Chapter 17.6, Problem 65E

Flux across concentric spheres Consider the radial fields $F=\frac{\langle x,y,z\rangle }{{\left(x^2+y^2+z^2\right)}^{p/2}}=\frac{r}{{\left|r\right|}^p}$, where p is a real number. Let S consist of the spheres A and B centered at the origin with radii 0 < a < b, respectively. The total outward flux across S consists of the flux out of S across the outer sphere B minus the flux into S across the inner sphere A.

 a.    Find the total flux across S with p = 0. Interpret the result.

b.    Show that for p = 3 (an inverse square law), the flux across S is independent of a and b.