plane wave

# plane wave - Plane Wave Maxwells Equations r v E = r r B E...

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Plane Wave

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Maxwell’s Equations v c E ρ ε ∇⋅ = r 0 H ⋅= r B E t ∇× =− r r D HJ t = + r rr The vector phasor of is E r { } (,,,) R e (,,) j t E xyzt Exyze ω = r % The Maxwell’s equations become: v c E = % 0 H % E jH ωμ %% c HJj E ωε = + % (1a) (1b) (1c) (1d)
Maxwell’s Equations 0 E ∇⋅ = % 0 H ⋅= % (2a) (2c) E jH ωμ ∇× =− %% c HJj E ωε = + % (2b) (2d) In charge free medium: Vector Calculus: () ( ) Ej H ∇× ∇× (3) Substitute (2d) into (3): 2 22 ( ) ( ) cc c j E E EE E E ω μω ε μ ωμε = =∇ ∇⋅ −∇ = % % % 0 0 c += Homogeneous wave equation for E %

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Maxwell’s Equations Propagation constant 22 c γ ωμε =− 0 EE −= %% Similarly, 0 HH (4)
Uniform Plane Wave in Lossless Medium Lossless Medium: '" " cr o jj j σ ε εε ω =− = = To be lossless Æ 0 or " 0 o = = == Uniform Plane Wave If x-y is the plane, & EH rr do not vary with x and y (z is propagation direction) 0 0 xx zz EE xy HE ∂∂ = = %% (Transmission line waves)

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Uniform Plane Wave in Lossless Medium When waves are in lossless media, we use wave number k 22 2 or kk γ ωμε ω με =− = Or 0 Ek E −= %% For uniform plane waves: ˆˆ ˆ x yz E xE y Ez E =++ % % Substitute into equation (4): () 222 2 0 xy z z xE yE zE k xE yE zE xyz ⎛⎞ ∂∂∂ ++ + = ⎜⎟
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plane wave - Plane Wave Maxwells Equations r v E = r r B E...

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