Chapter 8 Plane Electromagnetic Waves

1. Chapter 8 Plane Electromagnetic Waves Plane waves in perfect dielectric Plane waves in conducting media Polarizations of plane waves Normal incidence on a planar surface Plane waves in arbitrary directions Oblique incidence at boundary Plane waves in anisotropic media

2. 1. Wave Equations 2. Plane Waves in Perfect Dielectric 3. Plane Waves in Conducting Media 4. Polarizations of Plane Waves 5. Normal Incidence on A Planar Surface 6. Normal Incidence at Multiple Boundaries 7. Plane Waves in Arbitrary Directions 8. Oblique Incidence at Boundary between Perfect Dielectrics 9. Null and Total Reflections 10. Oblique Incidence at Conducting Boundary 11. Oblique Incidence at Perfect Conducting Boundary 12. Plane Waves in Plasma 13. Plane Waves in Ferrite

3. where is the impressed source. 1. Wave Equations In infinite, linear, homogeneous, isotropic media, a time-varying electromagnetic field satisfies the following equations: which are called inhomogeneous wave equations,and

4. The relationship between the charge density (r, t) and the conduction current is In a region without impressed source, J' = 0. If the medium is a perfect dielectric, then, = 0 . In this case, the conduction current is zero, and  = 0. The above equation becomes Which are called homogeneous wave equations. To investigate the propagation of plane waves, we first solve the homogeneous wave equations.

5. For a sinusoidal electromagnetic field, the above equation becomes which are called homogeneous vector Helmholtz equations, and here In rectangular coordinate system, we have which are called homogeneous scalar Helmholtz equations. All of these equations have the same form, and the solutions are similar.

6. In a rectangular coordinate system, if the field depends on one variable only, the field cannot have a component along the axis of this variable. If the field is related to the variable z only, we can show Since the field is independent of the variables x and y, we have

7. Considering Due to , from the above equations we obtain We find Substituting that into Helmholtz equations:

8. Where . Let , then the magnetic field intensity His 2. Plane Waves in Perfect Dielectric In a region without impressed source in a perfect dielectric, a sinusoidal electromagnetic field satisfies the following homogeneous vector Helmholtz equation If the electric field intensity E is related to the variable zonly, and independent of the variables x and y, then the electric field has no z-component.

9. Due to We have From last section, we know that each component of the electric field intensity satisfies the homogeneous scalar Helmholtz equation. Considering , we have which is an ordinary differential equation of second order, and the general solution is The first term stands for a wave traveling along the positive direction of the z-axis, while the second term leads to the opposite .

10.  The instantaneous value is Ex(z, t)  t1 = 0 O z Here only the wave traveling along with the positive direction of z-axis is considered where Ex0 is the effective value of the electric field intensity at z = 0 . An illustration of the electric field intensity varying over space at different times is shown in the left figure. The wave is traveling along the positivez-direction.

11. where  t accounts for phase change over time, and kz over space.The surface made up of all points with the same space phase is called the wave front. Here the plane z = 0 is a wave front, and this electromagnetic wave is called a plane wave. Since Ex(z) is independent of the x and ycoordinates, the field intensity is constant on the wave front. Hence, this plane wave is called a uniform plane wave.

12. Since , we have Since , we have And we have The time interval during which the time phase(t) is changed by 2is called the period, and it is denoted as T. The number of periods in one second is called the frequency, and it is denoted asf. The distance over which the space phase factor (kr) is changed by 2 is called the wavelength, and it is denoted as . The frequency describes the rate at which an electromagnetic wave varies with time, while the wavelength gives the interval in space for the wave to repeat itself. The constant k stands for the phase variation per unit length, and it is called the phase constant, and the constant k gives the numbers of full waves per unit length. Thus k is also called the wave number.

13. The speed of phase variationvp can be found from the locus of a point with the same phase angle. Let , and nothing that , then the phase velocityvp is Considering , we have Consider the relative permittivities of all media with , and with relative permeability . The phase velocity of a uniform plane wave in a perfect dielectric is usually less than the velocity of light in vacuum. It is possible to have . Therefore , the phase velocity must not be the energy velocity. In a perfect dielectric, the phase velocity is governed by the property of the medium.

14. From the above results, we find We find where Since , , and . Namely, the wavelength of a plane wave in a medium is less than that in vacuum. This phenomenon may be called the shrinkage of wavelength. The frequency of a plane wave depends on the source, and it is always the same as that of the source in a linear medium. However, the phase velocity is related to the property of the medium, and hence the wavelength is related to the property of the medium. where 0 is the wavelength of the plane wave with frequency f in vacuum.

15. Using , we find where Ex z Hy In perfect dielectrics, the electric field and the magnetic field of a uniform plane wave are in phase, and both have the same spatial dependence, but the amplitudes are constant. The left figure shows the variation of the electric field and the magnetic field in space at t = 0.

16. Ex z Or Hy The ratio of the amplitude of electric field intensity to that of magnetic field intensity is called the intrinsic impedance, and is denoted as Zas given by The intrinsic impedance is a real number. In vacuum, the intrinsic impedance is denoted as Z0 The above relationship between the electric field intensity and the magnetic field intensity can be written in vector form as follows:

17. The electric field and the magnetic field are transverse with respect to the direction of propagation and the wave is called a transverse electromagnetic wave, or TEM wave. We will encounter non-TEM wave that has the electric or the magnetic field component in the direction of propagation. A uniform plane wave is a TEM wave. Only non-uniform waves can be non-TEM waves, and TEM waves are not necessarily uniform plane waves. From the electric field intensity and the magnetic field intensity found, we can find the complex energy flow density vector Scas The complex energy flow density vector is real, while the imaginary part is zero. It means that the energy is traveling in the positive direction only,

18. A S l Obviously, the ratio stands for the displacement of the energy in time t, and it is called the energy velocity, denoted as ve. We obtain We construct a cylinder of long l and cross-section A along the direction of energy flow, as shown in the figure. Suppose the distribution of the energy is uniform in the cylinder. The average value of the energy density is wav , and that of the energy flow density is Sav. Then the total energy in the cylinder is wav Al , and the total energy flowing across the cross-sectional area Aper unit time is SavA. If all energy in the cylinder flows across the area A in the time interval t, then

19. Considering and , we find The wave front of a uniform plane wave is an infinite plane and the amplitude of the field intensity is uniform on the wave front, and the energy flow density is constant on the wave front. Thus this uniform plane wave carries infinite energy. Apparently, an ideal uniform plane wave does not exist in nature. If the observer is very far away from the source, the wave front is very large while the observer is limited to the local area, the wave can be approximately considered as a uniform plane wave. By spatial Fourier transform, a non-plane wave can be expressed in terms of the sum of many plane waves, which proves to be useful sometimes

20. Solution:(a) The frequency is The wavelength is Example. A uniform plane wave is propagating along with the positive direction of the z-axis in vacuum, and the instantaneous value of the electric field intensity is Find: (a) The frequency and the wavelength. (b) The complex vectors of the electric and the magnetic field intensities. (c) The complex energy flow density vector. (d) The phase velocity and the energy velocity.

21. (b) The electric field intensity is The magnetic field intensity is (c) The energy flow density vector is (d) The phase and energy velocities are

22. If let 3. Plane Waves in Conducting Media If  0,the first Maxwell’s equation becomes Then the above equation can be rewritten as where e is called the equivalent permittivity. In this way, a sinusoidal electromagnetic field then satisfies the following homogeneous vector Helmholtz equation:

23. Let We obtain If we let as before, and , then the solution of the equation is the same as that in the lossless case as long as k is replaced by kc, so that We find Becausekc is a complex number, we define

24. The phase velocity is In this way, the electric field intensity can be expressed as where the first exponent leads to an exponential decay of the amplitude of the electric field intensity in the z-direction, and the second exponent gives rise to a phase delay. The real part kis called the phase constant, with the unit of rad/m, while the imaginary part k is called the attenuation constant and has a unit of Np/m. It depends not only on the parameters of the medium but also on the frequency. A conducting medium is a dispersive medium.

25. The wavelength is The intrinsic impedance is The wavelength is related to the properties of the medium, and it has a nonlinear dependence on the frequency. which is a complex number. Since the intrinsic impedance is a complex number, and it leads to a phase shift between electric field and the magnetic field.

26. Ex z Hy The magnetic field intensity is The amplitude of the magnetic field intensity also decreases with z, but the phase is different from that of the electric field intensity. Since the electric and the magnetic field intensities are not in phase, the complex energy flow density vector has non-zero real and imaginary parts. This means that there is both energy flow and energy exchange when a wave propagates in a conductive medium.

27. (a) If , as in an imperfect dielectric, the approximation Then (b) If , as in good conductors, we take Two special cases： The electric and the magnetic field intensities are essentially in phase. There is still phase delay and attenuation in this case. The attenuation constant is proportional to the conductivity .

28. Then The skin depth集肤深度 is the distance over which the field amplitude is reduce by a factor of , mathematically determined from The electric and the magnetic field intensities are not in phase, and the amplitudes show a rapid decay due to a large . In this case, the electromagnetic wave cannot go deep into the medium, and it only exists near the surface. This phenomenon is called the skin effect集肤效应. The skin depth is inversely proportional to the square root of the frequencyf and the conductivity .

29. The skin depths at different frequencies for copper f MHz 0.05 1  mm 29.8 0.066 0.00038 The frequency for sets the boundary between an imperfect dielectric and a conductor, and it is called crossover frequency界限频率. stands for the ratio of amplitude of the conduction current to that of the displacement current. The skin depth deceases with increasing frequency. Several crossover frequencies for different materials: In imperfect dielectrics the dis-placement current位移电流dominates, while the converse is true for a good conductor.

30. Besides conductor loss there are other losses due to dielectric polarization and magnetization.As a result, both permittivity介电常数and permeability磁导率are complex, so that and . The attenuation of a plane wave is caused by the conductivity , resulting in power dissipation, and conductors are called lossy media有耗媒质. Dielectrics without conductivity are called lossless media. The imaginary part stands for dissipation, and they are called dielectric loss and magnetic loss, respectively. For non-ferromagnetic media, the magnetization loss can be neglected. For electromagnetic waves at lower frequencies, dielectric loss can be neglected.

31. Example.A uniform plane wave of frequency 5MHz is propagating along the positive direction of the z-axis. The electric field intensity is in the x-direction at , with an effective value of 100(V/m). If the region is seawater, and the parameters are , find: (a) The phase constant, the attenuation constant, the phase velocity, the wavelength, the wave impedance, and the skin depth in seawater. (b) The instantaneous values of the electric and the magnetic field intensities, and the complex energy flow density vector at z = 0.8m. Solution: (a) The seawater can be considered as a good conductor, and the phase constant k' and the attenuation constant k" are, respectively,

32. The phase velocity is The wavelength is The intrinsic impedance波阻抗is The skin depth is (b) The complex vector of the electric field intensity is The complex vector of the magnetic field intensity is

33. The instantaneous values of the electric and the magnetic field intensities at z = 0.8m as The complex energy flow density vector as The plane wave of frequency 5MHz is attenuated very fast in seawater. Therefore it is impossible to communicate between two submarines by using the direct wave in seawater.

34. 4. Polarizations极化 of Plane Waves The time-varying behavior of the direction of the electric field intensity is called the polarization of the electromagnetic wave. Suppose the instantaneous value of the electric field intensity of a plane wave is Obviously, at a given point in space the locus轨迹of the tip端点of the electric field intensity vector over time is a straight line parallel to the x-axis. Hence, the wave is said to have a linear polarization线极化. The instantaneous value of the electric field intensity of another plane wave of the same frequency is This is also a linearly polarized plane wave, but with the electric field along the y-direction.

35. y y E Ey y E Ey   x O x Ex Ex O O Ex x  Ey E If the above two orthogonal正交, linearly polarized plane waves with the same phase but different amplitudes coexist, then the instantaneous value of the resultant electric field is The time-variation of the magnitude of the resultant合成 electric field is still a sinusoidal function, and the tangent of the angle between the field vector and the x-axis is The polarization direction of the resultant electric field is independent of time, and the locus of the tip of the electric field intensity vector over time is a straight line at an angle of  to the x-axis. Thus the resultant field is still a linearly polarized wave线极化波.

36. If the above two linearly polarized plane waves have a phase difference of , but the same amplitude Em, i.e. If two orthogonal, linearly polarized plane waves of the same phase but different amplitudes are combined, the resultant合成wave is still a linearly polarized plane wave. Conversely, a linearly polarized plane wave can be resolved into分解two orthogonal, linearly polarized plane waves of the same phase but different amplitudes. If the two plane waves have opposite phases and different amplitudes, how about the resultant wave?

37. i.e. Then the instantaneous value of the resultant wave合成波is The direction of the resultant wave is at an angle of  to the x-axis, and At a given point zthe angle is a function of timet. The direction of the electric field intensity vector is rotating with time, but the magnitude is unchanged. Therefore, the locus of the tip of the electric field intensity vector is a circle, and it is called circular polarization圆极化.

38. Right Left y x E Ey  z O x O Ex y The angle will be decreasing with increasing time t . When the fingers of the left hand follow the rotating direction, the thumb points to the propagation direction and it is called the left-hand circularly polarized wave左旋圆极化波.

39. If Ey is lagging behindExby , the resultant wave is at an angle of to the x-axis. At a given point z, the angle will be increasing with increasing time t. The rotating direction and the propagation direction ez obey the right-hand rule, and it is called a right-hand circularly polarized wave右旋圆极化波. Two orthogonal, linearly polarized waves of the same amplitude and phase difference of result in a circularly polarized wave. Conversely, a circularly polarized wave can be resolved into two orthogonal, linearly polarized waves of the same amplitude and a phase difference of . A linearlypolarized wave can be resolved into two circularly polarized waves with opposite senses of rotation, and vice versa.

40. x ' y E y ' Ex m Ey m  x If two orthogonal, linearly polarized plane waves Exand Ey have different amplitudes and phases The components Ex and Ey of the resultant wave satisfy the following equation which describes an ellipse. At a given point z, the locus轨迹of the tip of the resultant wave vector over time is an ellipse, and it is called an elliptically polarized wave椭圆极化波.

41. If  < 0, Ey lags behind Ex, and the resultant wave vector is rotated in the counter-clockwise direction. It is a right-hand elliptically polarized wave左旋椭圆极化波. If  > 0, then the resultant wave vector is rotated in the clockwise direction, and it is a left-hand elliptically polarized wave左旋椭圆极化波. The linearly and the circularly polarized waves can both be considered as the special cases of the elliptically polarized wave. Since all polarized waves can be resolved into linearly polarized waves, only the propagation of linearly polarized wave will be discussed. The propagation behavior of an electromagnetic wave is a useful property with many practical applications. Since a circularly polarized electromagnetic wave is less attenuated by rain, it is used in all-weather radar.

42. In wireless communication systems, the polarization of the receiving antenna must be compatible with that of the wave to be received. In mobile satellite communications and globe positioning systems, because the position of the satellite changes with time, circularly polarized waves should be used. some microwave devices use the polarization of the wave to achieve special functions, as found in ferrite circulators, ferrite isolators, and others. Stereoscopic film is taken by using two cameras with two orthogonally polarized lenses. Hence, the viewer has to wear a pair of orthogonally polarized glasses to be able to see the three-dimensional effect.

43. Consider an infinite planar boundary between two homogeneous media with the parameters of the media and . x 111 222 Sr y z St Si 5. Normal Incidence正入射 on A Planar Surface平面边界 Let the boundary coincides with the plane z = 0. As an x-directed, linearly polarized plane wave is normally incident on the boundary from medium ①, a reflected and a transmitted wave are produced at the boundary. Since the tangential components of the electric field intensities must be continuous at any boundary, the sum of the tangential components of the electric field intensities of the incident入射and the reflected反射waves is equal to that of the transmitted透射wave.

44. x 111 222 Incident wave y Sr z Reflected wave Transmitted wave St Si where , , are the amplitudes at the boundary, respectively. The polarization cannot be changed during the reflection and the transmission of a linearly polarized wave. Assume that the electric field intensities of the incident, the reflected, and the transmitted waves are given in the figure, and they can be expressed as follows:

45. Incident wave Reflected wave x 111 222 Sr Transmitted wave y z St Si We have The magnetic field intensities as The tangential components of electric field intensities must be continuous at any boundary. Consider there is no surface current at the boundary with the limited conductivity有限电导率. Hence the tangential components of the magnetic field intensities are also continuous at the boundary z = 0.

46. We find We obtain The ratio of the electric field components of the reflected wave to that of the incident wave at the boundary is defined as the reflection coefficient反射系数, denoted as R. The ratio of the electric field components of the transmitted wave to that of the incident wave at the boundary is called the transmission coefficient透射系数, denoted as T. In medium ①, the resultant合成electric and magnetic field intensities are

47. (a) If medium ① is a perfect dielectric and medium ② is a perfect electric conductor , the intrinsic impedances波阻抗of the media, are respectively, We find The refection coefficient R = 1 implies that at the boundary so that the electric fields of the reflected and the incident waves have the same amplitude but opposite phase, resulting in a total electric field that is zero at the boundary. Two special boundaries as follows: All of the electromagnetic energy is reflected by the boundary, and no energy enters into medium ②. This case is called total reflection.

48. The instantaneous value is x 111 222 Sr In medium ① the phase of the resultant electric field is dependent on time only, and the amplitude has a sinusoidal dependence on z. when the electric field is zero all the time. y z St Si The propagation constant for medium ① is , and in medium ① the complex vector of the resultant合成electric field is

49. x 111 222 Sr y z St Si At the amplitude of the electric field is always the largest at any time. This means that the spatial phase of the resultant wave is fixed, and the amplitudes vary in a proportionate manner. This plane wave is not traveling, but stays at a fixed location with the field intensities varying periodically周期性地as time progresses. It is called a standing wave驻波.

50. 2 =  1 = 0 t1 = 0 2 =  2 =  2 =  2 =  1 = 0 1 = 0 1 = 0 1 = 0 Ex 0>0 Ex 0>0 t1 = 0 Ex 0>0 z z z z z 1 1 1 1 1 Ex 0>0 Ex 0>0 O O O O O The location at which the amplitude is always zero are called wave node, while the location at which the amplitude is always maximum are called wave loop. The plane wave in an open perfect dielectric as discussed previously is called a traveling wave行波, as opposed to a standing wave. The phase of a traveling wave is progressing in the propagation direction, while that of a standing wave does not move in space.