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Chapter 27: Problem 31

How many dark fringes will be produced on either side of the central maximum if light \((\lambda=651 \mathrm{nm})\) is incident on a single slit that is \(5.47 \times 10^{-6} \mathrm{m}\) wide?

Short Answer

Expert verified
There are 8 dark fringes on each side of the central maximum.

Step by step solution

01

Understanding the Concept

When light passes through a single slit, it creates a diffraction pattern composed of bright and dark fringes. The formula for determining the angle of the dark fringes in single-slit diffraction is given by \( a \sin \theta = m \lambda \), where \( a \) is the slit width, \( \lambda \) is the wavelength, and \( m \) is the fringe order (an integer). We are tasked with finding how many dark fringes exist on either side of the central maximum.
02

Setting Up the Inequality

For each dark fringe, we have the condition \( a \sin \theta = m \lambda \). The sine of any angle cannot exceed 1, so we know \( \sin \theta \leq 1 \). Therefore, the highest order of the dark fringe \( m \) must satisfy \( a \leq m \lambda \). Solving for \( m \), this becomes \( m \leq \frac{a}{\lambda} \).
03

Calculating Maximum Order of Fringe

Substitute the known values \( a = 5.47 \times 10^{-6} \) m and \( \lambda = 651 \times 10^{-9} \) m into the inequality \( m \leq \frac{a}{\lambda} \). Calculate the maximum integer \( m \):\[ m \leq \frac{5.47 \times 10^{-6}}{651 \times 10^{-9}} \approx 8.4 \]
04

Interpreting the Result

Since \( m \) must be an integer, the largest integer value is \( m = 8 \). Thus, there are 8 dark fringes on each side of the central maximum.

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Key Concepts

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

Central Maximum
The central maximum is the brightest part of a single-slit diffraction pattern. It is located directly in the center, where the amplitude of light waves overlaps most constructively. This happens because all rays from the slit arrive in phase at the central point, producing the greatest intensity.
The central maximum is significant because, unlike other maxima, this is not affected by dark fringes on either side disrupting the intensity. It is the broadest and most prominent feature of the diffraction pattern and stands out due to its high brightness and width, which is double that of any of the successive bright fringes.
  • The central maximum is pivotal in determining the characteristics of the entire diffraction pattern.
  • It's crucial for calculating fringe order and understanding the limits of dark fringe formation.
Diffraction Pattern
A diffraction pattern is the result of light bending around obstacles or passing through narrow openings, such as a single slit. The pattern is typically composed of a central bright fringe, surrounded by alternating dark and bright fringes on either side. This pattern occurs due to the interference of light waves as they pass through the slit and spread out.
The details of a diffraction pattern can tell us much about the properties of the light and the slit, including:
  • The slit width: Wider slits produce less spread out patterns.
  • The wavelength of light: Longer wavelengths result in more dispersed patterns.
The primary focus is often on where and how the dark fringes and maxima occur within this pattern. Understanding the interference of light waves is key to analyzing how the pattern is formed.
Dark Fringes
Dark fringes appear in the diffraction pattern as regions where the light intensity falls to zero. These occur due to destructive interference, where the peaks of light from one part of the slit cancel out the troughs from another.
The formula used to determine the position of dark fringes is the same as the one for finding the angle of dark fringes: \[ a \sin \theta = m \lambda \] where:
  • \( a \) is the slit width.
  • \( \lambda \) is the wavelength.
  • \( m \) is the fringe order.
Importantly, the concept of the sine of an angle not exceeding 1 is used to determine the maximum order. Dark fringes on either side of the central maximum help define the overall scope and boundary of the diffraction pattern.
Fringe Order
Fringe order, denoted as \( m \), is a number that indicates the position of a dark or bright fringe in relation to the central maximum. It is an integer, which means it only takes whole number values such as 0, 1, 2, and so on.
Each value of \( m \) corresponds to a specific dark fringe according to the equation:\[ a \sin \theta = m \lambda \] Here:
  • \( m = 0 \) refers to the central maximum itself.
  • \( m = 1 \) is the first dark fringe.
  • The pattern follows positively and negatively from the center.
To find how many dark fringes occur, you solve the inequality to find the largest integer \( m \), which tells how many can fit within the given physical limitations, like slit width and light wavelength. This explains the total number of fringes visible on each side of the central maximum.

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Most popular questions from this chapter

In Young's experiment a mixture of orange light \((611 \mathrm{nm})\) and blue light \((471 \mathrm{nm})\) shines on the double slit. The centers of the first- order bright blue fringes lie at the outer edges of a screen that is located \(0.500 \mathrm{m}\) away from the slits. However, the first-order bright orange fringes fall off the screen. By how much and in which direction (toward or away from the slits) should the screen be moved so that the centers of the first-order bright orange fringes will just appear on the screen? It may be assumed that \(\theta\) is small, so that \(\sin \theta \approx \tan \theta\).

A uniform layer of water \((n=1.33)\) lies on a glass plate \((n=1.52) .\) Light shines perpendicularly on the layer. Because of constructive interference, the layer looks maximally bright when the wavelength of the light is \(432 \mathrm{nm}\) in vacuum and \(a l s o\) when it is \(648 \mathrm{nm}\) in vacuum. (a) Obtain the minimum thickness of the film. (b) Assuming that the film has the minimum thickness and that the visible spectrum extends from 380 to \(750 \mathrm{nm},\) determine the visible wavelength(s) in vacuum for which the film appears completely dark.

A spotlight sends red light (wavelength \(=694.3 \mathrm{nm}\) ) to the moon. At the surface of the moon, which is \(3.77 \times 10^{\circ} \mathrm{m}\) away, the light strikes a reflector left there by astronauts. The reflected light returns to the earth, where it is detected. When it leaves the spotlight, the circular beam of light has a diameter of about \(0.20 \mathrm{m},\) and diffraction causes the beam to spread as the light travels to the moon. In effect, the first circular dark fringe in the diffraction pattern defines the size of the central bright spot on the moon. Determine the diameter (not the radius) of the central bright spot on the moon.

A flat observation screen is placed at a distance of \(4.5 \mathrm{m}\) from a pair of slits. The separation on the screen between the central bright fringe and the first-order bright fringe is \(0.037 \mathrm{m} .\) The light illuminating the slits has a wavelength of \(490 \mathrm{nm} .\) Determine the slit separation.

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?
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