Two metal plates form a parallel plate capacitor. The distance between the plates is d. A metal plate of thickness b=d/2 and same area is inserted completely between the plates. What is the ratio of the capacitance in the two cases?

When the metal plates of thickness b=d/2 is inserted between the parallel plate capacitor separated by a distance d, then the system is equivalent to the series combination of two capacitors, one is between the filled space by the metal plate and another is unfilled space between parallel plate capacitor.

The capacitance of the capacitor between metal plate of thickness d/2 is

 

And that between unfilled part will be

 

The equivalent capacitance will be

And the capacitance between the parallel plate capacitor when same area of metal plate is inserted  is

 

The required ratio of the capacitance is