Electrostatics problem (Spherical surface)

[ravager1987]

Homework Statement

Find the electric field a distance z above the center of a spherical surface of radius R, which carries a uniform charge density "sigma" Treat the case z<R(inside) as well z>R (outside). express your answers in terms of the total charge q on the sphere.[hint:use the law of cosines to write r interms of R and "theta". be sure to take the positive square root:
sqrt(R^2+z^2-2Rz)=(R-z) if R>z and (z-R) for R<z]


here is the solution for the problem:
http://img689.imageshack.us/img689/2492/questionr.jpg

even with the solution i have problem understanding...
what i don't understand is:
1) how did he get dq=sigma*R^2*sin(theta)d(theta)d(phi)
2)the reason for taking cos(curlyphi) ?
my guess: to get the vertical component? since horizontal components cancels out
3)is there a different way looking at this problem? i remember in college there is a simpler way doing this kind of problem...

 

[Delphi51]

1. R*sinθ*dφ and R*dθ are the length and width of the area da where the charge dq lies.
2. You are correct.
3. Yes, you could use Gauss' Law. This problem must be at a point in your course where you have not taken Gauss' Law - and it is good practise!