Work out the theory of TM modes for a rectangular wave

Problem 3P Use Eq. 9.19 to determine A3 and ?3 in terms of A1, A2, ?1, and ?2. Equation 9.19

Problem 1P By explicit differentiation, check that the functions f1, f2, and f3 in the text satisfy the wave equation. Show that f4 and f5 do not.

Problem 2P Show that the standing wave f (z, t) = A sin(kz) cos(kvt) 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). Equation 9.6 f ( z , t ) = g(z ? vt) + h(z + vt).

Problem 4P Obtain Eq. 9.20 directly from the wave equation, by separation of variables. Equation 9.20

Problem 6P (a) Formulate an appropriate boundary condition, to replace Eq. 9.27, for the case of two strings under tension T joined by a knot of mass m. (b) Find the amplitude and phase of the reflected and transmitted waves for the case where the knot has a mass m and the second string is massless. Reference equation 9.27

Problem 5P Suppose you send an incident wave of specified shape, gI (z ? v1t), down string number 1. It gives rise to a reflected wave, hR(z + v1t), and a transmitted wave, gT (z ? v2t). By imposing the boundary conditions 9.26 and 9.27, find hR and gT .

Problem 40P Consider the resonant cavity produced by closing off the two ends of a rectangular wave guide, at z = 0 and at z = d, making a perfectly conducting empty box. Show that the resonant frequencies for both TE and TM modes are given by for integers l,m, and n. Find the associated electric and magnetic fields.

Problem 7P Suppose string 2 is embedded in a viscous medium (such as molasses), which imposes a drag force that is proportional to its (transverse) speed: (a) Derive the modified wave equation describing the motion of the string. (b) Solve this equation, assuming the string vibrates at the incident frequency ?. That is, look for solutions of the form (c) Show that the waves are attenuated (that is, their amplitude decreases with increasing z). Find the characteristic penetration distance, at which the amplitude is 1/e of its original value, in terms of ?, T,?, and ?. (d) If a wave of amplitude AI , phase ?I = 0, and frequency ? is incident from the left (string 1), find the reflected wave’s amplitude and phase.

Problem 9P Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude E0, frequency ?, and phase angle zero that is (a) traveling in the negative x direction and polarized in the z direction; (b) traveling in the direction from the origin to the point (1, 1, 1), with polarization parallel to the xz plane. In each case, sketch the wave, and give the explicit Cartesian components of k and

Problem 12P In the complex notation there is a clever device for finding the time average of a product. Suppose f (r, t) = A cos (k · r – ? + ?a) and g(r, t) = B cos (k · r ? ?t + ?b). Show that where the star denotes complex conjugation. [Note that this only works if the two waves have the same k and ?, but they need not have the same amplitude or phase.] For example,

Problem 8P 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 ) 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 90° (say, ?v = 0, ?h = 90?), the result is a circularly polarized wave. In that case: (a) At a fixed point z, show that the string moves in a circle about the z axis. Does it go clockwise or counterclockwise, 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 counterclockwise, left circular polarization.)3 (b) Sketch the string at time t = 0. (c) How would you shake the string in order to produce a circularly polarized wave? Equation 9.36

Problem 10P The intensity of sunlight hitting the earth is about 1300 W/m2. If sunlight strikes a perfect absorber, what pressure does it exert? How about a perfect reflector? What fraction of atmospheric pressure does this amount to?

Problem 13P Find all elements of the Maxwell stress tensor for a monochromatic plane wave traveling in the z direction and linearly polarized in the x direction (Eq. 9.48). Does your answer make sense? (Remember that represents the momentum flux density.) How is the momentum flux density related to the energy density, in this case? Reference equation 9.48

Problem 14P Calculate the exact reflection and transmission coefficients, without assuming ?1 = ?2 = ?0. Confirm that R + T = 1.

Problem 17P Analyze the case of polarization perpendicular to the plane of incidence (i.e. electric fields in the y direction, in Fig. 9.15). Impose the boundary conditions (Eq. 9.101), and obtain the Fresnel equations for Sketch as functions of ?I, for the case ? = n2/n1 = 1.5. (Note that for this ? the reflected wave is always 180° out of phase.) Show that there is no Brewster’s angle for any n1 and n2: is never zero (unless, of course, n1 = n2 and ?1 = ?2, in which case the two media are optically indistinguishable). Confirm that your Fresnel equations reduce to the proper forms at normal incidence. Compute the reflection and transmission coefficients, and check that they add up to 1.

Problem 16P Suppose Aeiax + Beibx = Ceicx , for some nonzero constants A, B, C, a, b, c, and for all x. Prove that a = b = c and A + B = C.

Problem 15P In writing Eqs. 9.76 and 9.77, I tacitly assumed that the reflected and transmitted waves have the same polarization as the incident wave—along the x direction. Prove that this must be so. [Hint: Let the polarization vectors of the transmitted and reflected waves be and prove from the boundary conditions that ?T = ?R = 0.] Reference equation 9.76 Reference equation 9.77

The index of refraction of diamond is 2.42. Construct the graph analogous to Fig. 9.16 for the air/diamond interface. (Assume \(\mu_{1}=\mu_{2}=\mu_{0}\).) In particular, calculate (a) the amplitudes at normal incidence, (b) Brewster’s angle, and (c) the “crossover” angle, at which the reflected and transmitted amplitudes are equal.

Problem 19P (a) Suppose you imbedded some free charge in a piece of glass. About how long would it take for the charge to flow to the surface? (b) Silver is an excellent conductor, but it’s expensive. Suppose you were designing a microwave experiment to operate at a frequency of 1010 Hz. How thick would you make the silver coatings? (c) Find the wavelength and propagation speed in copper for radio waves at 1 MHz. Compare the corresponding values in air (or vacuum).

Problem 20P (a) Show that the skin depth in a poor conductor (independent of frequency). Find the skin depth (in meters) for (pure) water. (Use the static values of ? , ?, and ?; your answers will be valid, then, only at relatively low frequencies.) (b) Show that the skin depth in a good conductor (where ? is the wavelength in the conductor). Find the skin depth (in nanometers) for a typical metal (? ? 107(? m) ?1) in the visible range (? ? 1015/s), assuming ? ? ? 0 and ? ? ?0. Why are metals opaque? (c) Show that in a good conductor the magnetic field lags the electric field by 45?, and find the ratio of their amplitudes. For a numerical example, use the “typical metal” in part (b).

Problem 22P Calculate the reflection coefficient for light at an air-to-silver interface at optical frequencies (? = 4 × 1015/s).

Problem 21P (a) Calculate the (time-averaged) energy density of an electromagnetic plane wave in a conducting medium (Eq. 9.138). Show that the magnetic contribution always dominates. (b) Show that the intensity is Equation 9.138

Problem 23P (a) Shallow water is nondispersive; waves travel at a speed that is proportional to the square root of the depth. In deep water, however, the waves can’t “feel” all the way down to the bottom—they behave as though the depth were proportional to ?. (Actually, the distinction between “shallow” and “deep” itself depends on the wavelength: If the depth is less than ?, the water is “shallow”; if it is substantially greater than ?, the water is “deep.”) Show that the wave velocity of deep water waves is twice the group velocity. (b) In quantum mechanics, a free particle of mass m traveling in the x direction is described by the wave function where p is the momentum, and E = p2/2m is the kinetic energy. Calculate the group velocity and the wave velocity. Which one corresponds to the classical speed of the particle? Note that the wave velocity is half the group velocity.

Problem 24P If you take the model in Ex. 4.1 at face value, what natural frequency do you get? Put in the actual numbers. Where, in the electromagnetic spectrum, does this lie, assuming the radius of the atom is 0.5 Å? Find the coefficients of refraction and dispersion, and compare them with the measured values for hydrogen at 0 ? C and atmospheric pressure: A = 1.36 × 10?4, B = 7.7 × 10?15m2 .

Problem 27P (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. Reference equation 9.179 Reference equation 9.180 Reference equation 9.181

Problem 28P Show that the mode TE00 cannot occur in a rectangular wave guide. [Hint: In this case ?/c = k, so Eqs. 9.180 are indeterminate, and you must go back to Eq. 9.179. Show that Bz is a constant, and hence—applying Faraday’s law in integral form to a cross section—that Bz = 0, so this would be a TEM mode.] Reference equation 9.179 Reference equation 9.180

Problem 25P Find the width of the anomalous dispersion region for the case of a single resonance at frequency ?0. Assume ? ? ?0. Show that the index of refraction assumes its maximum and minimum values at points where the absorption coefficient is at half-maximum.

Problem 29P Consider a rectangular wave guide with dimensions 2.28 cm × 1.01 cm. What TE modes will propagate in this wave guide, if the driving frequency is 1.70 × 1010 Hz? Suppose you wanted to excite only one TE mode; what range of frequencies could you use? What are the corresponding wavelengths (in open space)?

Problem 31P Work out the theory of TM modes for a rectangular wave guide. In particular, find the longitudinal electric field, the cutoff frequencies, and the wave and group velocities. Find the ratio of the lowest TM cutoff frequency to the lowest TE cutoff frequency, for a given wave guide. [Caution: What is the lowest TM mode?]

Problem 30P Confirm that the energy in the TEmn mode travels at the group velocity. [Hint: Find the time averaged Poynting vector and the energy density (use Prob. 9.12 if you wish). Integrate over the cross section of the wave guide to get the energy per unit time and per unit length carried by the wave, and take their ratio.] Reference prob 9.12 In the complex notation there is a clever device for finding the time average of a product. Suppose f (r, t) = A cos (k · r – ? + ?a) and g(r, t) = B cos (k · r ? ?t + ?b). Show that where the star denotes complex conjugation. [Note that this only works if the two waves have the same k and ?, but they need not have the same amplitude or phase.] For example,

Problem 32P (a) Show directly that Eqs. 9.197 satisfy Maxwell’s equations (Eq. 9.177) and the boundary conditions (Eq. 9.175). (b) Find the charge density, ?(z, t), and the current, I (z, t), on the inner conductor. Reference equation 9.197 Reference equation 9.197 Reference equation 9.195

Problem 33P The “inversion theorem” for Fourier transforms states that Use this to determine , in Eq. 9.20, in terms of ?f ? ?) ?and a Reference equation 9.20

Problem 34P [The naive explanation for the pressure of light offered in Section 9.2.3 has its flaws, as you discovered if you worked Problem 9.11. Here’s another account, due originally to Planck.25] A plane wave traveling through vacuum in the z direction encounters a perfect conductor occupying the region z ? 0, and reflects back: (a) Find the accompanying magnetic field (in the region z < 0). (b) Assuming B = 0 inside the conductor, find the current K on the surface z = 0, by invoking the appropriate boundary condition. (c) Find the magnetic force per unit area on the surface, and compare its time average with the expected radiation pressure (Eq. 9.64). Reference prob 9.11 Consider a particle of charge q and mass m, free to move in the xy plane in response to an electromagnetic wave propagating in the z direction (Eq. 9.48—might as well set ? = 0). (a) Ignoring the magnetic force, find the velocity of the particle, as a function of time. (Assume the average velocity is zero.) (b) Now calculate the resulting magnetic force on the particle. (c) Show that the (time) average magnetic force is zero. The problem with this naive model for the pressure of light is that the velocity is 90 ? out of phase with the fields. For energy to be absorbed, there’s got to be some resistance to the motion of the charges. Suppose we include a force of the form ??mv, for some damping constant ? . (d) Repeat part (a) (ignore the exponentially damped transient). Repeat part (b), and find the average magnetic force on the particle.9

Problem 35P Suppose (This is, incidentally, the simplest possible spherical wave. For notational convenience, let (kr ? ?t) ? u in your calculations.) (a) Show that E obeys all four of Maxwell’s equations, in vacuum, and find the associated magnetic field. (b) Calculate the Poynting vector. Average S over a full cycle to get the intensity vector I. (Does it point in the expected direction? Does it fall off like r?2, as it should?) (c) Integrate I · da over a spherical surface to determine the total power radiated.

Problem 38P Light from an aquarium (Fig. 9.27) goes from water into air (n = 1). Assuming it’s a monochromatic plane wave and that it strikes the glass at normal incidence, find the minimum and maximum transmission coefficients (Eq. 9.199). You can see the fish clearly; how well can it see you? Equation 9.199 Figure 9.27

Problem 36P Light of (angular) frequency ? passes from medium 1, through a slab (thickness d) of medium 2, and into medium 3 (for instance, from water through glass into air, as in Fig. 9.27). Show that the transmission coefficient for normal incidence is given by [Hint: To the left, there is an incident wave and a reflected wave; to the right, there is a transmitted wave; inside the slab, there is a wave going to the right and a wave going to the left. Express each of these in terms of its complex amplitude, and relate the amplitudes by imposing suitable boundary conditions at the two interfaces. All three media are linear and homogeneous; assume ?1 = ?2 = ?3 = ?0.] Figure 9.27

Problem 39P According to Snell’s law, when light passes from an optically dense medium into a less dense one (n1 > n2) the propagation vector k bends away from the normal (Fig. 9.28). In particular, if the light is incident at the critical angle then ?T = 90?, and the transmitted ray just grazes the surface. If ?I exceeds ?c, there is no refracted ray at all, only a reflected one (this is the phenomenon of total internal reflection, on which light pipes and fiber optics are based). But the fields are not zero in medium 2; what we get is a so-called evanescent wave, which is rapidly attenuated and transports no energy into medium 2.26 A quick way to construct the evanescent wave is simply to quote the results of Sect. 9.3.3, with kT = ?n2/c and the only change is that is now greater than 1, and is imaginary. (Obviously, ?T can no longer be interpreted as an angle!) (a) Show that This is a wave propagating in the x direction (parallel to the interface!), and attenuated in the z direction. (b) Noting that ? (Eq. 9.108) is now imaginary, use Eq. 9.109 to calculate the reflection coefficient for polarization parallel to the plane of incidence. [Notice that you get 100% reflection, which is better than at a conducting surface (see, for example, Prob. 9.22).] (c) Do the same for polarization perpendicular to the plane of incidence (use the results of Prob. 9.17). (d) In the case of polarization perpendicular to the plane of incidence, show that the (real) evanescent fields are (e) Check that the fields in (d) satisfy all of Maxwell’s equations (Eq. 9.67). (f) For the fields in (d), construct the Poynting vector, and show that, on average, no energy is transmitted in the z direction. Prob 9.22 Calculate the reflection coefficient for light at an air-to-silver interface at optical frequencies (? = 4 × 1015/s). Prob 9.17 Analyze the case of polarization perpendicular to the plane of incidence (i.e. electric fields in the y direction, in Fig. 9.15). Impose the boundary conditions (Eq. 9.101), and obtain the Fresnel equations for Sketch as functions of ?I, for the case ? = n2/n1 = 1.5. (Note that for this ? the reflected wave is always 180? out of phase.) Show that there is no Brewster’s angle for any n1 and n2: is never zero (unless, of course, n1 = n2 and ?1 = ?2, in which case the two media are optically indistinguishable). Confirm that your Fresnel equations reduce to the proper forms at normal incidence. Compute the reflection and transmission coefficients, and check that they add up to 1. Equation 9.67

Problem 37P A microwave antenna radiating at 10 GHz is to be protected from the environment by a plastic shield of dielectric constant 2.5. What is the minimum thickness of this shielding that will allow perfect transmission (assuming normal incidence)? [Hint: Use Eq. 9.199.] Equation 9.199