4: (a) A narrow beam of light containing yellow (580 nm) and green (550 nm) wavelengths goes from polystyrene to air, striking the surface at a 30.0o incident angle. What is the angle between the colours when they emerge? (b) How far would they have to travel to be separated by 1.00 mm?

Any type of wave can exhibit dispersion. Sound waves, all types of electromagnetic waves, and water waves can be dispersed according to wavelength. Dispersion occurs whenever the speed of propagation depends on wavelength, thus separating and spreading out various wavelengths. Dispersion may require special circumstances and can result in spectacular displays such as in the production of a rainbow. This is also true for sound, since all frequencies ordinarily travel at the same speed. If you listen to sound through a long tube, such as a vacuum cleaner hose, you can easily hear it is dispersed by interaction with the tube. Dispersion, in fact, can reveal a great deal about what the wave has encountered that disperses its wavelengths. The dispersion of electromagnetic radiation from outer space, for example, has revealed much about what exists between the stars—the so-called empty space.

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2: A beam of white light goes from air into water at an incident angle of 75.0 degrees. At what angles are the red (660 nm) and violet (410 nm) parts of the light refracted?

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Have you ever found yourself in the middle of a DIY project, desperately searching for the right tool to tighten that pesky screw? We've all been there. And more often than not, that elusive tool is an Allen key. But what exactly is an Allen key, and what is the most common size you're likely to encounter? Let's dive into the world of Allen keys and unlock the mystery once and for all!

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8: A narrow beam of white light enters a prism made of crown glass at a 45.0o incident angle, as shown below in Figure 7. At what angles, θRed and θViolet do the red (660 nm) and violet (410 nm) components of the light emerge from the prism?

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We see about six colors in a rainbow—red, orange, yellow, green, blue, and violet; sometimes indigo is listed, too. Those colors are associated with different wavelengths of light, as shown in Figure 2. When our eye receives pure-wavelength light, we tend to see only one of the six colors, depending on wavelength. The thousands of other hues we can sense in other situations are our eye’s response to various mixtures of wavelengths. White light, in particular, is a fairly uniform mixture of all visible wavelengths. Sunlight, considered to be white, actually appears to be a bit yellow because of its mixture of wavelengths, but it does contain all visible wavelengths. The sequence of colors in rainbows is the same sequence as the colors plotted versus wavelength in Figure 2. What this implies is that white light is spread out according to wavelength in a rainbow. Dispersion is defined as the spreading of white light into its full spectrum of wavelengths. More technically, dispersion occurs whenever there is a process that changes the direction of light in a manner that depends on wavelength. Dispersion, as a general phenomenon, can occur for any type of wave and always involves wavelength-dependent processes.

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But why is the 5/32 inch Allen key so popular, you ask? Well, it's all about versatility. This size fits a wide range of screws and bolts, making it a go-to choice for many manufacturers. It's like the Swiss Army knife of Allen keys, ready to tackle any hexagonal challenge that comes its way.

Everyone enjoys the spectacle of a rainbow glimmering against a dark stormy sky. How does sunlight falling on clear drops of rain get broken into the rainbow of colors we see? The same process causes white light to be broken into colors by a clear glass prism or a diamond. (See Figure 1.)

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Refraction is responsible for dispersion in rainbows and many other situations. The angle of refraction depends on the index of refraction, as we saw in Chapter 25.3 The Law of Refraction. We know that the index of refraction n depends on the medium. But for a given medium, n also depends on wavelength. (See Table 2. Note that, for a given medium, n increases as wavelength decreases and is greatest for violet light. Thus violet light is bent more than red light, as shown for a prism in Figure 3(b), and the light is dispersed into the same sequence of wavelengths as seen in Figure 1 and Figure 2.

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7: A narrow beam of light containing red (660 nm) and blue (470 nm) wavelengths travels from air through a 1.00 cm thick flat piece of crown glass and back to air again. The beam strikes at a 30.0o incident angle. (a) At what angles do the two colours emerge? (b) By what distance are the red and blue separated when they emerge?

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1: (a) What is the ratio of the speed of red light to violet light in diamond, based on Table 2? (b) What is this ratio in polystyrene? (c) Which is more dispersive?

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Dispersion may produce beautiful rainbows, but it can cause problems in optical systems. White light used to transmit messages in a fiber is dispersed, spreading out in time and eventually overlapping with other messages. Since a laser produces a nearly pure wavelength, its light experiences little dispersion, an advantage over white light for transmission of information. In contrast, dispersion of electromagnetic waves coming to us from outer space can be used to determine the amount of matter they pass through. As with many phenomena, dispersion can be useful or a nuisance, depending on the situation and our human goals.

Rainbows are produced by a combination of refraction and reflection. You may have noticed that you see a rainbow only when you look away from the sun. Light enters a drop of water and is reflected from the back of the drop, as shown in Figure 4. The light is refracted both as it enters and as it leaves the drop. Since the index of refraction of water varies with wavelength, the light is dispersed, and a rainbow is observed, as shown in Figure 5 (a). (There is no dispersion caused by reflection at the back surface, since the law of reflection does not depend on wavelength.) The actual rainbow of colors seen by an observer depends on the myriad of rays being refracted and reflected toward the observer’s eyes from numerous drops of water. The effect is most spectacular when the background is dark, as in stormy weather, but can also be observed in waterfalls and lawn sprinklers. The arc of a rainbow comes from the need to be looking at a specific angle relative to the direction of the sun, as illustrated in Figure 5 (b). (If there are two reflections of light within the water drop, another “secondary” rainbow is produced. This rare event produces an arc that lies above the primary rainbow arc—see Figure 5 (c).)

5: A parallel beam of light containing orange (610 nm) and violet (410 nm) wavelengths goes from fused quartz to water, striking the surface between them at a 60.0o incident angle. What is the angle between the two colours in water?

So, there you have it! The most common Allen key size is the 5/32 inch (or 4mm) Allen key. Armed with this knowledge, you'll be well-equipped to tackle any hexagonal fastening challenge that comes your way. Just remember, when in doubt, reach for your trusty Allen key set and let the hexagonal superhero save the day!

While the 5/32 inch Allen key may be the most common size, it's not the only player in the game. Allen keys come in a variety of sizes, ranging from tiny 0.7mm versions to hefty 19mm monsters. So, depending on your project, you might encounter different sizes along the way.

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Allenkeysizeschart

Before we get into the nitty-gritty of sizes, let's start with the basics. An Allen key, also known as a hex key or hex wrench, is a simple yet versatile tool used to drive screws and bolts with hexagonal sockets. It's like a superhero with a hexagonal cape, ready to save the day when you need to tighten or loosen those hexagonal fasteners.

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Now, let's get to the heart of the matter: the size of the most common Allen key. Drumroll, please... It's the 5/32 inch (or 4mm for our metric friends) Allen key! This little guy is the superstar of the Allen key world, the MVP of hexagonal fastening. It's the size you're most likely to encounter in your everyday DIY adventures.

6: A ray of 610 nm light goes from air into fused quartz at an incident angle of 55.0o. At what incident angle must 470 nm light enter flint glass to have the same angle of refraction?

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But fear not! Most Allen key sets come with a range of sizes, ensuring you'll have the right tool for the job. It's like having a whole squad of hexagonal superheroes at your disposal, ready to save the day whenever a hexagonal fastener needs tightening or loosening.

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