The Force Between Two Charged Plates

Consider now two parallel flat surfaces, each with equal charge density a, separated by a distance D. Although charged, there is no net electrostatic repulsion of the surfaces, since the counterions in the gap between the plates cancel the charges on the plates. However, because the counterions must be drawn into the gap, there is an entropy cost that becomes greater as 

If the plates are greater than one Debye distance apart, an acceptable approximation is the superposition approximation in which the potential at the midplane is taken to be the sum of the potentials from each of two isolated surfaces at x = Evaluating from twice rjr x) in Eq. (2-49) with x = D/2, we get 

Inserting this into Eq. (2-54a) gives for flat plates 

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