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 between the Dielectrics??

Gauss Law applies, no matter if the inner surface is curved, or flat. It is the charge enclosed in the gaussian volume that affects the field through the gaussian surface.

To understand the situation better, let's break it down into two parts:

Part 1: Electric field between concentric cylinders (without dielectrics):
When using Gauss' Law for finding the electric field between concentric cylinders, we consider the charge enclosed by a Gaussian surface. Since the outer cylinder does not enclose any charge, we assume that it does not contribute to the electric field between the cylinders.

However, in the limiting case you described, where the inner cylinder has very large curvature and the outer cylinder is just larger than the inner cylinder, the electric field between the cylinders can be approximately treated as a flat plate capacitor analogy. In this case, the outer cylinder does contribute to the electric field between the cylinders. The reason for this is that as the curvature of the inner cylinder becomes larger, the electric field lines near the edges of the inner cylinder start to converge towards the outer cylinder, resulting in an influence from the outer cylinder on the electric field between them.

So, in this case, the assumption that the outer cylinder has no influence on the electric field between the cylinders is no longer valid. You would need to consider the electric field contribution from both cylinders to accurately determine the electric field between them.

Part 2: Electric field between dielectric cylinders:
If both the inner and outer cylinders are made of dielectric materials, then their presence will affect the analysis. Dielectrics have a property called the relative permittivity (εr) or dielectric constant, which affects the electric field in the material.

In this case, Gauss' Law is still applicable, but you would need to account for the effect of the dielectric materials on the electric field. The electric field inside a dielectric material will be reduced by a factor of εr compared to the electric field in vacuum. So, the electric field between the dielectric cylinders will be influenced by the dielectric constant of the materials.

The field inside an infinitely long uniformly charged cylinder will still be zero in the presence of dielectric materials. However, the dielectric constants of the cylinders will affect the electric field between them, and you will need to consider the relative permittivity of the dielectric materials in your calculations.

To summarize, in both cases, Gauss' Law can still be applied, but you need to consider the specific conditions, including the contribution of both cylinders and the effects of dielectric materials, to determine the electric field accurately.