Problem 12 (10 points) - Boundary Value Problem You receive a cylindrical birthday cake with a radius of and height of h as seen in the figure. The cake's permittivity is cake Ep. You place the cake on a round PEC plate. Unfortunately, you are so excited to eat the cake that when you cut yourself a wedge of cake you accidentally cut a wedge out of the PEC plate as well. When you are done getting your cake, you cut a wedge out Ⓒ=Yof another PEC plate and put it on top of the cake for storage. a) (9 points) If you connect a DC powers supply of V, to the top plate and grounded the bottom plate, the cake acts like a dielectric in a capacitor. Assuming that w » h, use the Laplace equation to find a particular solution for the electric potential inside the cake. b) (1 point) Using the potential function you derived in part a), find the electric field E and the electric flux density D inside the cake.

Fig: 1