A thick spherical shell (inner radius a, outer radius b) is made of dielectric material...

A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a “frozen-in” polarization

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:

⁶The polarization drops abruptly to zero outside the material, so its derivative is a delta function (see Prob. 1.46). The surface bound charge is precisely this term—in this sense it is actually included in ρ_b, but we ordinarily prefer to handle it separately as σ_b.

(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.]

Reference equation 4.23

Reference equation 4.21

Reference equation 2.13

Reference Prob. 1.46

(a) Show that

[Hint: Use integration by parts.]

(b) Let θ(x) be the step function:

Show that dθ/dx = δ(x).