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