(i) Starting with Poisson's equation in ,
derive Gauss' flux theorem
for and for any volume .
Show that if is the sphere , and that if bounds a volume that does not contain the origin.
(iii) Show that the electric field defined by
where is a surface bounding a closed volume and , and where the electric charge and permittivity of free space are constants. This is Gauss' law for a point electric charge.
(iv) Assume that is spherically symmetric around the origin, i.e., it is a function only of . Assume that is also spherically symmetric. Show that depends only on the values of inside the sphere with radius but not on the values of outside this sphere.