The beauty of physics is not only expressed in concise formulas. We also surprisingly find that many physical phenomena occur in a way that causes a certain variable to reach an extreme value. A typical example is Fermat’s Principle, which points out an important law of the propagation path of light: Light always travels along the path that takes the shortest time. Here, we will briefly introduce Fermat’s Principle.
Fermat’s Principle is commonly known as the "Principle of Least Time." In 1657, Fermat proposed:
From point P to point Q, among all feasible paths, light chooses the one that requires the shortest time.
From point P to point Q, among all feasible paths, light chooses the one where the required time is an extremum.
This is an extremely wonderful principle and one of the most miraculous extremes in nature. As a non-biological entity, light actually autonomously chooses the optimal path, becoming the most "efficient" thing in the world, which makes one admire the greatness of the universe. Is this a meticulous design by the Creator, or an unintentional act?
Fermat’s Principle is mainly reflected in:
1. The speed of light is the fastest speed in the universe
2. Light travels in a straight line (in a homogeneous medium)
3. The law of reflection of light
4. The law of refraction of light (Snell’s Law)
Among these, (1) is a topic in physics that we will not discuss here; (2) and (3) are content we are relatively familiar with, which even middle school students have encountered, so they will not be discussed in detail. We will only briefly talk about (4), and we can find that: (2) and (3) are both conclusions of (4).
As shown in the figure, on the straight line interface, there is a moving point O, and two fixed points P and Q. The necessary and sufficient condition for \frac{PO}{v_1}+\frac{QO}{v_2} to be minimized is \frac{v_1}{\sin\theta_1}=\frac{v_2}{\sin\theta_2}.
We can set P=(x_1,y_1), Q=(x_2,y_2), O=(x,0), then t=\frac{\sqrt{(x_1-x)^2+y_1^2}}{v_1}+\frac{\sqrt{(x_2-x)^2+y_2^2}}{v_2} Taking the derivative and setting it to 0, we get 0=\frac{x-x_1}{v_1 \sqrt{(x_1-x)^2+y_1^2}}+\frac{x-x_2}{v_2 \sqrt{(x_2-x)^2+y_2^2}} Which is \frac{v_1}{\sin\theta_1}=\frac{v_2}{\sin\theta_2} Proof complete.
Note that in the above proof, we did not mention that P and Q must be on opposite sides of the interface. Therefore, the conclusion also holds when P and Q are on the same side. Perhaps the following figure can help you better understand this (by means of mirror symmetry, it turns into a light refraction problem; do you find this method somewhat familiar?)
For the convenience of reference, let us tentatively call Snell’s Law for the case where P and Q are on the same side the "Law of Refraction-Reflection." It is not difficult to see that the straight-line propagation and the law of reflection are actually the cases when v_1=v_2. It should be noted that although we used the term "necessary and sufficient condition" above, that was only for a specific case. In more general cases, Fermat’s Principle is only a "necessary condition." In this article, we have briefly provided a demonstration of light propagation and the fastest light path, laying a good foundation for subsequent applications. However, next we will not immediately talk about the application of optical principles, but instead shift our focus to another "natural extreme," which is related to many phenomena we usually see...
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