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Unformatted text preview: 24 Wave reflections, standing waves, radiation pressure In this lecture we will examine the phenomenon of planewave reflections at an interface separating two homogeneous regions where Maxwell’s equations allow for traveling TEM wave solutions. The solutions will also need to ˆ n · ( D + D ) = ρ s ˆ n · ( B + B ) = 0 ˆ n × ( E + E ) = 0 ˆ n × ( H + H ) = J s satisfy the boundary condition equations repeated in the margin. We will consider a propagation scenario in which (see margin): Region 1 Region 2 H i x y z E i E t H t H r E r 1. Region 1 where z < is occupied by a perfect dielectric with medium parameters μ 1 , 1 , and σ 1 = 0 , 2. Region 2 where z > is homogeneous with medium parameters μ 2 , 2 , and σ 2 , 3. Interface z = 0 contains no surface charge or current except possibly in σ 2 → ∞ limit which will be considered separately at the end. • In Region 1 we envision an incident planewave with linearpolarized field phasors ˜ E i = ˆ xE o e jβ 1 z and ˜ H i = ˆ y E o η 1 e jβ 1 z , where – E o is the wave amplitude due to far away source located in z → ∞ region, – η 1 = μ 1 1 and β 1 = ω √ μ 1 1 . 1 Fields above satisfy Maxwell’s equations in Region 1, but if there were no other fields in Regions 1 and 2 boundary condition equations requiring continuous tangential E and H at the z = 0 interface would be violated. In order to comply with the boundary condition equations we postulate a set of reflected and transmitted wave fields in Regions 1 and 2 as follows: Incident: ˜ E i = ˆ xE o e jβ 1 z , ˜ H i = ˆ y E o η 1 e jβ 1 z , Reflected: ˜ E r = ˆ x Γ E o e jβ 1 z , ˜ H r = ˆ y Γ E o η 1 e jβ 1 z , Transmitted: ˜ E t = ˆ xτE o e γ 2 z , ˜ H t = ˆ y τE o η 2 e γ 2 z . • In Region 1 we postulate a reflected planewave with linearpolarized field phasors ˜ E r = ˆ x Γ E o e jβ 1 z and ˜ H r = ˆ y Γ E o η 1 e jβ 1 z including an unknown Γ that we will refer to as reflection coefficient . – Note that the reflected wave propagates in z direction (direction of ˜ H r and the exponential terms have been adjusted accordingly). • In Region 2 we postulate a transmitted planewave with linearpolarized field phasors ˜ E t = ˆ xτE o e γ 2 z and ˜ H t = ˆ y τE o η 2 e γ 2 z including an unknown τ that we will refer to as reflection coefficient . – Note that the transmitted wave propagates in z direction, and – since Region 2 is conducting we have η 2 = jωμ 2 σ 2 + jω 2 and γ 2 = ( jωμ 2 )( σ 2 + jω 2 ) = α 2 + jβ 2 . 2 • To determine the unknowns Γ and τ we enforce the following boundary conditions at z = 0 where the fields simplify as shown in the margin: Incident at z = 0 : ˜ E i = ˆ xE o , ˜ H i = ˆ y E o η 1 , Reflected at z = 0 : ˜ E r = ˆ x Γ E o , ˜ H r = ˆ y Γ E o η 1 Transmitted at z = 0 : ˜ E t = ˆ xτE o , ˜ H t = ˆ y τE...
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 Summer '10
 KUDEKI
 Transmission line, Standing wave

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