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When coherent electromagnetic waves with wavelength \(\lambda=120 \mu \mathrm{m}\) are incident on a single slit of width \(a,\) the width of the central maximum on a tall screen \(1.50 \mathrm{~m}\) from the slit is \(90.0 \mathrm{~cm}\). For the same slit and screen, for what wavelength of the incident waves is the width of the central maximum \(180.0 \mathrm{~cm},\) double the value when \(\lambda=120 \mu \mathrm{m} ?\)

Short Answer

Expert verified
The wavelength of the incident waves for which the width of the central maximum is double the value when \(\lambda=120 \mu m\) is \(\lambda ' = 60 \mu m\)

Step by step solution

01

Understand the problem and Gather information

From the given exercise, we know that \(\lambda = 120 \mu m\), \(a\) is the width of the slit, \(d = 1.50 m\) and the width of the central maximum is \(d_{1} = 90.0 cm = 0.9 m\). Now, for another incident wave, we don't know the wavelength, which we'll denote as \(\lambda '\), but the central maximum width has doubles to \(d_{2} = 180.0 cm = 1.8 m\)
02

Using the condition for minima

In a single slit diffraction pattern, minima occurs where \(a \sin(\theta) = \lambda m\). Here, we are only interested in the first minima, so \(m = 1\) and \(\sin(\theta) = \lambda / a\). The diffraction angle \(\theta\) can be approximated as \(tan(\theta) = d /2L \) when \(\theta\) is small, which simplifies to \(\theta = d / 2L \). Therefore our equation simplifies to \(a \lambda = d L\)
03

Apply this formula to both wavelengths

So, for the first wavelength, we have \(a \lambda = d_{1} L\) and for the new wavelength, we get \(a \lambda' = d_{2} L\). Both these equations actually represent the same slit, therefore, they can be set equal: \(a \lambda = a \lambda'\)
04

Solve for the unknown

The width of the central maximum doubled, so \(d_{2} = 2d_{1}\), therefore, substituting this, we get \(\lambda = 2 \lambda '\) or \(\lambda ' = \lambda / 2\)

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Key Concepts

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

Single Slit Diffraction
Single slit diffraction is a phenomenon that happens when light waves encounter an obstacle or opening, such as a slit, that is comparable in size to the wavelength of the light. This interaction causes the waves to spread out or 'diffract' as they pass through the slit and emerge on the other side. The classic physics experiment involving single slit diffraction allows us to observe and analyze patterns on a screen placed behind the slit. These patterns are a direct consequence of the wave nature of light, and they can provide useful information about the light such as its wavelength.

Understanding how the size of the slit, or aperture, affects the diffraction pattern is essential. The smaller the slit in comparison to the light's wavelength, the greater the spreading of the waves. Conversely, if the slit is much larger than the wavelength, the diffraction effects become less pronounced and might not be easily observable. This is because diffraction is most significant when the waves encounter barriers that are of similar scale to their own wavelengths.

To describe diffraction mathematically, we use the concept of an angle of diffraction, which quantifies how much the light has spread out. The first minima of the pattern — points where the light intensity is at a minimum — occur at specific angles that can be predicted using the single slit diffraction formula.
Wavelength
Wavelength is the distance between two consecutive peaks or troughs in a wave and is denoted by the Greek letter lambda (lambda). It is an intrinsic property of waves that determines many of their characteristics. In the context of light, different wavelengths correspond to different colors when we talk about the visible spectrum, ranging from red to violet. However, light waves can also exist outside the visible spectrum, such as in ultraviolet or infrared wavelengths.

Wavelength is not only central to explaining the diffraction patterns but also to understanding various other phenomena in optics, such as interference and refraction. The wavelength of light is inversely proportional to its frequency, meaning that as the wavelength increases, the frequency decreases, and vice versa. This relationship is crucial when investigating the diffraction through a single slit, as the pattern's dimensions are directly related to the incident light's wavelength. In practical terms, longer wavelengths will produce broader diffraction patterns, while shorter wavelengths result in narrower patterns.
Central Maximum
The central maximum is the brightest and widest part of the diffraction pattern produced by a single slit and can be observed on a screen as a band of light at the center of the pattern, flanked by alternating dark and light bands or fringes. It is a direct result of constructive interference of light waves that have passed through the slit and travel straight ahead without being deflected.

When light undergoes diffraction, most of the light energy is concentrated in this central bright fringe, causing it to have higher intensity compared to the other maxima. The width of the central maximum is an important characteristic of the diffraction pattern and can provide insights into the properties of the light and the slit. The width is determined by the point where the first minimum occurs on either side of the central maximum, and it increases as the wavelength of the incident light increases or as the slit width decreases. In an educational context, measuring the central maximum's width is a common way to calculate the wavelength of light, given the slit width and distance to the screen.
Diffraction Pattern
A diffraction pattern is the series of dark and bright regions observed on a screen due to the bending of light around an obstacle or through an aperture. The interference of the light waves coming from different parts of the slit leads to these characteristic patterns. Constructive interference, where wave peaks align, results in bright fringes, while destructive interference, where peaks coincide with troughs, causes the dark regions.

In the case of a single slit, the pattern consists of a central maximum flanked by several dimmer maxima and intervening minima. The specific geometry and intensity distribution of the diffraction pattern serve as a 'fingerprint' for the incident light and the diffraction setup. Parameters such as slit width, light wavelength, and the distance between the slit and the screen all influence the appearance of the diffraction pattern. By measuring the various aspects of the diffraction pattern, such as the width of the central maximum or the spacing between the fringes, one can deduce important information about these parameters.

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Most popular questions from this chapter

On December 26, 2004, a violent earthquake of magnitude 9.1 occurred off the coast of Sumatra. This quake triggered a huge tsunami (similar to a tidal wave) that killed more than 150,000 people. Scientists observing the wave on the open ocean measured the time between crests to be \(1.0 \mathrm{~h}\) and the speed of the wave to be \(800 \mathrm{~km} / \mathrm{h}\). Computer models of the evolution of this enormous wave showed that it bent around the continents and spread to all the oceans of the earth. When the wave reached the gaps between continents, it diffracted between them as through a slit. (a) What was the wavelength of this tsunami? (b) The distance between the southern tip of Africa and northern Antarctica is about \(4500 \mathrm{~km},\) while the distance between the southern end of Australia and Antarctica is about \(3700 \mathrm{~km}\). As an approximation, we can model this wave's behavior by using Fraunhofer diffraction. Find the smallest angle away from the central maximum for which the waves would cancel after going through each of these continental gaps.

Coherent electromagnetic waves with wavelength \(\lambda\) pass through a narrow slit of width \(a\). The diffraction pattern is observed on a tall screen that is \(2.00 \mathrm{~m}\) from the slit. When \(\lambda=500 \mathrm{nm}\), the width on the screen of the central maximum in the diffraction pattern is \(8.00 \mathrm{~mm}\). For the same slit and screen, what is the width of the central maximum when \(\lambda=0.125 \mathrm{~mm} ?\)

Different isotopes of the same element emit light at slightly different wavelengths. A wavelength in the emission spectrum of a hydrogen atom is \(656.45 \mathrm{nm} ;\) for deuterium, the corresponding wavelength is \(656.27 \mathrm{nm}\). (a) What minimum number of slits is required to resolve these two wavelengths in second order? (b) If the grating has 500.00 slits/mm, find the angles and angular separation of these two wavelengths in the second order.

You are asked to design a space telescope for earth orbit. When Jupiter is \(5.93 \times 10^{8} \mathrm{~km}\) away (its closest approach to the earth), the telescope is to resolve, by Rayleigh's criterion, features on Jupiter that are \(250 \mathrm{~km}\) apart. What minimum-diameter mirror is required? Assume a wavelength of \(500 \mathrm{nm}\).

When laser light of wavelength \(632.8 \mathrm{nm}\) passes through a diffraction grating, the first bright spots occur at \(\pm 17.8^{\circ}\) from the central maximum. (a) What is the line density (in lines/cm) of this grating? (b) How many additional bright spots are there beyond the first bright spots, and at what angles do they occur?

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