How to derive the electric field between parallel plates of negligible separation?

[spikelau]

I read from a website that the electric field is independent of the separation between the plates if the separation is negligibly small. But how to derive this?

[Doc Al]

Start by deriving the field close to a sheet of charge. Use Gauss's law, realizing that the electric field, by symmetry, will be perpendicular to the surface.

The field between two parallel plates will be the sum of the fields from each plate.

[Gokul43201]

In fact, this is true for a single, infinitely large plane too. And a hand waving argument for why this might be true is that there is no distance scale set by an infinite plane ....i.e. the result should be scale independent. I could use this same argument for a single point charge, or a line charge too...and of course it would be wrong, so don't take it too seriously

But go ahead with the Gaussian pillbox and see what you get.