# 25.5 Dispersion: the rainbow and prisms  (Page 2/4)

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Index of refraction n In selected media at various wavelengths
Medium Red (660 nm) Orange (610 nm) Yellow (580 nm) Green (550 nm) Blue (470 nm) Violet (410 nm)
Water 1.331 1.332 1.333 1.335 1.338 1.342
Diamond 2.410 2.415 2.417 2.426 2.444 2.458
Glass, crown 1.512 1.514 1.518 1.519 1.524 1.530
Glass, flint 1.662 1.665 1.667 1.674 1.684 1.698
Polystyrene 1.488 1.490 1.492 1.493 1.499 1.506
Quartz, fused 1.455 1.456 1.458 1.459 1.462 1.468

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 [link] . 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 [link] (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 [link] (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 [link] (c).)

## Rainbows

Rainbows are produced by a combination of refraction and reflection.

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.

## Phet explorations: geometric optics

How does a lens form an image? See how light rays are refracted by a lens. Watch how the image changes when you adjust the focal length of the lens, move the object, move the lens, or move the screen.

## Section summary

• The spreading of white light into its full spectrum of wavelengths is called dispersion.
• Rainbows are produced by a combination of refraction and reflection and involve the dispersion of sunlight into a continuous distribution of colors.
• Dispersion produces beautiful rainbows but also causes problems in certain optical systems.

## Problems&Exercises

(a) What is the ratio of the speed of red light to violet light in diamond, based on [link] ? (b) What is this ratio in polystyrene? (c) Which is more dispersive?

A beam of white light goes from air into water at an incident angle of $\text{75}\text{.}0º$ . At what angles are the red (660 nm) and violet (410 nm) parts of the light refracted?

$\text{46}\text{.}5º\text{, red; 46}\text{.}0º\text{, violet}$

By how much do the critical angles for red (660 nm) and violet (410 nm) light differ in a diamond surrounded by air?

(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 $\text{30}\text{.}0º$ incident angle. What is the angle between the colors when they emerge? (b) How far would they have to travel to be separated by 1.00 mm?

(a) $0\text{.}\text{043º}$

(b) $1\text{.}\text{33 m}$

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 $\text{60}\text{.}0º$ incident angle. What is the angle between the two colors in water?

A ray of 610 nm light goes from air into fused quartz at an incident angle of $\text{55}\text{.}0º$ . At what incident angle must 470 nm light enter flint glass to have the same angle of refraction?

$\text{71.3º}$

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 $\text{30}\text{.}0º$ incident angle. (a) At what angles do the two colors emerge? (b) By what distance are the red and blue separated when they emerge?

A narrow beam of white light enters a prism made of crown glass at a $\text{45}\text{.}0º$ incident angle, as shown in [link] . At what angles, ${\theta }_{\text{R}}$ and ${\theta }_{\text{V}}$ , do the red (660 nm) and violet (410 nm) components of the light emerge from the prism?

$\text{53.5º}\text{, red;}$ $\text{55.2º}\text{, violet}$

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