What happens if you introduce an infinitely large copper plate in a charge distribution? Suppose you have a positive charge fixed at a point in space.
Now introduce an infinitely large metal plate close to the charge.
Would there be electric field on the other side of the metal plate where there is no charge?
According to the superposition principle of electric fields, there must be an electric field (as the copper plate does not contribute anything to electric field at any point outside the copper plate)
But the metal plate has infinite electric permittivity. So no electric field lines pass through the metal plate.
A contradiction.
Please explain where I have gone wrong.
 A: The copper plate is a conductor, and as such has free charges that can rearrange themselves. The presence of the positive charge $Q$ would induce an overall negative charge on the plate with magnitude $-Q$.
Let's imagine your infinite sheet to have a "top" and a "bottom", with the top being the side of the sheet closest to $Q$. The conductor being neutral has both positive and negative charges, and the negative charges move to the top and distribute themselves on the top surface in such a way as to cancel out the field of the positive charge. (To show this is a common exercise and is done in most books, including Griffiths.) As a result, the field lines from the positive charge do not enter the sheet.
But what about the remaining positive charges on the sheet? They will stay on the "bottom" of the sheet. However, because the charges on the top rearranged themselves so conveniently, these "bottom" charges will not "know" about the field of the original positive charge. As a result, they are only affected by each other, and so will arrange themselves in a uniform distribution, and this will lead to a constant electric field below the sheet.
Of course, the total charge on the bottom of the sheet is just $+Q$ (since I'm assuming that we started off with a neutral copper sheet), which is a finite number. The area of the sheet, however, is infinite. Since the charge density is uniform, it's just the ratio of total charge to total area, and so the surface charge density is zero! As a result, the field below the sheet is zero. You can actually see this in action using this amazing applet.
This is the principle of Electromagnetic Shielding, and you can read up on it in more detail here.
