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Chapter 27: Problem 63

In a single-slit diffraction pattern on a flat screen, the central bright fringe is \(1.2 \mathrm{cm}\) wide when the slit width is \(3.2 \times 10^{-5} \mathrm{m} .\) When the slit is replaced by a second slit, the wavelength of the light and the distance to the screen remaining unchanged, the central bright fringe broadens to a width of \(1.9 \mathrm{cm} .\) What is the width of the second slit? It may be assumed that \(\theta\) is so small that \(\sin \theta \approx \tan \theta\)

Short Answer

Expert verified
The width of the second slit is approximately \( 2.02 \times 10^{-5} \, \text{m} \).

Step by step solution

01

Understand the Formula

The formula for the width of the central bright fringe in a single slit diffraction pattern is given by: \[ W = \frac{2 \lambda D}{a} \]where \( W \) is the width of the central bright fringe, \( \lambda \) is the wavelength of light, \( D \) is the distance from the slit to the screen, and \( a \) is the slit width.
02

Calculate the Value of \( \lambda D \)

Using the initial condition with slit width \( a_1 = 3.2 \times 10^{-5} \, \text{m} \) and central fringe width \( W_1 = 1.2 \, \text{cm} = 0.012 \, \text{m} \), we can rearrange the formula as:\[ 0.012 = \frac{2 \lambda D}{3.2 \times 10^{-5}} \]This gives us:\[ \lambda D = \frac{0.012 \times 3.2 \times 10^{-5}}{2} = 1.92 \times 10^{-7} \, \text{m}^2 \]
03

Use New Conditions to Find New Slit Width

For the new slit width \( a_2 \), the width of the central fringe is \( W_2 = 1.9 \text{cm} = 0.019 \text{m} \). We use the previous calculation of \( \lambda D \):\[ 0.019 = \frac{2 \times 1.92 \times 10^{-7}}{a_2} \]Rearranging gives:\[ a_2 = \frac{2 \times 1.92 \times 10^{-7}}{0.019} \]
04

Calculate the Width of the Second Slit

Now, calculate \( a_2 \):\[ a_2 = \frac{3.84 \times 10^{-7}}{0.019} \approx 2.02 \times 10^{-5} \, \text{m} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Central Bright Fringe
The central bright fringe is a key feature in the single-slit diffraction pattern. It refers to the widest and brightest band that appears directly in line with the light source. To calculate its width, we use a vital formula:
  • The formula is: \[ W = \frac{2 \lambda D}{a} \] Here, \(W\) represents the width of the central bright fringe.
  • This formula relies on knowing the wavelength of the light (\(\lambda\)), distance to the screen (\(D\)), and the slit width (\(a\)).
This central bright fringe is larger than the others because it results from overlapping wavefronts in the diffraction pattern. Recognizing its dimensions is crucial in diverse applications, from optical experiments to understanding the properties of light itself.
Slit Width
The slit width, denoted by \(a\), is a fundamental parameter in determining the characteristics of a diffraction pattern. It simply refers to the physical opening of the slit through which light passes. This value directly influences how light spreads out and the consequent appearance of the diffraction pattern.
  • A smaller slit width increases the spread of the diffraction pattern, resulting in a wider central bright fringe.
  • Conversely, a larger slit width constrains the spread, making the central fringe narrower.
In exercises involving single-slit diffraction, accurately measuring and understanding the slit width help in predicting how light will behave as it encounters the slit. Specifically, adjusting the slit width impacts the visibility and clarity of the diffraction fringes.
Diffraction Pattern
A diffraction pattern is the combination of dark and light bands that form on a screen when light passes through a slit. This occurs because light behaves as a wave, causing it to spread out when it encounters an obstacle or an opening. When light passes through a single slit, the most notable feature is the pattern of alternating dark and bright fringes, with the central bright fringe being the largest and most visible.
  • The diffraction pattern is predictable, governed by factors like the slit width and the wavelength of light.
  • With this understanding, scientists and students predict how light will interact with different materials and openings, showcasing the wave nature of light.
The ability to explain and anticipate what occurs in a diffraction pattern is fundamental to studies in physics and essential for developing precise optical instruments.
Wavelength of Light
The wavelength of light, symbolized by \(\lambda\), plays a critical role in shaping the diffraction pattern. Wavelength is the distance between consecutive peaks of a wave, and it determines many characteristics of how light interacts with different environments.
  • In the context of single-slit diffraction, longer wavelengths result in broader diffraction patterns, including wider central bright fringes.
  • Shorter wavelengths produce narrower diffraction patterns. Thus, calculating the wavelength is essential in projecting how patterns will develop when light interacts with a slit.
In the given exercise, though the wavelength remains constant, its relationship with the slit width and distance to the screen determines the size of the central fringe. This understanding is not only fundamental to mastering single-slit diffraction, but also imperative in various scientific disciplines and technologies.

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Most popular questions from this chapter

Light shines through a single slit whose width is \(5.6 \times 10^{-4} \mathrm{m}\). A diffraction pattern is formed on a flat screen located \(4.0 \mathrm{m}\) away. The distance between the middle of the central bright fringe and the first dark fringe is \(3.5 \mathrm{mm} .\) What is the wavelength of the light?

Two gratings A and B have slit separations \(d_{\lambda}\) and \(d_{\mathrm{B}},\) respectively. They are used with the same light and the same observation screen. When grating A is replaced with grating \(\mathrm{B},\) it is observed that the first-order maximum of \(A\) is exactly replaced by the second-order maximum of B. (a) Determine the ratio \(d_{\mathrm{B}} / d_{\mathrm{A}}\) of the spacings between the slits of the gratings. (b) Find the next two principal maxima of grating A and the principal maxima of B that exactly replace them when the gratings are switched. Identify these maxima by their order numbers.

A slit has a width of \(W_{1}=2.3 \times 10^{-6} \mathrm{m} .\) When light with a wavelength of \(\lambda_{1}=510 \mathrm{nm}\) passes through this slit, the width of the central bright fringe on a flat observation screen has a certain value. With the screen kept in the same place, this slit is replaced with a second slit (width \(W_{2}\) ), and a wavelength of \(\lambda_{2}=740 \mathrm{nm}\) is used. The width of the central bright fringe on the screen is observed to be unchanged. Find \(W_{2}\)

A diffraction grating is \(1.50 \mathrm{cm}\) wide and contains 2400 lines. When used with light of a certain wavelength, a third-order maximum is formed at an angle of \(18.0^{\circ} .\) What is the wavelength (in \(\mathrm{nm}\) )?

The central bright fringe in a single-slit diffraction pattern has a width that equals the distance between the screen and the slit. Find the ratio \(\lambda / W\) of the wavelength \(\lambda\) of the light to the width \(W\) of the slit.
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