Flux of a Vector Field on a Sphere

[Pyroadept]

Homework Statement

Consider the vector field:
F = r/r³

where r = xi + yj + zk

Compute the flux of F out of a sphere of radius a centred at the origin.

Homework Equations

The Attempt at a Solution

Hi everyone,

I have: flux = $$\int$$F.dA

I can't use Gauss' Law, because the field will not be defined at the origin.

Instead, I want to use F.n, where n is the normal vector to the sphere.

Is it correct that the normal vector is the div of the equation of the sphere?

ie. n = div (x^2 + y^2 + z^2 = a^2)

= 2x i + 2y j + 2z k

and then F.n = 2/r

Is this correct so far?
Thanks for any help

 

[dacruick]

Looks good. the divergence is how fast the field falls off, so it has to be in the direction of the normal vector.