# Flux Integral

## 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 Im getting a^3*pi.

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tiny-tim
Homework Helper
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 Im 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? )

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
Homework Helper
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