# Potential in the midway between parallel plates

When we think of parallel plates with uniform charge density and charges $+q$ and $-q$, and if we set the potential to be zero at infinity we find the negative charged plate's potential to be zero. But when we think of the charge on each plate as collection of point charges and try to find the potential just in the midway between plates by using $kq/r$ formula, the potential there seems to be zero. But this cannot be true because when we set zero potential to be at infinity and bring a test charge to the negative plate, we dont feel any force acting on the test charge hence the plate is at zero potential. What is the reason behind this contradiction?

## 2 Answers

I think the problem is in the first part itself where you take potential to be zero at infinity in case of a uniformly charged plate. Check this paragraph from Griffith.

You are mixing up "potential" and "force" or "electric field". Force results from a gradient of potential: a change in potential vs distance traveled. kq/r is the formula for potential around a point charge. To find the potential around two infinite, parallel, charged plates it is necessary to do an integral of kq/r over the whole surface of both plates. I won't go into the details. The end result, though, is that the potential actually comes out to be kz, where z is the distance from one plate to the point where you want to find the potential.

The electric field strength is, in that case, constant between the plates; and the field points along lines perpendicular to the two plates.

On either side outside the two plates, the potential is the same everywhere. Because it doesn't change vs distance traveled, there is no electric field there. Another way to look at it is that a test charge sees two infinite planes, both in the same direction but charged oppositely. The electric fields from the two planes are equal and opposite, so they cancel out. Between the plates, though, the two electric fields add together.

• We need a drawing to refer to. But if a point positive charge on one plate is paired up with a point negative charge on the other plate, the two potentials add; they don't cancel. If both plates carried the same charge, then between the two plates the potentials would cancel and outside the plates they would add. – S. McGrew Apr 13 '18 at 23:46