Explicit Calculation of Gauss's Law

[Momentous]

Homework Statement

Consider a sphere centered at origin with radius R > z0. By calculating the total flux
ϕ = ∫E . da through the sphere, explicitly show that ϕ = q/ϵ0

Homework Equations

Gauss's Law

The Attempt at a Solution

I have a general idea of what to do, but I just want to make sure I don't screw up from the start. I think that I have to do a surface integral. The instructions say to integrate over spherical coordinates.

I'm just confused because I usually think of spherical coordinates as 3 dimensions, but a surface integral is 2-D.

I figured that I would change dA into r^2 sinθ dr dϕ dθ, and integrate from there.
I'd use 0 to R for radius, 0 to 2∏ for ϕ, and 0 to pi for θ. I figure you can take E out of the integral because it is constant and always normal to the Surface Area.

Am I on the right track with this one?

 

[Momentous]

Oh yeah, I should probably mention this too. It goes w/o saying, but there is a charge located at (0, 0, z0) as the initial condition.

 

[Dick]

dA=R^2 sinθ dϕ dθ. You don't integrate over r. It's just a surface. And dA should also include the outward pointing unit normal which you need to dot with E. Finally, it doesn't quite go without saying but if the charge is located at (0,0,z0) instead of the origin then the magnitude of the E field isn't constant either. There is some work to do here.