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Chapter 4: Problem 34

Two slits of width \(2 \mu \mathrm{m},\) each in an opaque material, are separated by a center-to-center distance of \(6 \mu \mathrm{m}\). A monochromatic light of wavelength \(450 \mathrm{nm}\) is incident on the double- slit. One finds a combined interference and diffraction pattern on the screen. (a) How many peaks of the interference will be observed in the central maximum of the diffraction pattem? (b) How many peaks of the interference will be observed if the slit width is doubled while keeping the distance between the slits same? (c) How many peaks of interference will be observed if the slits are separated by twice the distance, that is, \(12 \mu \mathrm{m}\), while keeping the widths of the slits same? (d) What will happen in (a) if instead of 450-nm light another light of wavelength \(680 \mathrm{nm}\) is used? (e) What is the value of the ratio of the intensity of the central peak to the intensity of the next bright peak in (a)? (f) Does this ratio depend on the wavelength of the light? (g) Does this ratio depend on the width or separation of the slits?

Short Answer

Expert verified
= \(\dfrac{\sin^{-1}(\frac{450 \times 10^{-9} m}{2 \times 2 \times 10^{-6} m})}{\sin^{-1}(\frac{450 \times 10^{-9} m}{6 \times 10^{-6} m})}\) After calculating the values, we find that there are approximately 4 peaks in the central maximum of the diffraction pattern.

Step by step solution

01

a) How many peaks of the interference will be observed in the central maximum of the diffraction pattern?

First, we need to find the angular position of the first minimum of the diffraction pattern, which occurs when the path length difference between the waves coming from the two slits is equal to half the wavelength: \(\sin \theta = \dfrac{\lambda}{2w}\) Next, we need to find the angular positions of the consecutive maxima in the interference pattern, which occur when the path length difference is an integer multiple of the wavelength: \(\sin \theta_n = \dfrac{n \lambda}{d}\) Now, we need to count how many interference peaks are observed within the central maximum of the diffraction pattern, which corresponds to the angular range between \(0\) and \(\frac{\lambda}{2w}\): Number of peaks = \(\dfrac{\sin^{-1}(\frac{\lambda}{2w})}{\sin^{-1}(\frac{\lambda}{d})}\)

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