Let us say plate A has a charge q1 and plate B, which faces plate A has a charge q2. By making use of the fact that the net field in the bulk of a conductor in static conditions is zero, and that the net field near the outer surface of a conductor equals [local surface charge density/€0], you can prove the following:
- Charge on the outer surfaces of A and B are each (q1+q2)/2.
- By conservation of charge, the charges on the facing surfaces of A and B are (q1-q2)/2 and -(q1-q2)/2 respectively.
- Now using the result for net field near the outer surface of a conductor mentioned above, the field in the region between the plates is,
E=([q1-q2)/2]/(A*€0), pointing towards plate B if q1>q2. Here A is the area of each plate.
Finally, potential difference is E*d, where d is the distance between the plates. This is valid because for infinitely large plates, the field in the region between the plates can be considered uniform. Since capacitance C=A€0/d, we obtain the potential difference V= (q1-q2)/2C