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Chapter 9: Q37P (page 433)
A microwave antenna radiating at is to be protected from the environment by a plastic shield of dielectric constant . What is the minimum thickness of this shielding that will allow perfect transmission (assuming normal incidence)? [Hint: Use Eq. 9.199.]
The minimum thickness of the shielding
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Suppose, for some nonzero constants A, B, C, a, b, c, and for all x. Prove that a = b = cand A + B = C.
By explicit differentiation, check that the functions , , and in the text satisfy the wave equation. Show that and do not.
Question:Equation 9.36 describes the most general linearly polarized wave on a string. Linear (or "plane") polarization (so called because the displacement is parallel to a fixed vector n) results from the combination of horizontally and vertically polarized waves of the same phase (Eq. 9.39). If the two components are of equal amplitude, but out of phase by (say,,), the result is a circularly polarized wave. In that case:
(a) At a fixed point, show that the string moves in a circle about the axis. Does it go clockwise or counter clockwise, as you look down the axis toward the origin? How would you construct a wave circling the other way? (In optics, the clockwise case is called right circular polarization, and the counter clockwise, left circular polarization.)
(b) Sketch the string at time t =0.
(c) How would you shake the string in order to produce a circularly polarized wave?
Question: Show that the standing wave satisfies the wave equation, and express it as the sum of a wave traveling to the left and a wave traveling to the right (Eq. 9.6).
a) Derive Eqs. 9.179, and from these obtain Eqs. 9.180.
(b) Put Eq. 9.180 into Maxwell's equations (i) and (ii) to obtain Eq. 9.181. Check that you get the same results using (i) and (iv) of Eq. 9.179.
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