The ratio of the amplitude of electric field intensity to that ofmagnetic field intensity is called the intrinsic impedance, and isdenoted as Z as given byuE.1H8The intrinsicimpedanceis a real number.In vacuum,the intrinsicimpedanceis denoted as Z40=377~120元QZo60The aboverelationship between the electric field intensityand themagnetic field intensity can be written in vector form as follows:EH-e.xEOrE.= ZH,xeHu
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 by = = y x H E Z The intrinsic impedance is a real number. In vacuum,the intrinsic impedance is denoted as Z0 377Ω 120π Ω 0 0 0 = = Z The above relationship between the electric field intensity and the magnetic field intensity can be written in vector form as follows: y z x Z H = e E 1 x Z y z Or E = H e Ex Hy z
The electric field and the magnetic field are transverse withrespect to the direction of propagation and the wave is called atransverse electromagnetic wave, or TEM waveWe will encounter non-TEM wave that has the electric or themagnetic field componentin the direction of propagationA uniform plane wave is a TEM wave. Only non-uniform wavescan be non-TEM waves, and TEM waves are not necessarily planewaves.From the electric field intensityand the magnetic field intensityfound, we can find the complex energy flow density vectorS.asE-e.ZH?S.=E.xH-1The complex energy flow density vectoris real, while theimaginary part is zero. It means that the energy is traveling inthepositivedirectiononlyUV
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 TEM wave. Only non-uniformwaves 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 2 0 2 * 0 c z y x x y z ZH Z E S = E H = e = e 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, We will encounter non-TEM wave that has the electric or the magnetic field componentin the direction of propagation
We constructa cylinderof long l and cross-sectionAalongthedirection ofenergy flow,as shown in the figure.Suppose the distribution of the energy isuniform in the cylinder. The average valueSof the energy density is way , and that of theenergy flowdensityis SavThen the total energy in the cylinder is way Al, and the total energyflowing across the cross-sectionalarea A per unit time is Say A.If all energy in the cylinder flows across the area A in the timeintervalt, thenWaylASavAt=waylASAObviously, the ratio - stands for the displacement of the energy in timet, and it is called the energy velocity, denoted as ve. We obtainWaU
We construct a cylinder of long l and cross-section A along the direction of energy flow, as shown in the figure. l S A 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. t l w A t w lA S A av av av = = 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 t l av av e w S v = SavAt = wavlA 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
Eroand w = 2w.= cE, we findConsiderings,zVeuThe wavefront of a uniform planewaveis an infiniteplaneandtheamplitudeof thefieldintensitvisuniform onthe wavefront,andthe energy flow densityis constanton thewave front.Thus thisuniform plane wave carries infinite energy. Apparently, an idealuniformplanewavedoes notexistinnatureIf the observer is very far away from the source, the wave frontisvery large while the observeris limited to the localarea,the wave canbe approximately considered as a uniform plane wave.By spatialFouriertransform,a non-plane wave can be expressedin terms of the sumof many planewaves, which provestobe usefulsometimesuV
Considering and , we find Z E S x 2 0 av = 2 av eav 0 2 w w Ex = = e p 1 v = = v 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
Example. A uniform plane wave is propagating along with thepositive direction of the z-axis in vacuum, and the instantaneousvalueoftheelectricfieldintensityisE(z, t) = e, 20 /2 sin( 6元×10°t - 2元z) V/mFind:(a)The frequencyandthe wavelength(b) The complex vectors of the electric and the magnetic fieldintensities.(c)Thecomplexenergyflowdensityvector.(d) Thephase velocityand the energy velocity6元×1080Solution: (a) The frequency is= 3×108 Hz2元2元2元2元The wavelength is:=1mk2元u7
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 ( , ) 20 2 sin( 6π 10 2π ) V/m 8 z t = t − z x E e 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. Solution:(a) The frequency is 3 10 Hz 2π 6π 10 2π 8 8 = = = f 1m 2π 2π 2π = = = k The wavelength is