Problem 13 (10 points) - Boundary Value Problem Z The parallel-plate capacitor shown is filled with a dielectric with €₁. As the figure also shows, the capacitor has a square hole cut all the way through it. Assume that the sides of the capacitor are much larger than the distance between the two plates. Also assume that a voltage source of Vs is connected to the top plate, and the bottom plate is grounded. Derive the capacitance of this capacitor. a) (4 points) Solve the Laplace equation to find the electric potential function in the region between the two plates. b) (4 points) Use either Gauss's law or the normal electric flux boundary condition to find the total charge on one plate. c) (2 points) Use the capacitance definition to find this capacitor's capacitance.

Fig: 1