How to Calculate Electric Field at the Center of Curvature of a Hemisphere?

[captainjack2000]

Homework Statement

Charge Q is distributed uniformly over surface of a hemisphere of radius r. Calculate electric field at centre of curvature of hemisphere.

Homework Equations

I realize that this should be quite straightforward but I am a bit confused.
The total charge Q is distributed over surface of hemisphere with surface charge density sigma where
sigma = Q/(2*PI*r^2)
I think you would then divide the hemisphere into a series of rings where each ring would carry a charge
dQ=sigma*2*PI*dr ?

At the centre of curvature the components of electric field perpendicular to the axis will cancel so you will only have electric field along the axis z.

Not quite sure where to go from here.

The Attempt at a Solution

 

[Cyosis]

You're correct in saying that there only is a z-component of the field so express dE_z in terms of dE.

You also know that dQ=\sigma dA. What is dA in spherical polar coordinates? What are the values of \theta and \phi for half a sphere?