# Parallel plates with differing charges

I'm given a problem involving two parallel conducting plates a small distance apart. One has charge Q, the other is neutral. I'm asked to find the surface charges on all faces.

I began by creating a Gaussian cylinder with each face terminating within one of the two plates. Since the electric field inside a conductor is zero, the net charge must be zero and whatever charge on the interior surface of the charged plate, the uncharged plate must have the same one, but opposite in sign.

Furthermore, the surface charges on the charged plate must sum to Q and on the uncharged plate to 0 by conservation of charge. Here's where I'm stuck -- aren't there many possible charge configurations that satisfy these conditions?

There many configurations that statisfy your assumptions. But you had forgot about many other constraints like potential difference, charge density, end effects etc.

When you consider PD you come to know, ignoring end effects, that only one configuration is possible.

Since 1st plate has charge Q its surface charge density of 1st side is Q/2A 2nd side again Q/2A. For second plate -Q/2A and 3rd +Q/2A

First consider that two plates are not much thick. Plate 2 has 0 net charge, so potential due to it on Plate 1 is 0. It means charge can uniformly distribute over 1st plate. Half of charge moves to one side of plate and half to other. If charges will distribute non uniformly, it will create a potential difference and charges would move to counterbalance it, so that whole plate is uniformly charged.

Charge Q/2 on one side gives charge density of Q/2A.

Now, at plate 2 1st side of plate gets charged -Q/2 due to plate1 and 2nd side gets charged+Q/2.

Remember you said plates are near, if they were not charge on plate 2 one side would decrease.

• Could you elaborate on how to use PD to get the correct charges? – Why-Seven-Six Feb 2 '16 at 6:27