E,(z,t) = ~2Ero sin(ot - kz)where o t accounts for phasechange over time, and kz over spaceThe surface made up of all points with the same space phase is calledthe wave front.Here the plane z = O is a wave front, and this electromagnetic waveis called a plane waveSince E,(z)is independent ofthexand y coordinates,the field intensityis constant on the wave front. Hence.this plane wave is called a uniformplane wave.U
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 y coordinates, the field intensity is constant on the wave front. Hence, this plane wave is called a uniform plane wave. ( , ) 2 sin( ) 0 E z t E t kz x = x −
The time interval during which the time phase (t) is changed by 2元is called the period, and it is denoted as T. The number of periods inone secondis called the frequency,and it is denoted as fSince oT =2元, we haveT_ 2元_ 1QThe distance over which the space phase factor (kr)is changed by 2元is calledthe wavelength, anditis denotedas a2元Since ka = 2元, we have2kThe frequency describes the rate at which an electromagnetic wavevaries with time, while the wavelength gives the intervalin space forthewaveto repeatitself.2元And we havek=元The constant k stands for the phase variation per unit length, and itis called the phase constant, and the constant k gives the numbers offull waves per unit length.Thus k is also called the wave numberuaV
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 as f. Since , we have f T 2π 1 = = T = 2π The distance over which the space phase factor (kr) is changed by 2 is called the wavelength, and it is denoted as . Since , we have k = 2π k 2π = 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 2π k = 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
The speed of phasevariationy,can be found fromthe locus of apoint with the same phase angle.Let ot-kz= const, and nothing thato dt - kdz = O, then the phase velocity y, isdzのdtkConsidering k=oeu, we haveeoloeesuIn a perfect dielectric, the phase velocity is governed by the propertyof the mediumConsiderthe relative permittivities of all media withe,>l,and withrelative permeabilityμ - l. The phase velocity of a uniform plane wavein a perfect dielectric is usually less than the velocity of lightin vacuumIt is possible to have y, > c . Therefore, the phase velocity must notbetheenergyvelocityUV
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 is t −kz = const dt − kdz = 0 t k z v = = d d p Considering , we have k = c c = = 0 0 r r r r 1 1 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. r 1 r 1 In a perfect dielectric, the phase velocity is governed by the property of the medium. 1 p = = k v It is possible to have . Therefore , the phase velocity must not be the energy velocity. v c p
Fromtheaboveresults,wefindp=afThe frequency of a plane wave depends on the source,and itisalways the same as that of the sourcein a linearmedium.Howeverthe phase velocityis related to the property of the medium, and hencethe wavelength is related to the property of the medium2Wefind福.feooVe,M+ou2wherefJeoMowhere Zois the wavelength of the planewavewith freguencyfin vacuumSince s, >1,u, ~1 , and a< Z.. Namely, the wavelength of a planewave in a medium is less than that in vacuum.This phenomenon maybe calledthe shrinkageof wavelengthUEV
v f From the above results, we find p = 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. r r 0 0 0 r r p 1 = = = f f v We find where 0 0 0 1 f = where 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 wavelength. r 1 0 r 1
jQEUsing Hwe findμ ozC= Hroe-ikoe-jk-HHL8whereHE0uIn perfect dielectrics, the electric field and the magnetic field of auniform plane wave are in phase, and both have the same spatialdependence,butthe amplitudesare constantEThe left figure shows thevariation of the electric fieldand the magnetic field in spaceHat t = 0.UV
z kz y kz Hy Ex H j 0 j 0 e e − − = = Using , we find z E H x y = j H y0 Ex0 where = 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. Hy Ex