$(15$ points$)$ Three capacitors are connected as shown in the below figure. The capacitor $C_1$ has plate area $0\mathrm{.}0043m^2$ and plate separation $1\mathrm{.}6$ mm and half of $C_1$ is filled with a material of dielectric constant $K=2.$ Capacitances of $C_2$ and $C_3$ are $5{10}^{-12}F$ and $15{10}^{-12}F,$ respectively.

a$)$ Calculate the capacitance of $C_1.$

b$)$ Find the equivalent capacitance of the three capacitors.

c$)$ If a $2{10}^{-10}J$ energy is stored in capacitor $C_2,$ what is the potential difference between the plates of capacitor $C_3?$

d$)$ What is the charge on capacitor $C_1?$

e$)$ What is the potential difference supplied by the battery?