Chapter 15.6, Problem 33ES

Let $\text{F=}{‖\text{r}‖}^k\text{r,}\text{where r}=x\text{i+}y\text{j+}z\text{k}\text{and }k\text{is a constant}\text{.}$ (Note that if $k=-3,$ this is an inverse-square field.) Let $\sigma$ be the sphere of radius a centered at the origin and oriented by the outward normal $\text{n}=r/‖\text{r}‖=r/a.$

(a) Find the flux of F across $\sigma$ without performing any integration.

(b) For what value of k is the flux independent of the radius of the sphere?