P37_045 - 1 4 cos φ 4 cos 2 φ E 1 E 2 E 3 E φ φ φ φ ωt

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45. Since the slit width is much less than the wavelength of the light, the central peak of the single-slit diFraction pattern is spread across the screen and the diFraction envelope can be ignored. Consider three waves, one from each slit. Since the slits are evenly spaced, the phase diFerence for waves from the ±rst and second slits is the same as the phase diFerence for waves from the second and third slits. The electric ±elds of the waves at the screen can be written E 1 = E 0 sin( ωt ), E 2 = E 0 sin( ωt + φ ), and E 3 = E 0 sin( ωt +2 φ ), where φ =(2 πd/λ ) sin θ .H e r e d is the separation of adjacent slits and λ is the wavelength. The phasor diagram is shown below. It yields E = E 0 cos φ + E 0 + E 0 cos φ = E 0 (1 + 2 cos φ ) for the amplitude of the resultant wave. Since the intensity of a wave is proportional to the square of the electric ±eld, we may write I = AE 2 0 (1 + 2 cos φ ) 2 ,where A is a constant of proportionality. If I m is the intensity at the center of the pattern, for which φ =0 ,then I m =9 AE 2 0 .W etak e A to be I m / 9 E 2 0 and obtain I = I m 9 (1 + 2 cos φ )
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Unformatted text preview: 1 + 4 cos φ + 4 cos 2 φ ) . ..... ..... ..... ..... ..... ..... ..... . . . . . . . . . . . . . . ..... ..... ..... ..... ..... ..... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E 1 E 2 E 3 E φ φ φ φ ωt...
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This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.

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