p36_029 - 29. We take the electric field of one wave, at...

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Unformatted text preview: 29. We take the electric field of one wave, at the screen, to be E1 = E0 sin(ωt) and the electric field of the other to be E2 = 2E0 sin(ωt + φ) , where the phase difference is given by φ= 2πd λ sin θ . .... . . ...... ...... ..... .. .. ... ... ... . ........ .. .... .. ... .. .... .. .... .. ... . .. .... . ... . .. ... . .. ... ... .. ... .. . ... . ... .. . .. . ... 0 ... .. ... .. . ... . ... .. . .. ... . ... .. ... .. ... . ... . .. .. . ... . ... . .. ... .. ... ... . .. .. ... . .... .. ... .. . . .. .. .. .. .. .... .. ..... .. .. . .. . .. . . .. . .. . . .. . .. . . . .. .. . . 0 . .. . .. . .. . . .. .. .. . . .. . . ... ... . . 2E E φ αE ωt Here d is the center-to-center slit separation and λ is the wavelength. The resultant wave can be written E = E1 + E2 = E sin(ωt + α), where α is a phase constant. The phasor diagram is shown above. The resultant amplitude E is given by the trigonometric law of cosines: 2 2 2 E 2 = E0 + (2E0 )2 − 4E0 cos(180◦ − φ) = E0 (5 + 4 cos φ) . The intensity is given by I = I0 (5 + 4 cos φ), where I0 is the intensity that would be produced by the first wave if the second were not present. Since cos φ = 2 cos2 (φ/2) − 1, this may also be written I = I0 1 + 8 cos2 (φ/2) . ...
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