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Section2Fermat's Principle

In our study of mirrors we discovered the Principle of Extreme Distance. The Principle of Extreme Distance states that the path light will travel between two points, when reflected once off a planar mirror, will take the path with an extreme distance. This can be either the shortest possible path, or the longest possible path.

Because refraction obviously does not travel either the longest or shortest possible path, we can not apply the Principle of Extreme Distance as is, however we will determine that light will travel along the path that minimizes the “time” taken. To this effect we will utilize a model where our light will be represented by Joe, who can run 2 meters/second in the grass, but only 1 meter/second in the mud.

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Figure2.1Example path from A to B passing from Grass to Mud.

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Figure2.2\(\theta_i\) vs \(\theta\) for the optimal \(A \rightarrow B\)

The paths in the six problems together show a strong relationship that fits with our conception of the index of refraction. We would expect that for a medium in which travel is slowed by half that the index of refraction would be 0.5, which it was.

Seeing as we have applied principles of extremes (distance and time) to both reflection and refraction, it follows that we should be able to summarize these principles into a single rule, thus unifying geometric optics. To this effect, we can state that light will travel the path that takes an extreme amount of time to traverse.

For light traveling within a homogeneous medium, fastest path between two points will be the direct path between them as there is no change in speed within the medium, thus that will be the fastest path.

With a mirror, the path bouncing off of the mirror with equal angle in and angle out will exibit either the shortest, and thus fastest, or longest, and thus slowest, path; this holds for when the light is staying in one medium, further complications occur if there is refraction involved as well.

These examples uphold the Principle of Extreme Time. Further, we can determine, from the Principle of Extreme Time, that light travels faster in air than it does in water, and that is why it bends towards the normal in accordance with Snell's law. We could then, with proper experiments, further validate the Principle of Extreme Time by measuring directly the speed of light in air and in water. However, we cannot determine the speed of light using this principle as we are simply utilizing the relative change in speed, not the absolute speed of light.