Problem 4.15 A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a "frozen-in" polarization k P(r) ==ŕ, r where k is a constant and r is the distance from the center (Fig. 4.18). (There is no free charge in the problem.) Find the electric field in all three regions by two different methods: P AP b a FIGURE 4.18 (c) Wafer (a) Sphere (b) Needle FIGURE 4.19 P (a) Locate all the bound charge, and use Gauss's law (Eq. 2.13) to calculate the field it produces. (b) Use Eq. 4.23 to find D, and then get E from Eq. 4.21. [Notice that the second method is much faster, and it avoids any explicit reference to the bound charges.]