1026_PartUniversity Physics Solution

1026_PartUniversity Physics Solution - Diffraction 36-9...

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Diffraction 36-9 EVALUATE: Note that / 3.0 da = . Figure 36.27 36.28. IDENTIFY: The maxima are located by sin dm θλ = . SET UP: The order corresponds to the values of m . EXECUTE: First-order: 1 sin d = . Fourth-order: 4 sin 4 d = . 4 41 4 1 sin 4 , sin 4sin 4sin8.94 and 38.4 sin d d θθ θ == = ° = ° . EVALUATE: We did not have to solve for d . 36.29. IDENTIFY and SET UP: The bright bands are at angles given by sin . = Solve for d and then solve for for the specified order. EXECUTE: (a) 78.4 for 3 m = and 681 nm, λ = so 4 /sin 2.086 10 cm λθ × The number of slits per cm is 1/ 4790 slits/cm d = (b) 1st order: 1, m = so 96 sin / (681 10 m)/(2.086 10 m) d −− ==× × and 19.1 2nd order: 2, m = so sin 2 / d = and 40.8 (c) For 4, sin 4 / md is greater than 1.00, so there is no 4th-order bright band. EVALUATE: The angular position of the bright bands for a particular wavelength increases as the order increases. 36.30. IDENTIFY: The bright spots are located by sin = . SET UP: Third-order means 3 m = and second-order means 2 m = . EXECUTE: constant sin m d , so rr vv rv sin sin mm λλ = . vr 24 0 0 n m sin sin (sin65.0 ) 0.345 37 0 0 n m m m ⎛⎞ = ⎜⎟ ⎝⎠ ° and v 20.2 = ° . EVALUATE: The third-order line for a particular occurs at a larger angle than the second-order line. In a given order, the line for violet light (400 nm) occurs at a smaller angle than the line for red light (700 nm). 36.31. IDENTIFY and SET UP: Calculate d for the grating. Use Eq.(36.13) to calculate for the longest wavelength in the visible spectrum and verify that is small. Then use Eq.(36.3) to relate the linear separation of lines on the screen to the difference in wavelength. EXECUTE: (a) 5 1 cm 1.111 10 m 900 d × For 2 700 nm, / 6.3 10 . d × The first-order lines are located at sin / ; d = sin is small enough for sin to be an excellent approximation. (b) / , yxd = where 2.50 m. x = The distance on the screen between 1st order bright bands for two different wavelengths is () / , yx d Δ= Δ so 53 ( )/ (1.111 10 m)(3.00 10 m)/(2.50 m) 13.3 nm dyx = × × = EVALUATE: The smaller d is (greater number of lines per cm) the smaller the Δ that can be measured. 36.32. IDENTIFY: The maxima are located by sin = . SET UP: 6 51 1 350 slits mm 2.86 10 m 3.50 10 m d × ×
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36-10 Chapter 36 EXECUTE: 7 400 6 4.00 10 m 1: arcsin arcsin 8.05 2.86 10 m m d λ θ ⎛⎞ × == = = ⎜⎟ × ⎝⎠ D . 7 700 6 7.00 10 m arcsin arcsin 14.18 2.86 10 m d × = × D . 1 14.18 8.05 6.13 Δ= ° ° = ° . 7 400 6 33 ( 4 . 0 0 1 0 m ) 3: arcsin arcsin 24.8 2.86 10 m m d × = = ° × . 7 700 6 ( 7 . 0 0 1 0 m ) arcsin arcsin 47.3 2.86 10 m d × = ° × . 1 47.3 24.8 22.5 ° ° = ° . EVALUATE: Δ is larger in third order. 36.33. IDENTIFY: The maxima are located by sin dm θλ = . SET UP: 6 1.60 10 m d EXECUTE: 7 6 [6.328 10 m] arcsin arcsin arcsin([0.396] ) mm m d × = × . For 1 1, 23.3 m ° . For 2 m = , 2 52.3 = ° . There are no other maxima. EVALUATE: The reflective surface produces the same interference pattern as a grating with slit separation d . 36.34. IDENTIFY: The maxima are located by sin = .
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1026_PartUniversity Physics Solution - Diffraction 36-9...

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