Question

two metal plates form a parallel plate condenser . the distance between the plates in d. now a metal plate of thicknes d+2 of same area is inserted ocmpletelty between platesd the capacitance

Answer 1

Capacitance C = K ε A / d ,    K = 1

here since a metal plate of thickness d/2 is introduced between them, the capacitor is converted into two equal capacitors of capacity 4 times and connected in series.  The plates in the center will also have the same charge as on the end plates.

Capacitance becomes 4 times for each of the two new capacitors as d becomes d/4 = thickness of air between plates.

The equivalent capacitance for series combination = 2 times the original value.