Electric Field between concentric cylinders

I wanted the electric field between concentric cylinders, and got anexpression using Gauss' Law that considers only charge enclosed. Thus, there was no influence due to the outer cylinder.

But, if we consider the limiting case of the inner cylinder having very large curvature and the outer cylinder not too far off from it..i.e a is very large and b is just larger than a.

I assume that then the electric field between the cylinders can be considered as a flat plate analogy, which gives some other answer of the field between(for flat plate analogy, the outer cylinder also supplies electric field but it had no influence watsoever when we considered the normal cases)...Please help. I cant figure out what's wrong.

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And suppose now, I consider both the cylinders to be dielectrics. Does that affect the analysis? Will the Gauss' Law still be applicable? Will the field inside an infinitely long uniformly charged cylinder still be zero, and have no affect on the electric field .

Electric field due to infinite cylinder

So, can we suggest that electric field inside an infinite hollow cylinder with charge on the inner surface is 0 ?

I tried to this problem analytically assuming the cylinder to be made of many infinite lines of thickness R*d(theta) , of which i know the contribution to the electric field. Adding the vector components and relating them through the projected triangle, and using cosine law, i get an integral that is very complex to solve..Solving it online, gave me 2 different answers (Doh). I just cant figure out the method to do this.

Simply put, my question is what is the electric field inside an infinite hollow cylinder with charge on inner surface?