Section 26–4 Applications of Fermat’s principle

(Least-time path / Same time paths / Relative refractive index)

 

In this section, Feynman applies Fermat’s principle from the perspective of least-time path and same-time paths, and apply the principle to deduce the relationship between relative refractive index and absolute refractive index.

 

1. Least-time path:

“… instead of the light deciding to come in the straightforward way, it also has a least-time path by which it goes into the region where it goes faster for awhile, in order to save time (Feynman et al., 1963, section 26–4 Applications of Fermat’s principle).”

 

Feynman mentions that when the sun appears to go below the horizon, it is actually already well below the horizon. He offers a one-line explanation: light travels more slowly in air than in a vacuum, and so the sunlight can reach point S beyond the horizon more quickly if it avoids the dense regions where it goes slowly by getting through them at a steeper tilt. One may add that a greater refraction occurs for light rays from the lower parts of the sun that travels through the denser air in the lower atmosphere. Thus, the sun appears to be flattened vertically. Perhaps Feynman should explain that the position of the sun can be deduced from the moon’s eclipse. The fact that we can see the sun and partial eclipse of the moon (well above the horizon) at the same time implies that the sun should be well below the horizon.

 

According to Feynman, the mirage we see on the road is a reflection of light from the sky. Essentially, light goes faster in the hotter air near the ground than the cooler air that is higher because thinner air has a lower refractive index. Because the temperature change in the air and the change in its refractive index are gradual, the rays later slowly curve upward (to achieve the least-time path). Furthermore, we can see some objects in the original positions, with an inverted image below it, as if there is a horizontal mirror. Another point is: the hot road due to the sunlight also warms the air and creates turbulence, thereby resulting in a blurry mirage (or shimmering effect).

 

2. Same-time paths:

“Therefore the problem of making a focusing system is merely to arrange a device so that it takes the same time for the light to go on all the different paths! (Feynman et al., 1963, section 26–4 Applications of Fermat’s principle).”

 

Feynman says that the sum of the two distances r₁ and r₂ traveled by the light (in the air) within the ellipsoidal mirror must be a constant. Then, he defines an ellipse as a  curve which has the property that the sum of the two distances from the two foci is a constant for every point on the curve. In a lost lecture, Feynman defines the ellipse as “the curve that can be made, by taking one string and two tacks and putting a pencil here and going around (Feynman et al., 1997, p. 149).” He also explains the reflection property of the ellipse and says that “… light is reflected as though the surface here were a plane tangent to the actual curve (Feynman et al., 1997, p. 150).” However, the paths traveled by the light rays within the ellipsoidal mirror may be considered as a relative maximum (Hecht, 2002, p. 110).

 

Feynman explains that the mirror is made in the shape of a parabola for gathering the light of a star. In addition, he defines a parabola as a curve which has the property that the sum of the distances XX′+X′P′ is a constant, no matter where X is chosen. One may elaborate that a parabola is a limiting case of an ellipse and thus, it has the same reflection property as the ellipse. On the other hand, the usefulness of the parabolic mirror to focus parallel lights is related to the focal property of the parabola. The focal property can be stated as “The tangent and normal at any point P of a parabola bisect the angles formed by the line joining P with the focus and the diameter through P (Korn & Korn, 2013, p. 53)” and can be proved without words (Ayoub, 1991).

 

3. Relative refractive index:

“So if we measure the speed of light in all materials, and from this get a single number for each material, namely its index relative to vacuum, called n_i…, then our formula is easy... (Feynman et al., 1963, section 26–4 Applications of Fermat’s principle).”

 

Feynman says that we can predict the index for a new pair of materials from the indexes of the individual materials that are both against air or against vacuum. It could be clearer to some students if he uses the two terms absolute refractive index and relative refractive index. Firstly, the absolute refractive index of a medium n₁ is a measured value that determines the angle of refraction when a ray of light passed from a vacuum into the medium 1. Mathematically, it is the ratio of the speed of light in vacuum to its speed in the medium 1. Secondly, the relative refractive index n₂₁ of a second medium relative to the first medium is the ratio of the speeds of light, v₁ and v₂, in the first and second medium respectively. In other words, we can predict the relationship between absolute refractive index and relative refractive index using Fermat’s principle.

 

Feynman elaborates that it is easy to get our formula (refractive index) if we can measure the speed of light in all materials and get its index relative to vacuum. In the next chapter, he explains the concept of chromatic aberration and then clarifies that “[t]he principle of least time is only an approximation, and it is interesting to know how much error can be allowed...” Strictly speaking, the definition of refractive index assumes the condition of no dispersion because different colors of light have different speeds in different refraction indices. It is related to the phenomenon of chromatic aberration in which the image of a white spot will have colors such that when we focus the red color, the violet color is out of focus, and vice versa. However, it is not easy as it may seem to measure the speed of light (different colors) in all materials.

 

Review Questions:

1. How would you explain the sun is actually well below the horizon during sunset?

2. How would you explain the reflection property of an ellipsoidal mirror?

3. Is it easy to measure the speed of light in different materials in order to determine the relative refractive index?

 

The moral of the lesson: we can apply Fermat’s principle to understand the position of the sun during sunset, the nature of mirage, and the principle of ellipsoidal mirror and parabolical mirror.

 

References:

1. Ayoub, A. B. (1991). Proof without Words: The reflection property of the parabola. Mathematics Magazine, 64(3), 175-175.

2. Feynman, R. P., Goodstein, D. L., & Goodstein, J. R. (1997). Feynman’s lost lecture: the motion of planets around the sun. London: Vintage.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Hecht, E (2002). Optics (4th edition). San Francisco: Addison Wesley.

5. Korn, G. A. & Korn, T. M. (2013). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover.