Lecture 26 - 26 Standing waves radiation pressure We...

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26 Standing waves, radiation pressure We continue in this lecture with our studies of wave reflection and transmis- sion at a plane boundary between two homogeneous media. In case of total reflection from a perfectly conducting mirror placed at z = 0 surface, the incident and reflected waves found in z < 0 combine to produce standing waves of electric and magnetic field: Region 1 Region 2 H i x y z E i E t = 0 H t = 0 H r E r – Incident wave (a traveling wave going in z -direction): ˜ E i = ˆ xE o e - j β 1 z and ˜ H i = ˆ y E o η 1 e - j β 1 z , – Reflected wave (a traveling wave going in - z -direction): ˜ E r = - ˆ xE o e j β 1 z and ˜ H i = ˆ y E o η 1 e j β 1 z , – Standing wave : ˜ E = ˜ E i + ˜ E r = ˆ xE o ( e - j β 1 z - e j β 1 z ) and ˜ H = ˜ H i + ˜ H r = ˆ y E o η 1 ( e - j β 1 z + e j β 1 z ) which simplify as Standing waves ˜ E = - j ˆ x 2 E o sin( β 1 z ) and ˜ H = ˆ y 2 E o η 1 cos( β 1 z ) . 1
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These are called standing wave phasors because when we go to the time-domain (by multiplying with e j ω t and taking the real time as usual) we obtain: E ( z, t ) = ˆ x 2 E o sin( β 1 z ) sin( ω t ) and H ( z, t ) = ˆ y 2 E o η 1 cos( β 1 z ) cos( ω t ); these, unlike d’Alembert solutions of the format f ( t z v ) , describe os- cillations in time t , with di ff erent amplitudes at di ff erent positions z (see margin and the animation linked in class calendar). λ 2 z E x ( z, t ) sin( β z ) sin( ω t ) z H y ( z, t ) cos( β z ) cos( ω t ) λ = 2 π β Note: Nulls in Ex and Hy are separated by half wavelength. Adjacent nulls of Ex and Hy are separated by quarter wavelength. It is useful to think of nulls of Ex as "shorts" in analogy to shorts in circuits where v=0. Conductor shorts Ex on its surface where a current flows.
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