emw - Electromagnetic Waves EEL3472 EEL3472...

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EEL 3472 EEL 3472 Electromagnetic Electromagnetic Waves Waves
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EEL 3472 EEL 3472 2 Electromagnetic Waves Electromagnetic Waves Spherical Wavefront Direction of Propagation Plane-wave approximation
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EEL 3472 EEL 3472 3 Electromagnetic Waves In the case of              (fields inside a good insulator such as air or vacuum) we have  vector  Helmholtz    equation This equation has a rich variety of solutions. Let us assume that     has only an x component and varies  only in the z direction.   In this case the vector Helmholtz  equation simplifies to scalar  Helmholtz equation The latter equation is similar to the voltage wave equation for a lossless transmission line.  E j j E E ) ( 2 ϖε σ ϖ μ + = ϖε << σ E E 0 2 2 = + E E με ϖ 0 2 2 2 = + x E z x E Electromagnetic Waves Electromagnetic Waves
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EEL 3472 EEL 3472 4 The solution of the scalar Helmholtz equation is where                 , to be found from boundary conditions;  the minus and the plus correspond to waves moving in the +z and –z directions, respectively. In terms of propagation constant k  The position of a field maximum is given by    z z j o x e E e E E γ με ϖ ± ± = = 0 o j e A E φ 0 0 = ) cos( ] Re[ ) , ( o o t j z j o x z t A e e E t z E + - = = - με ϖ φ + με = = φ + με ϖ - ϖ o max o max t z 0 z t 1 max = = dt dz U p s / m 10 x 99793 . 2 / 1 c U   ,    If 8 o o p o o = ε μ = = ε = ε μ = μ ) cos( ) , ( o o x kz t A t z E + - = με ϖ = k Electromagnetic Waves Electromagnetic Waves k
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EEL 3472 EEL 3472 5 Characteristics of  Plane Waves For a plane wave which propagates in the +z direction and  has an electric field directed in the x direction where  The time-varying electric field of the wave must, according  to Faraday’s law, be accompanied by a magnetic field.      Thus, when                              and is called the characteristic impedance of vacuum. x jkz o e e E E - = λ π = με ϖ = / 2 k y jkz o y x e e E jk e z E E H j - - = = × = ϖ μ - y jkz e e E H - = η 0 = η μ = μ ε = ε 377      , o o ε μ η= wave impedance (intrinsic impedance of  the medium ( 29 dt dB - Electromagnetic Waves Electromagnetic Waves
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emw - Electromagnetic Waves EEL3472 EEL3472...

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