Calculate the field within a hole on a charged sphere

[Aesteus]

Homework Statement

If you have a uniformly-charged negative hollow sphere of charge, with a positive circular piece of charged shell superimposed on the outside, what is the electric field in the center of the circular piece? Also what part of the field going through the piece's center is locally generated?

Homework Equations

EA=q/ε ... E=(q/ε)*1/(4*pi*r^2)

The Attempt at a Solution

I assume that none of the field will be locally generated, because the circular-positive piece will totally cancel out with the field of the sphere. I'm not sure where to go from there though, as I would think the remaining sphere's field would extend radially.

 

[Dickfore]

Hint:

The circular piece creates the same magnitude, but opposite direction electric field on the two faces. The rest of the sphere would create the same field in these infinitesimally separated points.

What is the total field inside the charged sphere?

 

[Aesteus]

Alright so I'm guessing that because the fields point in opposite directions, they superimpose on each other, and because the circular-charged regions are essentially flat that we can treat them as planes of charge. Thus the field going through the disk is essentially the same at any arbitrary point such that the field through its center is E=E(disk)+E(sphere). What do you think?