Question

Let F be an inverse square field, that is, F(r) = cr/|r|3 for some constant c, where r = xi + yj + zk. Show that the flux of F across a sphere S with center the origin is independent of the radius of S.

Answer 1

$\vec r(x,y,z)=x\,\vec\imath+y\,\vec\jmath+z\,vec k$

$\vec F(\vec r)=\dfrac{cx\,\vec\imath+cy\,\vec\jmath+cz\,\vec k}{(x^2+y^2+z^2)^{3/2}}$

$\vec F$ has divergence 0, so the flux of $\vec F$ across a sphere of any radius will always be 0 and thus independent of the sphere's radius.