Q$4(15\%)$

A parallel$-$plate capacitor has square plates of edge lengths $1$ cm $,$ and a plate separation of $1$ mm $.$ The plates are separated by air and it is connected to a $12$ V battery

a$)$ What is the capacitance and the energy stored inside the capacitor?

b$)$ The battery is disconnected, and you insert a dielectric material between the plates, $k=2.$ What is the new capacitance of the capacitor? What is the new voltage? What is the new energy stored in the capacitor?

c$)$ The battery is reconnected, but you keep the dielectric in the capacitor. What is the new voltage? What is the new energy stored in the capacitor?

Hint in this problem, you are always looking for the quantity that is conserved, and the quantity that changes.