Snell’s Law and the Limitations of Adequality:

Section 3.7 Snell’s Law and the Limitations of Adequality:

When light passes from one medium to another, say from air to water, it bends. This property is called refraction. The physical law governing refraction is known as Snell’s Law, named after the Dutch Astronomer Willebrord Snell

¹⁹

https://mathshistory.st-andrews.ac.uk/Biographies/Snell/

(1580–1626) though it was accurately described before that time. In modern terms, Snell’s Law can be stated as follows.

Theorem 3.7.1. Snell’s Law of Refraction.

Suppose that light travels with a velocity of \(v_1\) in the first medium and velocity \(v_2\) in second medium, and that \(\theta_1\) and \(\theta_2\) are as seen in the diagram below.

Then the path that light follows satisfies

\begin{equation*} \frac{\sin(\theta_1)}{v_1} = \frac{\sin(\theta_2)}{v_2 }. \end{equation*}

Notice that if \(v_1=v_2\) this says that \(\sin(\theta_1)=\sin(\theta_2)\) and the path of the line would be a straight line.

A number of mathematicians, including both Snell and Fermat, gave derivations of Snell’s Law, but we will focus on Fermat’s method. To attack this problem, Fermat refined the Nature is Lazy assumption a bit. Instead of assuming that that light would follow the shortest path, he assumed that light would follow the path that takes the least time.

Notice that this is consistent with our observations concerning reflections from a mirror. In that case, the speed of light was constant so the shortest path was, in fact, the fastest path. But refraction occurs because the velocity of light changes when the surrounding medium changes.

Let’s see what happens when we try to use Fermat’s Method of Adequality to find this fastest path.

Problem 3.7.2.

Assuming that the velocity of light is \(v_1\) in the first medium and \(v_2\) in the second, use the following diagram to show the time, \(t\text{,}\) for light to travel along the path from \(A\) to \(C\) to \(B\) is \(\frac{\sqrt{a^2+x^2}}{v_1}+\frac{\sqrt{b^2+(c-x)^2}}{v_2}.\)

Mathematical Terminology.

Is it clear to you that we’ve used velocity in Problem 3.7.2 when we really mean speed? If not, think about it for a moment. We’re really only interested in how fast light travels in either medium, not in the direction it is going. This was the language used in the seventeenth century so it is accurate historically, but not mathematically. Unfortunately, it is still common to use these words interchangeably, even among folks who know better. This can be quite confusing for the beginner but there isn’t much that can be done except be aware of the problem. Whenever you encounter either word pause of a moment, and ask yourself if the speaker or author means velocity, which includes the direction of travel, or speed, which does not. It should always be clear from the context.

Of course the next step would be to use Fermat’s Method of Adequality to minimize \(t\text{,}\) but the square roots involved make the Algebra daunting to say the least. Despite these difficulties Fermat was able to show that \(t\) is minimized precisely when Snell’s Law holds.

Although it is possible to derive Snell’s Law with the tools we now have it is very difficult so we won’t attempt it here. Instead we will return to this problem in Chapter 9 where we will complete the derivation of Snell’s Law using the rules of Calculus. That will be much simpler.

It seems that our efforts to solve optimization problems using pre–Calculus techniques has hit an impasse. To be sure those techniques, and others of a similar nature, were very clever and it must be admitted that they yielded interesting and correct results. However for the most part they were ad hoc methods and quite limited in their applicability. We need a way to overcome these limitations.

This is exactly what the invention of Calculus in the mid–seventeenth century did. The title of Leibniz’ first published work on Calculus makes this very clear:

A New Method for Maxima and Minima, as Well as Tangents, Which is Impeded Neither by Fractional nor Irrational Quantities, and a Remarkable Type of Calculus for This.

It is time for us to learn about this Remarkable Type of Calculus.

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