Question

A capacitor is connected across a battery why does each plate receive a charge of same magnitude? Will it be turn even if the plates of different sizes?

Answer 1

Capacitance is determined by the area between the plates among other things such as distance of the plates, dielectric etc. The battery will supply an amount of charge as determined by C = Q/V As long as nothing is done to change the capacitor configuration the Voltage will charge up to a value equal to the charging source (battery), irrespective to the shape and size of the plates. Now if the battery is disconnected, and the shapes or configuration is changed, the value of V will be altered.