Section 26–2 Reflection and refraction

 (Law of reflection / Law of refraction / Ptolemy’s experimental results)

 

In this section, Feynman discusses the law of reflection, the law of refraction, and Ptolemy’s experimental results.

 

1. Law of reflection:

“For some reason it is customary to measure the angles from the normal to the mirror surface. Thus the so-called law of reflection is θ_i = θ_r (Feynman et al., 1963, section 26–2 Reflection and refraction).”

 

Feynman states the law of reflection simply using the equation θ_i = θ_r. In Volume II, Feynman adds that “[t]he angle of reflection is equal to the angle of incidence. With the angles defined as shown in Fig. 33-1, θ_r = θ_i (Feynman et al., 1964, section 33–1 Reflection and refraction of light).” Some authors prefer the term “plane mirror reflection” and include the second law of reflection: “the incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane.” To be precise, we can have a vectorial law of reflection using the cross product and dot product. For example, the incident ray, reflected ray, and normal can be represented by i = u_ix_i + u_iy_j + u_iz_k, r = u_ix_i + u_iy_j − u_iz_k and k respectively. The triple product of these three vectors (i × r).k is zero because they all lie in the same plane (Tkaczyk, 2012).

 

Feynman explains that when the light hits the mirror, it does not continue in a straight line, but bounces off the mirror into a new straight line. It is analogous to the collision of an elastic ball on a hard surface (neglecting the influence of gravity). One may clarify that the law of reflection is an empirical law, but it can be proved by using Maxwell’s equations. Historically, the first law of reflection, “the angle of incidence is equal to the angle of reflection,” appeared in Euclid’s Catoptrics. Euclid is also one of the very intelligent men that has proposed “something comes out of the eye” (mentioned in the last section).

 

2. Law of refraction:

“…θ_i is the angle in air and θ_r is the angle in the water, then it turns out that the sine of θ_i is equal to some constant multiple of the sine of θ_r: sin θ_i = n sin θ_r ... Equation (26.2) is called Snell’s law (Feynman et al., 1963, section 26–2 Reflection and refraction).”

 

Feynman states Snell’s law as “if θ_i is the angle in air and θ_r is the angle in the water, then it turns out that the sine of θ_i is equal to some constant multiple of the sine of θ_r: sin θ_i = n sin θ_r. In Volume II, he makes a revision of the law to “[t]he product n sin θ is the same for the incident and transmitted beams (Snell’s law): n₁sin θ_i = n₂sin θ_t (Feynman et al., 1964, section 33–1 Reflection and refraction of light).” Some authors prefer to include the second law of refraction: “the incident ray, refracted ray, and the normal to the interface of two media at the point of incidence all lie on the same plane.” Better still, one may express a vector form of Snell’s law that encompasses the two laws of refraction. For small values of i (in radians), Snell’s law can be approximated by using the equation n i = n¢ i¢.

 

Feynman says that the rule (Snell’s law) was found by Willebrord Snell, a Dutch mathematician. Specifically, Snell used the cosecant of the angles instead of the sine of the angles in his law of refraction (Smith, 1987). Historically, Descartes was the first to publish the law of refraction using the sine of the angles in 1637 and he was initially regarded as the originator of this sine law. On the other hand, some have argued that the first discovery of the sine law was made by the sixteenth-century English scientist Thomas Harriot (Dudley & Kwan, 1997). In addition, the Persian scientist Ibn Sahl applied his law of refraction to demonstrate the optical properties of hyperbolic lenses (Mark, 1999). Thus, it is debatable who was the first to discover the law of refraction.

 

3. Ptolemy’s experimental results:

“Claudius Ptolemy made a list of the angle in water for each of a number of different angles in air (Feynman et al., 1963, section 26–2 Reflection and refraction).”

 

Feynman presents Table 26–1 to show the angles in the air, the corresponding angle measured in the water, and concludes that there is a remarkable agreement with Snell’s law. Based on Ptolemy’s experimental results, one may deduce Snell’s law to be governed by the quadratic equation r = ai – bi² or r = R – (n²d₂ – nd₂)/2 where i is the angle of incidence, r is the angle of refraction, and n is the refractive index (Smith, 1996). That is, it is more difficult to deduce Snell’s sine law of refraction using Ptolemy’s results that fit a parabolic curve. Furthermore, it was also difficult for Ptolemy to formulate the quadratic equation because of the limitations of the Euclidean theory of proportionality (Smith, 1996). Thus, we should not conclude that the numbers based on Snell’s law are in a remarkable agreement with Ptolemy’s results.

 

Surprisingly, Feynman includes the following statement: “it should be noted, however, that these do not represent independent careful measurements for each angle but only some numbers interpolated from a few measurements, for they all fit perfectly on a parabola.” This statement presents Ptolemy’s results in Table 26–1 positively, but the numbers were manipulated to fit the parabolic curve. Importantly, Feynman criticizes the manipulation of experimental results in a talk titled Cargo Cult Science. In Feynman’s words, “[w]hen they got a number closer to Millikan's value they didn’t look so hard. And so they eliminated the numbers that were too far off, and did other things like that (p. 342).” Perhaps Feynman should explain how the manipulation of numbers could make it impossible to discover Snell’s law.

Review Questions:

1. How would you state the law(s) of reflection?

2. Was the law of refraction first discovered by Willebrord Snell?

3. Should Feynman conclude that the numbers based on Snell’s law are in a remarkable agreement with Ptolemy’s list?

 

The moral of the lesson: the adjusted Ptolemy’s experimental results (Table 26–1) are slightly different from the numbers based on Snell’s law (Table 26–2), and thus, it could have posed difficulties to discover Snell’s sine law of refraction.

 

References:

1. Dudley, J. M., & Kwan, A. M. (1997). Snell’s law or Harriot's?. Physics Teacher, 35(3), 158-159.

2. 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.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

4. Smith, A. M. (1987). Descartes's theory of light and refraction: a discourse on method. Transactions of the American Philosophical Society, 77(3), 1-92.

5. Smith, A. M. (1996). Ptolemy's theory of visual perception: an English translation of the Optics (Vol. 82). Philadelphia: American Philosophical Society.

6. Smith, A. M. (1999). Ptolemy and the foundations of ancient mathematical optics: A source based guided study. Transactions of the American Philosophical Society, 89(3), 1-172.

7. Tkaczyk, E. R. (2012). Vectorial laws of refraction and reflection using the cross product and dot product. Optics Letters, 37(5), 972-974.