Phys 110A - Electromagnetism
Homework 6
Spring 2024
Problems:
1. Consider a long teflon rod, (a dielectric cylinder), of radius a. Imagine that we could set up a permanent polarization P(s, θ, z) = ks where s is the usual cylindrical radial vector from the z-axis, and k is a constant). Neglect end effects, the cylinder is long.
(a) Calculate the bound charges σb and ρb (on the surface and interior of the rod, respectively). What are the units of “k”?
(b) Next, use these bound charges (along with Gauss law) to find the electric field inside and outside the cylinder. (Direction and magnitude)
(c) Find the electric displacement field (D) inside and outside the cylinder, and verify that Griffiths Eq 4.23 works. Explain briefly in words why your answer might be what it is.
2. For a dielectric sphere in a uniform. external field, Griffiths found the electric field inside the sphere in Example 4.7 using separation of variables and applying boundary conditions. There’s another approach, which in some ways is perhaps conceptually simpler.
(a) You’ve put the object into a uniform. external field E0, so it would be logical (but wrong) to assume that the polarization of the dielectric would just be simply P = 0χeE0. Briefly, why is that wrong?
(b) Go ahead and assume that anyway, as a first approximation. Call this approximation for the polarization P0 ≡ 0χeE0. Now this polarized sphere generates its own additional (induced) E field, which we will call E1. What is that field? (You don’t have to rederive it from scratch; the textbook and lectures both discuss the field inside a uniformly polarized sphere.)
(c) OK, but now E1 will modify the polarization by an additional amount, which we will call P1. What’s P1?
(d) That in turn will add to the electric field an additional amount E2. And so on. The final, total, real field will just be E0 + E1 + E2 + · · · . Work out that infinite sum, and check your answer with Griffiths’ result at the end of Example 4.7.
Hint: P ∞n=0xn = 1 1−x
. If the hint looks unfamiliar, convince yourself that it is true by Taylor expanding 1/(1 − x).
(e) If the sphere were made of silicon (see Table 4.2 in Griffiths), compare the “first approximation” for P0 with the true result for polarization. How important was it to go through the summation? In what limit, large or small r, does this summing procedure make a big difference?
3. Griffiths 4.21 (its the same problem in both the third and fifth edition).
4. We discussed the polarizability of an individual atom and the dielectric susceptibility of a solid. Do Griffiths 4.41 (problem 4.38 in the third edition and problem 4.42 in the fifth edition), which shows you how these quantities are related.