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Problems 555 (b) Obtain a single equation in E,. (c) If the fields vary sinusoidally with time, i t] Ex = Re [PE(z) e" what are the spatial dependences of the fields? (d) Specialize (c) to the (i) low loss limit (o/sE << 1) and (ii) large loss limit (o•/e >1). (e) Repeat (a)-(c) if the medium is a plasma with constitutive law aJ= eE at (f) A current sheet Ko cos wti, is placed at z = 0. Find the electric and magnetic fields if the sheet is placed within an Ohmic conductor or within a plasma. 9. A uniformly distributed volume current of thickness 2d, Jo cos wti., is a source of plane waves. e0, 1O o6, 00 < 2d (a) From Maxwell's equations obtain a single differential equation relating E, to J.. (b) Find the electric and magnetic fields within and outside the current distribution. (c) How much time-average power per unit area is delivered by the current? (d) How does this generated power compare to the electromagnetic time-average power per unit area leaving the volume current at z = ±d? 10. A TEM wave (E., H,) propagates in a medium whose permittivity and permeability are functions of z, e(z), and 1A(Z). (a) Write down Maxwell's equations and obtain single partial differential equations in E. and H,. (b) Consider the idealized case where e(z)=eee " Z and L(z)=L e- a l zI.A current sheet Koe"'i. is at z =0. What are 556 Eklctrody~amics-Fieldsand Waves the resulting electric and magnetic fields on each side of the sheet? (c) For what values of a are the solutions spatially evanescent or oscillatory? 11. We wish to compare various measurements between two observers, the second moving at a constant velocity vi, with respect to the first. (a) The first observer measures simultaneous events at two positions z and z2 so that tL =lt. What is the time interval between the two events t' -t as measured by the second observer? (b) The first observer measures a time interval At = tl - t2 between two events at the same position z. What is the time interval as measured by the second observer? (c) The first observer measures the length of a stick as L = z -z. What is the length of the stick as measured by the second observer? 12. A stationary observer measures the velocity of a particle as u = ni. + u,i, + ui,. (a) What velocity, u'=u'i.+u'i,+u 'i,, does another observer moving at constant speed vi, measure? (b) Find u' for the following values of u where co is the free space speed of light: (i) (ii) (iii) (iv) u = co0i. u = coi, u = coi u = (co/F)[i1 +i, +i,] (c) Do the results of (a) and (b) agree with the postulate that the speed of light for all observers is co? Section 7.4 13. An electric field is of the form E= 100 ej(2wx 10t-2wx 10-i2)iX volts/m (a) What is the frequency, wavelength, and speed of light in the medium? (b) If the medium has permeability lo = 47r x 10-7 henry/m, what is the permittivity e, wave impedance 'i, and the magnetic field? (c) How much time-average power per unit area is carried by the wave? 14. The electric field of an elliptically polarized plane wave in a medium with wave impedance 'i is ' t- X E = Re (E.oi, + E,o esi,) et(w A where E=o and E,o are real. Problems 557 (a) What is the magnetic field? (b) What is the instantaneous and time-average power flux densities? 15. In Section 3-1-4 we found that the force on one of the charges Q of a spherical atomic electric dipole of radius Ro is Q1 1d S41reoRJ where d is the dipole spacing. (a) Write Newton's law for this moveable charge with mass M assuming that the electric field varies sinusoidally with time as Eocoswt and solve for d. (Hint: Let wo = Q2I(M47oRE).) (b) What is the polarization P as a function of E if there are N dipoles per unit volume? What is the frequency dependent permittivity function e(w), where D(r) = e (w)E(r) This model is often appropriate for light propagating in dielectric media. (c) Use the results of (b),in Maxwell's equations to find the relation between the wavenumber k and frequency w. (d) For what frequency ranges do we have propagation or evanescence? (e) What are the phase and group velocities of the waves? (f) Derive the complex Poynting's theorem for this dispersive dielectric. 16. High-frequency wave propagation in the ionosphere is partially described by the development in Section 7-4-4 except that we must include the earth's dc magnetic field, which we take to be Hoi,. (a) The charge carriers have charge q and mass m. Write the three components of Newton's force law neglecting collisions but including inertia and the Coulomb-Lorentz force law. Neglect the magnetic field amplitudes of the propagating waves compared to Ho in the Lorentz force law. (b) Solve for each component of the current density J in terms of the charge velocity components assuming that the propagating waves vary sinusoidally with time as ey' Hint: Define 2 qn mE qijoHo m (c) Use the results of (b) in Maxwell's equations for fields of to solve for the wavenumber k in terms of o. (d) At what frequencies is the wavenumber zero or infinite? Over what frequency range do we have evanescence or propagation? the form ei(t-kz) 558 Electrodynamics-Fieldsand Waves (e) For each of the two modes found in (c), what is the polarization of the electric field? (f) What is the phase velocity of each wave? Since each mode travels at a different speed, the atmosphere acts like an anisotropic birefringent crystal. A linearly polarized wave Eo ei(dt-h°o>i, is incident upon such a medium. Write this field as the sum of right and left circularly polarized waves. Hint: Eoi.= 2 (i. +ji,)+ 2 (i -i,) (g) If the transmitted field at z = 0 just inside the medium has amplitude E, eai,, what are the electric and magnetic fields throughout the medium? 17. Nitrobenzene with 1A= to and e = 35eo is placed between parallel plate electrodes of spacing s and length I stressed by a dc.voltage Vo. Measurements have shown that light polarized parallel to the dc electric field travels at the speed c1l, while light polarized perpendicular to the dc electric field travels slightly faster at the speed c,, being related to the dc electric field Eo and free space light wavelength as 1 1 - ABE C11 CL where B is called the Kerr constant which for nitrobenzene is B -4.3 x 10 - 12 sec/V 2 at A = 500 nm. (a) Linearly polarized light with free space wavelength A= 500 nm is incident at 450 to the dc electric field. After exiting the Kerr cell, what is the phase difference between the field components of the light parallel and perpendicular to the dc electric field? (b) What are all the values of electric field strengths that allow the Kerr cell to act as a quarter- or half-wave plate? (c) The Kerr cell is placed between crossed polarizers (polariscope). What values of electric field allow maximum light transmission? No light transmission? Section 7.5 18. A uniform plane wave with y-directed electric field is normally incident upon a plasma medium at z = 0 with constitutive law 8Jf/at = (peE. The fields vary sinusoidally in time as e. (a) What is the general form of the incident, reflected, and transmitted fields? (b) Applying the boundary conditions, find the field amplitudes. (c) What is the time-average electromagnetic power density in each region for w > w, and for w )i.)is located in free space at z = 0. A dielectric medium (e, g) of semi-infinite extent is placed at z = d. Re[Koe j(wt - kz )iz I II C0,,A CO, oAO ,x '2 I 0 :·nw I d z (a) For what range of frequency can we have a nonuniform plane wave in free space and a uniform plane wave in the dielectric? Nonuniform plane wave in each region? Uniform plane wave in each region? (b) What are the electric and magnetic fields everywhere? (c) What is the time-average z-directed power flow density in each region if we have a nonuniform plane wave in free space but a uniform plane wave in the dielectric? Section 7.8 25. A uniform plane wave Re (Eo ei("-ik=-kc)i,) is obliquely incident upon a right-angled perfectly conducting corner. The wave is incident at angle 90to the z = 0 wall. 562 Electrodynamics-Fieldsand Waves DD E i (a) Try a solution composed of the incident and reflected waves off each surface of the conductor. What is the general form of solution? (Hint: There are four different waves.) (b) Applying the boundary conditions, what are the electric and magnetic fields? (c) What are the surface charge and current distributions on the conducting walls? (d) What is the force per unit area on each wall? (e) What is the power flow density? Section 7.9 26. Fermat's principle of least time states that light, when reflected or refracted off an interface, will pick the path of least time to propagate between two points. I t LA LAC 4, 4I (a) A beam of light from point A is incident upon a dielectric interface at angle 90 from the normal and is reflected through the point B at angle 0,.In terms of 6O,0,, hi and h2, and the speed of light c, how long does it take light to travel from A to B along this path? What other relation is there between O6,0,, LAB, hi and h 2 ? (b) Find the angle Oithat satisfies Fermat's principle. What is 0,? Problems 563 (c) In terms of Oi, 0,, hI, hs, and the light speeds c 1 and c 2 in each medium, how long does it take light to travel from A to C? (d) Find the relationship between Oi and 0, that satisfies Fermat's principle. 27. In many cases the permeability of dielectric media equals that of free space. In this limit show that the reflection and transmission coefficients for waves obliquely incident upon dielectric media are: E parallel to the interface R= 6 - 0t) sin (O sin (Oi + 0,)' T= 2 cos 0i sin 0, sin (60+ 0,) H parallel to the interface - 0,) R= tan (Oi tan (0i + 0)' T= 2 cos 0i sin 0, sin (0i + 0,) cos (0i - 0,) 28. White light is composed of the entire visible spectrum. The index of refraction n for most materials is a weak function of wavelength A, often described by Cauchy's equation n = A + B/A 2 Pe t A beam of white light is incident at 30" to a piece of glass with A = 1.5 and B = 5 x 10- 5 s m 2. What are the transmitted angles for the colors violet (400 nm), blue (450 nm), green (550 nm), yellow (600 nm), orange (650 nm), and red (700 nm)? This separation of colors is called dispersion. 29. A dielectric slab of thickness d with speed of light c2 is placed within another dielectric medium of infinite extent with speed of light c1 , where cl < c2 . An electromagnetic wave with H parallel to the interface is incident onto the slab at angle O6. (a) Find the electric and magnetic fields in each region. (Hint: Use Cramer's rule to find the four unknown field amplitudes in terms of Ei.) 564 Electodynamics-Fieldsand Waves el.A I el, JAI Ei C 'Vi dI I 0 d (b) For what range of incident angle do we have uniform or nonuniform plane waves through the middle region? (c) What is the transmitted time-average power density with uniform or nonuniform plane waves through the middle region. How can we have power flow through the middle region with nonuniform plane waves? Section 7.10 30. Consider the various prisms shown. (a) What is the minimum index of refraction n1 necessary for .no time-average power to be transmitted across the hypotenuse when the prisms are in free space, n 2 = 1, or water, n2 = 1.33? (b) At these values of refractive index, what are the exiting angles 0.?