p37_037 - 37. (a) Maxima of a diffraction grating pattern...

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Unformatted text preview: 37. (a) Maxima of a diffraction grating pattern occur at angles θ given by d sin θ = mλ, where d is the slit separation, λ is the wavelength, and m is an integer. The two lines are adjacent, so their order numbers differ by unity. Let m be the order number for the line with sin θ = 0.2 and m + 1 be the order number for the line with sin θ = 0.3. Then, 0.2d = mλ and 0.3d = (m + 1)λ. We subtract the first equation from the second to obtain 0.1d = λ, or d = λ/0.1 = (600 × 10−9 m)/0.1 = 6.0 × 10−6 m. (b) Minima of the single-slit diffraction pattern occur at angles θ given by a sin θ = mλ, where a is the slit width. Since the fourth-order interference maximum is missing, it must fall at one of these angles. If a is the smallest slit width for which this order is missing, the angle must be given by a sin θ = λ. It is also given by d sin θ = 4λ, so a = d/4 = (6.0 × 10−6 m)/4 = 1.5 × 10−6 m. (c) First, we set θ = 90◦ and find the largest value of m for which mλ < d sin θ. This is the highest order that is diffracted toward the screen. The condition is the same as m < d/λ and since d/λ = (6.0 × 10−6 m)/(600 × 10−9 m) = 10.0, the highest order seen is the m = 9 order. The fourth and eighth orders are missing, so the observable orders are m = 0, 1, 2, 3, 5, 6, 7, and 9. ...
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