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Sun light incident on a spherical water droplet gets refracted at various angles. The refracted rays further undergo total internal reflection and refract again when they leave the water droplet. As a result,  a sequence of colors ranging from violet to red is formed by the exiting light. The exiting light is on the same side of the drop as the incident sunlight.
A ray of light falling on this water drop enters and is reflected from the back of the drop. This light is refracted and dispersed both as it enters and as it leaves the drop.
In figure a, sunlight is incident on two water droplets close to one another. The incident rays undergo refraction and total internal reflection. Red light emerges from the upper drop, making an angle theta with the original direction of the ray of sunlight. Violet light emerges at a smaller angle.  Red and violet also emerge from the lower droplet at slightly different angles. A woman with her back to the sun and facing the droplets observes from a distance.  The red from the upper droplet and the violet from the lower droplet reach the observer’s eyes from different directions. The observer sees a band of color with violet at the bottom and red at the top. In figure b, a man looks at the rainbow, which is in the shape of an arc. Parallel rays from behind the man fall on the outside of the rainbow at different positions, reflect and refract and then reach the observer, each ray making the same angle theta with the incident ray. The rays reaching the observer are red. Figure c shows a photograph of a double rainbow in the sky.
(a) Different colors emerge in different directions, and so you must look at different locations to see the various colors of a rainbow. (b) The arc of a rainbow results from the fact that a line between the observer and any point on the arc must make the correct angle with the parallel rays of sunlight for the observer to receive the refracted rays. (c) Double rainbow. (credit c: modification of work by “Nicholas”/Wikimedia Commons)

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.

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.

Conceptual questions

Is it possible that total internal reflection plays a role in rainbows? Explain in terms of indices of refraction and angles, perhaps referring to that shown below. Some of us have seen the formation of a double rainbow; is it physically possible to observe a triple rainbow?

A photograph of a double rainbow.
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A high-quality diamond may be quite clear and colorless, transmitting all visible wavelengths with little absorption. Explain how it can sparkle with flashes of brilliant color when illuminated by white light.

In addition to total internal reflection, rays that refract into and out of diamond crystals are subject to dispersion due to varying values of n across the spectrum, resulting in a sparkling display of colors.

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Problems

(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?

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

46.5 ° for red, 46.0 ° for violet

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

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(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 30.0 ° 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. 0.04 ° ; b. 1.3 m

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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 60.0 ° incident angle. What is the angle between the two colors in water?

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

72.8 °

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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 30.0 ° incident angle. (a) At what angles do the two colors emerge? (b) By what distance are the red and blue separated when they emerge?

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A narrow beam of white light enters a prism made of crown glass at a 45.0 ° incident angle, as shown below. At what angles, θ R and θ V , do the red (660 nm) and violet (410 nm) components of the light emerge from the prism?

A blue incident light ray at an angle of incidence equal to 45 degrees to the normal falls on an equilateral triangular prism whose corners are all at angles equal to 60 degrees. At the first surface, the ray refracts and splits into red and violet rays. These rays hit the second surface and emerge from the prism. The red light with 660 nanometers bends less than the violet light with 410 nanometers.

53.5 ° for red, 55.2 ° for violet

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Practice Key Terms 1

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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