325_Sp2011_14_Plane_waves_updated

# 325_Sp2011_14_Plane_waves_updated - 14 Uniform plane waves...

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 1 14. Uniform plane waves EE325 Mikhail Belkin

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 2 Summary of electromagnetics: Maxwell’s equations summarizing everything we have so far, valid even if things are changing in time plus material properties 0 B = r r g { r o B H μ μ μ = r r v D ρ = r r g { r o D E ε ε ε = r r J E σ = r r D H J t ∇× = + r r r B E t ∇× = - r r Faraday’s law Ampere’s law Gauss’s law
© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 3 Wave equations Let’s derive equations that will determine how E and H fields should change in space and time in case of time-varying fields Remember that D= ε E, B= μ H, and J= σ E; assume there is no charge 0 H = r r g 0 E = r r g E H E t σ ε ∇× = + r r r H E t μ ∇× = - r r Take curl of Eq. (1) to get an equation for E-field and curl of Eq. (2) to get an equation for H-field: (1) (2) (3) (4) 2 2 H E E E H t t t t μ μ μσ με ∇×∇× = -∇× = - ∇× = - - r r r r r 2 2 E H H H E E E t t t t σ ε σ ε μσ εμ ∇×∇× = ∇× + ∇× = ∇× + ∇× = - - r r r r r r r

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 4 Wave equations (2) We know the expression for ∇ × ∇ × F: where We also know that and 2 2 H E E E H t t t t μ μ μσ με ∇×∇× = -∇× = - ∇× = - - r r r r r 2 2 E H H H E E E t t t t σ ε σ ε μσ εμ ∇×∇× = ∇× + ∇× = ∇× + ∇× = - - r r r r r r r ( 29 2 F F F ∇×∇× = ∇ ∇• -∇ r r r 2 2 2 2 x y z F x A y A z A + + ∇ r ˆ ˆ ˆ 0 H = r r g 0 E = r r g 2 2 2 E E E t t μσ με = + r r r 2 2 2 H H H t t μσ εμ = + r r r and When σ =0 (dielectrics): 2 2 2 0 E E t με - = r r and 2 2 2 0 H H t εμ - = r r Wave equations:
© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 5 Solutions of wave equations: use phasor form E and H are functions of both space and time: E (t,x,y,z), H (t,x,y,z) Assume harmonic functions to separate space and time We can always use Fourier series to satisfy any other time form of the initial conditions Any Fourier component of E and H field that oscillates at frequency ϖ must satisfy these equations! ( 29 ( 29 , , , , , j t s E x y z t E x y z e ϖ = r r ( 29 ( 29 , , , , , j t s H x y z t H x y z e ϖ = r r 2 2 s s s E j E E ϖμσ ϖ με = - r r r 2 2 s s s H j H H ϖμσ ϖ εμ = - r r Our wave equations become: and

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 6 Quick review: phasor notation and complex numbers complex numbers complex conjugate magnitude cos sin j e j θ θ θ = + ( 29 ( 29 arctan 2 2 cos sin y j j x Z x j y Z e Z j x y e θ θ θ ÷ = + × = = + = + ( 29 ( 29 2 2 Re Im Z Z Z = + ( 29 ( 29 Im arctan Re Z Z Z = ÷
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