Finding Flux of $\vec{F}$ Outward Across Sphere

[sandy.bridge]

Homework Statement

Find the flux of \vec{F}=(x, y, z) outward across the sphere x^2+y^2+z^2=a^2.


I am able to get it to this point:
\int\int_Cadxdy and I then convert it to polar coordinates, and integrate rdr from 0 to a, and theta from zero to 2pi. However, this does not give me the correct result, as the answer is 4a^3*pi, and I am getting a^3*pi.

 

[tiny-tim]

hi sandy.bridge!

I am able to get it to this point:
\int\int_Cadxdy and I then convert it to polar coordinates, and integrate rdr from 0 to a, and theta from zero to 2pi. However, this does not give me the correct result, as the answer is 4a^3*pi, and I am getting a^3*pi.

it's the surface of a sphere …

what does r have to do with it? ​

(and why are you integrating? surely you know the surface area of a sphere? )

 

[sandy.bridge]

Okay, well I know that once I get it to this point, it's right:
\int\int_CadS=\int\int_Ca(1)dxdy

The projection of the sphere on the xy-plane is a circle, no? So why can I not use
dxdy=rdrd\theta?

 

[tiny-tim]

The projection of the sphere on the xy-plane is a circle, no?

projection? are you treating F as if it was a parallel field along one of the axes?

in that case, yes, the projection perpendicular to the field would be a circle

but the given F is radial ((x,y,z) = t), and constant in magnitude over the sphere,

so you just need the amount of surface it cuts through, which is 4πa²