Lect27_[Compatibility_Mode]

Lect27_[Compatibility_Mode] - Physics 344 Foundations of...

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Physics 344 Foundations of 21 st Century Physics: Relativity, Quantum Mechanics and heir Applications Their Applications Instructor: Dr. Mark Haugan Office: PHYS 282 haugan@purdue.edu TA: Dan Hartzler Office: PHYS 7 dhartzle@purdue.edu Grader: Shuo Liu Office: PHYS 283 liu305@purdue.edu Office Hours: If you have questions, just email us to make an ppointment. e enjoy talking about physics! appointment. We enjoy talking about physics! Notices: Midterm II at 8:00pm, Thursday, December 2 in MSEE B012
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Two-Source Interference Last lecture I referred you to the discussion of this phenomenon in chapter 24 of the 2 nd edition of Matter & Interactions. In section 24.6 you will find a very nice discussion of the relationship between two-source and two-slit intereference. We are interested in interference because it is quintessentially a wave phenomenon which we must account for in our blend of the wave and photon models of light. The following is a slightly different treatment than you find in M&I just to emphasize that we are superposing electric and magnetic fields.
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In this case, we arrange for the observer to be located on the x axis far from the two sources at x = X >> d . The sources have equal amplitudes and are in phase, but they are separated along a line that makes an angle θ relative to the x axis. in( 2 z θ 2 d sin( θ )/2 d 1 e’ve chosen to orient our coordinate system in this way so that the We ve chosen to orient our coordinate system in this way so that the representations of the fields emitted by the antennae will have the vector components we usually consider at the observer’s location. Our familiar real-valued representations of these fields are ( ) 12 ˆ cos( / 2) cos( / 2) E EE E k X t k X t j ωδ φ =+ = −+ + −− GG G ˆ s( / 2) cos( / 2) E B B k X t k X t + = ++ G p ( ) cos( BB c η + + where sin( ) kd δ φθ =
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Continuing, for the moment, to work with the real-valued field representations z 2 d sin( θ )/2 x d θ 1 We use the trig identity to re-express the fields as 2 cos( ) cos( ) cos( ) cos( ) ab a b a b =− ++ ˆˆ s( / 2) cos( / 2) 2 cos( ) cos( / 2) E kX t kX t j E kX t j G ( ) cos( E ωδ φ ω δ −− = () 2 cos( / 2) cos( / 2) cos( ) cos( / 2) EE B k Xt k k cc δφ η + + = G 2 22 14 ˆ cos ( )cos ( / 2) E SE B k X t i = G G G So, the energy flux density at the observers location is given by the Poynting vector, 00 c μμ As we noted last time, we generally measure the time average of this quantity when working with high-frequency radiation like visible light.
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Our prediction for the time-averaged energy flux density at the observers 22 2 00 14 2 ˆˆ cos ( / 2) cos ( ) cos ( / 2) TT EE SS i k X t d t i Tc T c δφ ω μμ ≡= −= ∫∫ GG location is, therefore, where we’ve used another trig identity 2 1 cos ( ) (1 cos(2( ))) 2 kX t kX t ωω + Recall that sin( ) kd δ φθ = 2 2 1 2 2 cos ( sin( )) E Sk d θ = G So, 0 c μ which predicts the interference patterns like this.
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This note was uploaded on 11/26/2010 for the course PHYS 344 taught by Professor Garfinkel during the Spring '08 term at Purdue University-West Lafayette.

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Lect27_[Compatibility_Mode] - Physics 344 Foundations of...

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