# Charge Redistribution on Capacitor Plates

Say, I've got two Large Conducting Sheets (intention : to neglect end effects).
My question is quite fundamental and relies on the basic understanding of Charges.
Now, I bring these two plates together with some dielectric and so basically make a capacitor with these two neutral large conducting sheet.
Let's connect it to a battery. So, Obviously Charge will flow from the outer surface of the plate connected to the positive terminal of the battery to the end of the second plate connected to the negative terminal of the battery.

When it reaches a steady state, the charges resides on the inner surfaces of the capacitor.

But then, How can charge flow from the outer surface of the plate to its inner one?
As it's a conductor, Electric field inside is always zero. So, electrons are not supposed to move through the inner area.
Also, It doesn't seem plausible that electrons will flow from the outer surface to inner using the long path from outer to inner as it itself extends to infinity and so it would take infinite time just to pass!
So, How does this happen?
Is it all a matter of fact that in reality nothing extends to infinity?