Electric field in non-uniformly charged hollow sphere My full question is 

"A total charge $Q$ is uniformly distributed within a spherical shell with negligible thickness compared to its radius. A point charge $q$ ($q<< Q$) is taken away from the shell's north pole and replaced on its south pole without affecting the rest of the charge distribution. What is the electric field at the centre of the shell?"

I don't really know how to apply symmetry to this, in a normal uniformly charged shell the field inside would be zero, but for this..? Would using Gauss's law with non-uniform charge distribution be appropriate at all (even though no charge is enclosed at the centre)? Shall I treat the poles like point charges of 0 and +2q, is there some type of superposition thing I'm missing?
This has really stumped me.
 A: You can imagine the second case to be a superposition of the uniform spherical shell and two opposite point charges -q and q at the north and south pole respectively. The field inside the sphere would then be the field created by the -q/q dipole.
A: The field inside the sphere is the superposition of two point charges with opposite signs, and a uniform spherical shell. The latter gives zero field so we just need the field due to two point sources.
For a given distance $r$ from the "pole", the field is
$$E=\frac{q}{4\pi \epsilon_0 r^2}$$
Pointing away from the positive charge. For the negative charge you get a field pointing towards the charge.
The total field is the vector sum of these two. Along the plane midway between the charges this field is perpendicular to the plane - at other points it will curve.
For a given point you should be able to write down the X and Y components of the field due to these two point charges and add them together. I believe this should be sufficient to get you to your answer.
A: The solution follows by considering the superposition of two cases.


*

*when we just remove the charge (and the whole other body is kept as it is)

*when we place that charge diametrically opposite to chargeless space.(this time only this charge is considered and not the sphere) 

