let $ \Large \vec{F(r)}=\frac{c\vec{r}}{r^3} $ , $\large \vec{r} = x\vec{i}+y\vec{j}+z\vec{k} $,

Prove $ \Large \iint_{S} \vec{F}.d\vec{S} $, which $ S $ is a sphere, does not depend on the radius of the sphere.

Question

let $ \Large \vec{F(r)}=\frac{c\vec{r}}{r^3} $ , $\large \vec{r} = x\vec{i}+y\vec{j}+z\vec{k} $,

Prove $ \Large \iint_{S} \vec{F}.d\vec{S} $, which $ S $ is a sphere, does not depend on the radius of the sphere.

anonymous · Answer

it's not a hard problem, but i'm looking for some innovative and easy solutions ;)

irishboy123 · Answer

divergence theorem nails it

anonymous · Answer

$\Large  \iint_{S} \vec{F}.d\vec{S} = \iiint_{E} div\vec{F}dV$, therefore,

 $\Large \iint_{S} \vec{F}.d\vec{S}=\frac{ 3c}{a^3} \iiint_{E} dV =4 \pi c $

anonymous · Answer

@IrishBoy123 , correct ;) how about using Stokes' thm or other ideas?