Derive an expression for effective capacitance of three parallel plate capacitors connected in series.

A parallel plate air capacitor has capacitance $10\mu \mathrm{F}.$ It is charged by connecting to a $50 \mathrm{V}$ battery. Then it is disconnected from the battery and the space between the two plates of the capacitor is filled with a material having dielectric constant $5.$ Compare

capacitance,

The space between the plates of a parallel plate capacitor is completely filled in two ways. In the first case, it is filled with a slab of dielectric constant $K$. In the second case, it is filled with two slabs of equal thickness and dielectric constants $K_1$ and $K_2$ respectively as shown in the figure. The capacitance of the capacitor is same in the two cases. Obtain the relationship between $K$, $K_1$ and $K_2$.