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.

In the case of concentric cylinders, when the inner cylinder has a very large curvature and the outer cylinder is just slightly larger than the inner one, the electric field between the cylinders cannot be accurately treated as a flat plate analogy. The assumption that the outer cylinder has no influence in this case is incorrect.

As you correctly mentioned earlier, when using Gauss' Law to calculate the electric field between concentric cylinders, only the charge enclosed within the Gaussian surface is considered. If the outer cylinder has charge on its surface, it will contribute to the electric field between the cylinders.

In the limiting case you described, with a very large curvature for the inner cylinder and the outer cylinder just slightly larger, the electric field between the cylinders cannot be simplified as a flat plate analogy. The behavior of the electric field in this scenario is more complex and cannot be accurately represented by a simple flat plate model.

To analyze the electric field in this particular situation, you would need to consider the entire system of concentric cylinders, including the electric field contributions from both the inner and outer cylinders. This requires solving the electrostatics problem using the appropriate boundary conditions and formulas, such as applying Laplace's equation or using the method of images.

It's important to keep in mind that simplifications and analogies may work well in certain scenarios, but may not be valid in others. In this case, the flat plate analogy is not applicable, and a more comprehensive analysis is required to accurately determine the electric field between the concentric cylinders.

In the normal case of concentric cylinders, when applying Gauss' Law, we consider only the charge enclosed within the Gaussian surface. This means that the outer cylinder does not contribute to the electric field between the cylinders because the Gaussian surface does not enclose any charges from the outer cylinder.

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, we need to consider the influence of the outer cylinder as well.

To understand this situation, we can use the concept of a "parallel-plate capacitor" analogy. In this analogy, instead of considering the cylinders, we imagine the inner surface of the outer cylinder as one plate of a capacitor, and the outer surface of the inner cylinder as another plate of the same capacitor.

Now, let's analyze the situation step by step:

1. First, consider the outer cylinder as the positive plate and the inner cylinder as the negative plate of a parallel-plate capacitor. The electric field between the cylinders due to the outer cylinder is given by the formula for a parallel-plate capacitor: E = σ / (2ε₀), where σ is the surface charge density and ε₀ is the permittivity of free space.

2. Next, consider the inner cylinder as a Gaussian surface. Since the inner cylinder does not have any charges enclosed within it (assuming no charge is present inside the inner cylinder), Gauss' Law tells us that the electric field inside the inner cylinder should be zero.

3. Therefore, if we subtract the electric field due to the outer cylinder from the zero electric field inside the inner cylinder, the resulting electric field between the cylinders will be in the opposite direction to the electric field due to the outer cylinder.

In summary, in the limiting case you described, the electric field between the cylinders can be considered as a superposition of the electric field due to the outer cylinder and the electric field due to the inner cylinder in the opposite direction.

It is essential to note that this analysis is only valid for the specific case where the curvature of the inner cylinder is very large compared to the separation between the cylinders. In other cases, the electric field between the cylinders may not follow this simple analogy, and the detailed geometry and charge distribution need to be considered for an accurate calculation.