Simple Nature - Light and Matter

of memorization. The positive sign is used when both surfaces arecurved outward or both are curved inward; otherwise a negativesign applies. The proof of this equation is left as an exercise tothose readers who are sufficiently brave and motivated.12.4.4 DispersionFor most materials, we observe that the index of refraction dependsslightly on wavelength, being highest at the blue end of thevisible spectrum and lowest at the red. For example, white lightdisperses into a rainbow when it passes through a prism, q. Evenwhen the waves involved aren’t light waves, and even when refractionisn’t of interest, the dependence of wave speed on wavelengthis referred to as dispersion. Dispersion inside spherical raindrops isresponsible for the creation of rainbows in the sky, and in an opticalinstrument such as the eye or a camera it is responsible for a type ofaberration called chromatic aberration (subsection 12.3.3 and problem28). As we’ll see in subsection 13.3.2, dispersion causes a wavethat is not a pure sine wave to have its shape distorted as it travels,and also causes the speed at which energy and information aretransported by the wave to be different from what one might expectfrom a naive calculation. The microscopic reasons for dispersion oflight in matter are discussed in optional subsection 12.4.6.q / Dispersion of white lightby a prism. White light is amixture of all the wavelengths ofthe visible spectrum. Waves ofdifferent wavelengths undergodifferent amounts of refraction.12.4.5 ⋆ The principle of least time for refractionWe have seen previously how the rules governing straight-linemotion of light and reflection of light can be derived from the principleof least time. What about refraction? In the figure, it is indeedplausible that the bending of the ray serves to minimize the timerequired to get from a point A to point B. If the ray followed the unbentpath shown with a dashed line, it would have to travel a longerdistance in the medium in which its speed is slower. By bendingthe correct amount, it can reduce the distance it has to cover in theslower medium without going too far out of its way. It is true thatSnell’s law gives exactly the set of angles that minimizes the timerequired for light to get from one point to another. The proof ofthis fact is left as an exercise (problem 38, p. 802).r / The principle of least timeapplied to refraction.Section 12.4 Refraction 779