文本參考成果8.plane

1、Chapter 8Plane Electromagnetic Waves Plane waves in perfect dielectric Plane waves in conducting media Polarizations of plane waves Normal incidence on a planar surfacePlane waves in arbitrary directionsOblique incidence at boundaryPlane waves in anisotropic media1. Wave Equations2. Plane Waves in P

2、erfect Dielectric3. Plane Waves in Conducting Media4. Polarizations of Plane Waves5. Normal Incidence on A Planar Surface6. Normal Incidence at Multiple Boundaries7. Plane Waves in Arbitrary Directions8. Oblique Incidence at Boundary between Perfect Dielectrics 9. Null and Total Reflections10. Obliq

3、ue Incidence at Conducting Boundary 11. Oblique Incidence at Perfect Conducting Boundary12. Plane Waves in Plasma13. Plane Waves in Ferrite1. Wave Equations In infinite, linear, homogeneous, isotropic media, a time-varying electromagnetic field satisfies the following equations:which are called inho

4、mogeneous wave equations,andwhere is the impressed source. 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 es Which are called homogeneous wave equations. To investigate the prop

5、agation of plane waves, we first solve the homogeneous wave equations. The relationship between the charge density (r, t) and the conduction current is For a sinusoidal electromagnetic field, the above equation eswhich are called homogeneous vector Helmholtz equations, and here In rectangular coordi

6、nate system, we havewhich are called homogeneous scalar Helmholtz equations. All of these equations have the same form, and the solutions are similar. 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

7、the field is related to the variable z only, we can showSince the field is independent of the variables x and y, we haveDue to , from the above equations we obtain ConsideringSubstituting that into Helmholtz equations:We find2. Plane Waves in Perfect Dielectric In a region without impressed source i

8、n 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 z only, and independent of the variables x and y, then the electric field has no ponent. Let , then the magnetic field

9、 intensity H is Where .Due toWe have From last section, we know that each component of the electric field intensity satisfies the homogeneous scalar Helmholtz equation. Considering , we havewhich is an ordinary differential equation of second order, and the general solution is The first term stands

10、for a wave traveling along the positive direction of the z-axis, while the second term leads to the opposite . Here only the wave traveling along with the positive direction of z-axis is consideredwhere Ex0 is the effective value of the electric field intensity at z = 0 .The instantaneous value is A

11、n illustration of the electric field intensity varying over space at different times is shown in the left figure.Ez(z, t)zOt1 = 0 The wave is traveling along the positive z-direction.where t accounts for phase change over time, and kz over space. The surface made up of all points with the same space

12、 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 y coordinates, the field intensity is constant on the wave front. Hence, this plane wave is called a uniform plane wave. The time inter

13、val 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 as f. Since , we have The distance over which the space phase factor (kr) is changed by 2 is called the wavelength, and it

14、is denoted as . Since , we have 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.And we have The constant k stands for the phase variation per unit length, and it is called the phase con

15、stant, and the constant k gives the numbers of full waves per unit length. Thus k is also called the wave number. The speed of phase variation vp can be found from the locus of a point with the same phase angle. Let , and nothing that , then the phase velocity vp isConsidering , we have Consider the

16、 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. In a perfect dielectric, the phase velocity is governed by the property of the medium. It is possible

17、 to have . Therefore , the phase velocity must not be the energy velocity.From the above results, we find 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

18、 hence the wavelength is related to the property of the medium. We findwherewhere 0 is the wavelength of the plane wave with frequency f in vacuum. 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 wavelengt

19、h. zUsing , we findwhere 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

20、= 0.HyEx The ratio of the amplitude of electric field intensity to that of magnetic field intensity is called the intrinsic impedance, and is denoted as Z as given byThe intrinsic impedance is a real number. In vacuum, the intrinsic impedance is denoted as Z0 The above relationship between the elect

21、ric field intensity and the magnetic field intensity can be written in vector form as follows:OrExHyz 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. A uniform plane wave is a

22、 TEM wave. Only non-uniform waves can be non-TEM waves, and TEM waves are not necessarily plane waves. From the electric field intensity and the magnetic field intensity found, we can find the complex energy flow density vector Sc as The complex energy flow density vector is real, while the imaginar

23、y part is zero. It means that the energy is traveling in the positive direction only, We will encounter non-TEM wave that has the electric or the magnetic field component in the direction of propagation. We construct a cylinder of long l and cross-section A along the direction of energy flow, as sho

24、wn in the figure.lSA 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. Obviously, the ratio stands for the displacement of the energy in time t, and it is called the energy velocity, denoted

25、 as ve. We obtain If all energy in the cylinder flows across the area A in the time interval t, then Then the total energy in the cylinder is wav Al , and the total energy flowing across the cross-sectional area A per unit time is Sav A.Considering and , we find The wave front of a uniform plane wav

26、e 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

27、 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 so

28、metimesExample. 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 isFind: (a) The frequency and the wavelength. (b) The complex vectors of the electric and the magnetic field intensities. (c) The

29、 complex energy flow density vector. (d) The phase velocity and the energy velocity.Solution: (a) The frequency is The wavelength is(b) The electric field intensity isThe magnetic field intensity is(c) The energy flow density vector is(d) The phase and energy velocities are3. Plane Waves in Conducti

30、ng Media If 0 , the first Maxwells equation esIf let 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:LetWe obtain If we let as before, and , t

31、hen the solution of the equation is the same as that in the lossless case as long as k is replaced by kc, so thatBecause kc is a complex number, we define We findIn this way, the electric field intensity can be expressed aswhere the first exponent leads to an exponential decay of the amplitude of th

32、e electric field intensity in the z-direction, and the second exponent gives rise to a phase delay. The phase velocity isIt depends not only on the parameters of the medium but also on the frequency. A conducting medium is a dispersive medium. The real part k is called the phase constant, with the u

33、nit of rad/m, while the imaginary part k is called the attenuation constant and has a unit of Np/m.The wavelength is The wavelength is related to the properties of the medium, and it has a nonlinear dependence on the frequency.The intrinsic impedance iswhich is a complex number. Since the intrinsic

34、impedance is a complex number, and it leads to a phase shift between electric field and the magnetic field.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.ExHyz Since the electri

35、c 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. Two special cases ： (a) If , as in an imperfect dielectric

36、, the approximationThen 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 .(b) If , as in good conductors, we takeThen The electric and the magnetic field int

37、ensities 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. The skin depth is the distance over which the field amplitude is re

38、duce by a factor of , mathematically determined from The skin depth is inversely proportional to the square root of the frequency f and the conductivity .The skin depths at different frequencies for copper f MHz0.051 mm29.80.0660.00038The skin depth deceases with increasing frequency. The frequency

39、for sets the boundary between an imperfect dielectric and a conductor, and it is called crossover frequency. stands for the ratio of amplitude of the conduction current to that of the displacement current. In imperfect dielectrics the dis-placement current dominates, while the converse is true for a

40、 good conductor. Several crossover frequencies for different materials: MediaFrequencies MHzDry Soil 2.6 (Short Wave)Wet Soil6.0 (Short Wave)Pure Water0.22 (Medium Wave)Sea Water 890 (Super Short Wave)Silicon15103 (Microwave)Germanium 11104 (Microwave)Platinum 16.91016 (Light Wave)Copper 104.41016 (

41、Light Wave) The attenuation of a plane wave is caused by the conductivity , resulting in power dissipation, and conductors are called lossy media. Dielectrics without conductivity are called lossless media. Besides conductor loss there are other losses due to dielectric polarization and magnetizatio

42、n. As a result, both permittivity and permeability are complex, so that and . 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 frequ

43、encies, dielectric loss can be neglected. 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:(

44、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

45、can be considered as a good conductor, and the phase constant k and the attenuation constant k are, respectively, The wavelength isThe intrinsic impedance isThe phase velocity isThe skin depth is(b) The complex vector of the electric field intensity is The complex vector of the magnetic field intens

46、ity is The instantaneous values of the electric and the magnetic field intensities at z = 0.8m asThe 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 i

47、n seawater. The time-varying behavior of the direction of the electric field intensity is called the polarization of the electromagnetic wave.4. Polarizations of Plane Waves Suppose the instantaneous value of the electric field intensity of a plane wave isObviously, at a given point in space the loc

48、us of the tip 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. The instantaneous value of the electric field intensity of another plane wave of the same frequency is This is also a linearly polarized pl

49、ane wave, but with the electric field along the y-direction. If the above two orthogonal, 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 electric

50、 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 o

51、f to the x-axis. Thus the resultant field is still a linearly polarized wave. EyExEyxOEyExEyxOEyExEyxO If two orthogonal, linearly polarized plane waves of the same phase but different amplitudes are combined, the resultant wave is still a linearly polarized plane wave. If the above two linearly pol

52、arized plane waves have a phase difference of , but the same amplitude Em, i.e. Conversely, a linearly polarized plane wave can be resolved into two orthogonal, linearly polarized plane waves of the same phase but different amplitudes. If the two plane waves have opposite phases and different amplit

53、udes, how about the resultant wave? The direction of the resultant wave is at an angle of to the x-axis, andThen the instantaneous value of the resultant wave isi.e. At a given point z the angle is a function of time t. The direction of the electric field intensity vector is rotating with time, but

54、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. 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 propa

55、gation direction and it is called the left-hand circularly polarized wave.EyExEyxOLeftRightzy x O Two orthogonal, linearly polarized waves of the same amplitude and phase difference of result in a circularly polarized wave. A linearly polarized wave can be resolved into two circularly polarized wave

56、s with opposite senses of rotation, and vice versa. If Ey is lagging behind Ex by , 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

57、 called a right-hand 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 . If two orthogonal, linearly polarized plane waves Ex and Ey have different amplitudes and phases The

58、 components Ex and Ey of the resultant wave satisfy the following equationwhich describes an ellipse. At a given point z, the locus of the tip of the resultant wave vector over time is an ellipse, and it is called an elliptically polarized wave.yxEx y Ey mEx m The linearly and the circularly polariz

59、ed 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 propert

60、y with many practical applications. If 0 , then the resultant wave vector is rotated in the clockwise direction, and it is a left-hand elliptically polarized wave. Since a circularly polarized electromagnetic wave is less attenuated by rain, it is used in all-weather radar. Stereoscopic film is take