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Chapter 4: Q4.14P (page 179)
When you polarize a neutral dielectric, the charge moves a bit, but the total remains zero. This fact should be reflected in the bound charges and · Prove from Eqs. 4.11 and 4.12 that the total bound charge vanishes.
The value of total charge of a piece of neutral dielectric is .
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A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a "frozen-in" polarization
Where a constant and 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:
Figure 4.18
(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 , and then get from Eq. 4.21. [Notice that the second method is much faster, and it avoids any explicit reference to the bound charges.]
At the interface between one linear dielectric and another, the electric field lines bend (see Fig. 4.34). Show that
Assuming there is no free charge at the boundary. [Comment: Eq. 4.68 is reminiscent of Snell's law in optics. Would a convex "lens" of dielectric material tend to "focus’’ or "defocus," the electric field?]
A point charge is imbedded at the center of a sphere of linear dielectric material (with susceptibilityand radius ).Find the electric field, the polarization, and the bound charge densities, and .What is the total bound charge on the surface? Where is the compensating negative bound charge located?
Find the field inside a sphere of linear dielectric material in an otherwise uniform electric field (Ex. 4.7) by the following method of successive approximations: First pretend the field inside is just , and use Eq. 4.30 to write down the resulting polarization . This polarization generates a field of its own, (Ex. 4.2), which in turn modifies the polarization by an amount . which further changes the field by an amount , and so on. The resulting field is . Sum the series, and compare your answer with Eq. 4.49.
A very long cylinder, of radius a, carries a uniform polarization P perpendicular to its axis. Find the electric field inside the cylinder. Show that the field outside the cylinder can be expressed in the form
[Careful: I said "uniform," not "radial"!]
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