Fermat's Principle
An interesting way to derive both the law of reflection and the law of refraction (Snell's law) is to consider a principle credited to the the Frenchman Pierre De Fermat who lived in the 17th century.
The principle states that the path that light follows in undergoing reflection or refraction is the one that allows the light to travel from point A to B in the least amount of time. This is to be distinguished from the path of least distance in the case of refraction, since the speed of light propagation varies in media with different indices of refraction.
I should note that a more correct form of Fermat's principle is to say that light will travel along paths that are stationary in time with respect to variation in path. To fully understand this you might consider looking into the subject of calculus of variations, but the idea is this: Light will follow a path that satisfies one of three criteria:
1. The path of least time.
2. The path of greatest time.
3. The path which is a constant in time with variation in path.
We can prove both the law of reflection and Snell's law using the least time version of the principle.
Lifeguard Analogy
To explain the principle of least time and how it relates to a choice of path, consider the case of a lifeguard at the beach. He sits some distance from the shoreline. Imagine there is a drowning swimmer out in the water in a direction that is not normal to the interface between the water and sand. If the lifeguard wishes to get to the swimmer as soon as possible, should he follow a straight line path? At first that seems reasonable, but consider the fact that running on sand is much faster than swimming in water. Therefore it would seem reasonable to take a longer path along the sand in order to shorten the swim. The optimum path is found using Fermat's principle.
Mathematical Statement of Fermat's Principle
This may look odd to you since there are so many times that we take derivatives with respect to time. Now, however, we are taking a derivative of time with respect to path. The idea is that the transit time depends on the path chosen, and we want to find the path that leads to the minimum of the transit time. If we call the transit time t and the path s, then we want .
Animation of Fermat's Principle
In the animation below you can see a path between a lifeguard and a drowning swimmer. The path can be changed based on the location where the lifeguard passes from sand to water. The ratio of n2/n1 defines how much faster the lifeguard runs on sand vs swims in water. For instance, when n1=1 and n2=1.3, we should understand that running on sand is 1.3 times faster than swimming in water.
As the point of transition from sand to water is varied, notice that a plot is formed. The points on the plot are based on its x-coordinate being the location along the shoreline and its y-coordinate is the time it will take the lifeguard from chair to swimmer. Notice that there is a specific position that minimizes this time. That value of X for which time is a minimum is found mathematically using Fermat's principle.
The Math for the Lifeguard Case
As stated above, we wish for the minimum time with respect to path: Let's suppose we call the point where the lifeguard starts A, the point of transition from sand to water B, and the swimmer's location C. The path will be made of two straight line segments given by the segment AB and the segment BC. The length of segment AB is Likewise the length of the segment BC is The time that it takes to travel along those segments is and The speed of travel along the sand will be considered to be and in the water will be Writing the total time to reach the swimmer, we get
It is worth checking whether this is a minimum as compared with a maximum or stationary state, but I will leave that to you to check - perhaps by taking a second derivative. It turns out that any of those will in fact work, as all that's really required is that the derivative be zero, so it really isn't important to check in this case.