Suppose the field inside a large piece of dielectric is E _{0}, so that the electric displacement is D _{0}=\epsilon_{0} E _{0}+ P
(a) Now a small spherical cavity (Fig. 4.19a) is hollowed out of the material. Find the field at the center of the cavity in terms of E _{0} and P. Also find the displacement at the center of the cavity in terms of D _{0} . Assume the polarizationis “frozen in,” so it doesn’t change when the cavity is excavated.
(b) Do the same for a long needle-shaped cavity running parallel to P (Fig. 4.19b) .
(c) Do the same for a thin wafer-shaped cavity perpendicular to P (Fig. 4.19c).
Assume the cavities are small enough that P, E _{0}, and D _{0} are essentially uniform. [Hint: Carving out a cavity is the same as superimposing an object of the same shape but opposite polarization.]
