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# 329lect22 - 22 Phasor form of Maxwells equations and damped...

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22 Phasor form of Maxwell’s equations and damped waves in conducting media When the fields and the sources in Maxwell’s equations are all monochro- · D = ρ · B = 0 ∇ × E = - B t ∇ × H = J + D t . matic functions of time expressed in terms of their phasors, Maxwell’s equations can be transformed into the phasor domain. In the phasor domain all t j ω and all variables D , ρ , etc. are replaced by their phasors ˜ D , ˜ ρ , and so on. With those changes Maxwell’s equations take the form shown in the margin. · ˜ D = ˜ ρ · ˜ B = 0 ∇ × ˜ E = - j ω ˜ B ∇ × ˜ H = ˜ J + j ω ˜ D Also in these equations it is implied that ˜ D = ˜ E ˜ B = μ ˜ H ˜ J = σ ˜ E where , μ , and σ could be a function of frequency ω (as, strictly speaking, they all are as seen in Lecture 11). We can derive from the phasor form Maxwell’s equations shown in the margin the TEM wave properties obtained earlier on using the time-domain equations by assuming ˜ ρ = ˜ J = 0 . 1

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We will do that, and and after that relax the requirement ˜ J = 0 with ˜ J = σ ˜ E to examine how TEM waves propagate in conducting media. With ˜ ρ = ˜ J = 0 the phasor form Maxwell’s equation take their simpli- fied forms shown in the margin. · ˜ E = 0 · ˜ H = 0 ∇ × ˜ E = - j ω μ ˜ H ∇ × ˜ H = j ω ˜ E Using ∇ × [ ∇ × ˜ E = - j ω μ ˜ H ] -∇ 2 ˜ E = - j ω μ ∇ × ˜ H which combines with the Ampere’s law to produce 2 ˜ E + ω 2 μ ˜ E = 0 . For x -polarized waves with phasors ˜ E = ˆ x ˜ E x ( z ) , the phasor wave equation above simplifies as 2 z 2 ˜ E x + ω 2 μ ˜ E x = 0 . Try solutions of the form ˜ E x ( z ) = e - γ z or e γ z where γ is to be determined.
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