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Chapter 25: Problem 49

Light from a red laser passes through a single slit to form a diffraction pattern on a distant screen. If the width of the slit is increased by a factor of two, what happens to the width of the central maximum on the screen?

Short Answer

Expert verified
Answer: The width of the central maximum becomes half of its original value when the width of the slit is increased by a factor of two.

Step by step solution

01

Understand the basics of single-slit diffraction patterns

Single-slit diffraction is an interference pattern formed when light passes through a single, narrow slit and spreads out in a series of bright and dark fringes on a distant screen. The central maximum is the brightest part of the pattern, and its width depends on the width of the slit and the wavelength of the light.
02

Use the formula for the angular width of the central maximum

The angular width (胃) of the central maximum in a single-slit diffraction pattern is given by the formula: 胃 鈮� \(\frac{2位}{a}\) where 位 is the wavelength of the light and a is the width of the slit.
03

Calculate the new angular width of the central maximum with the increased slit width

Let's denote the original slit width as a and the new slit width(after increasing by a factor of two) as a'. Then, a' = 2a Now, we will plug this new slit width into the formula for the angular width of the central maximum: 胃' 鈮� \(\frac{2位}{a'}\) 胃' 鈮� \(\frac{2位}{2a}\) 胃' 鈮� \(\frac{位}{a}\)
04

Compare the new angular width with the original angular width

Now, let's compare the new angular width (胃') with the original angular width (胃). We have, 胃 鈮� \(\frac{2位}{a}\) and 胃' 鈮� \(\frac{位}{a}\) Dividing 胃' by 胃, we get: \(\frac{胃'}{胃}\) = \(\frac{\frac{位}{a}}{\frac{2位}{a}}\) \(\frac{胃'}{胃}\) = \(\frac{1}{2}\) This means that the width of the central maximum on the screen will be half of what it originally was when the width of the slit is increased by a factor of two.

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Most popular questions from this chapter

Find the height \(h\) of the pits on a CD (Fig. \(25.6 \mathrm{a}\) ). When the laser beam reflects partly from a pit and partly from land (the flat aluminum surface) on either side of the "pit," the two reflected beams interfere destructively; \(h\) is chosen to be the smallest possible height that causes destructive interference. The wavelength of the laser is \(780 \mathrm{nm}\) and the index of refraction of the poly carbonate plastic is \(n=1.55.\)
A thin film of oil \((n=1.50)\) of thickness \(0.40 \mu \mathrm{m}\) is spread over a puddle of water \((n=1.33) .\) For which wavelength in the visible spectrum do you expect constructive interference for reflection at normal incidence?
White light containing wavelengths from \(400 \mathrm{nm}\) to \(700 \mathrm{nm}\) is shone through a grating. Assuming that at least part of the third-order spectrum is present, show that the second-and third-order spectra always overlap, regardless of the slit separation of the grating.
The diffraction pattern from a single slit is viewed on a screen. Using blue light, the width of the central maximum is \(2.0 \mathrm{cm} .\) (a) Would the central maximum be narrower or wider if red light is used instead? (b) If the blue light has wavelength \(0.43 \mu \mathrm{m}\) and the red light has wavelength \(0.70 \mu \mathrm{m},\) what is the width of the central maximum when red light is used?
A Michelson interferometer is set up using white light. The arms are adjusted so that a bright white spot appears on the screen (constructive interference for all wavelengths). A slab of glass \((n=1.46)\) is inserted into one of the arms. To return to the white spot, the mirror in the other arm is moved $6.73 \mathrm{cm} .$ (a) Is the mirror moved in or out? Explain. (b) What is the thickness of the slab of glass?
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