniform surface charge density ${?}_S$. a. Using Coulomb's law and integration, find the electric field a distance $z$ from the center of the sphere. Treat the case $z<a$ (inside) as well as $z>a$ (outside). Express your answers in terms of the total charge $Q$ on the sphere. b. Find the potential inside and outside of the spherical shell, directly by integration (similar to part a), and by using the electric field obtained in the previous part. Hint: Use the law of cosines to find the distance from the surface to pint $\mathrm{P}$ in terms of $a,z$, and $?\left(\sqrt{a^2+z^2?2az\cos ?}\right)$. Be sure to take the positive square root, that is, $\sqrt{a^2+z^2?2az}=?z?a?$.