Answer to Question #87950 in Optics for Nicholas Legault

Question #87950
Red light with a wavelength of 725 nm shines through a single slit, and forms a pattern on a screen placed 1.25 m away. If the width of the slit is 18.5 micrometers, find the width of the central maximum (a) in degrees and (b) in centimeters
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Expert's answer
2019-04-18T10:51:04-0400

The angular position θm\theta_m of the first minimum is determined by the equation sinθm=λ/d\sin \theta_m = \lambda / d, where λ=725×109m\lambda = 725 \times 10^{-9}\, \text{m} is the wavelength, and d=18.5×106md = 18.5 \times 10^{-6}\, \text{m} is the width of the slit. We obtain sinθm=0.725/18.50.0392\sin \theta_m = 0.725 / 18.5 \approx 0.0392. In view of the smallness of this number, we have sinθmθm0.0392\sin \theta_m \approx \theta_m \approx 0.0392. The angular width of the central maximum is twice this angle: θ=2θm0.0784\theta = 2 \theta_m \approx 0.0784. The angular width in degrees is then 360θ/2π4.5360\, \theta / 2 \pi \approx 4.5^\circ. The width in centimetres is proportional to the distance to the screen R=125cmR = 125\, \text{cm} and is given approximately by θR=0.0784×125cm=9.8cm\theta R = 0.0784 \times 125\, \text{cm} = 9.8\, \text{cm}.


Answer: (a) 4.54.5^\circ, (b) 9.8cm9.8\, \text{cm}.


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