How Does a Cavity and Geometry Affect Electric Fields in Linear Dielectrics?

[mathlete]

Here's the problem:

http://img41.imageshack.us/my.php?image=efield3od.png

Can anyone give me a push in the right direction? I'm really stuck on two things - how does the cavity affect the problem (the hint doesn't really help me), and what does the geometry of the problem (since I'm giving 3 shapes) have to do with it?

 

[Physics Monkey]

Let me discuss the spherical case and hopefully then you can generalize. The hint says that you can replace the spherical cavity with an oppositely polarized spherical object superimposed on the original dielectric. Now, what is the field inside a uniformly polarized sphere? With this information, you can use superposition to find the complete field at the center of the sphere. One more word of caution, make sure you still satisfy that dielectric boundary conditions.