Fermat's Time Principle

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Fermat’s Principle of Least Time. Imagine that we want to analyze the trajectory of light reflecting off of a surface. Consider the following diagram below: Here the horizontal black line is the medium that the light is bouncing off of. The red line represents the path that the light would travel if it continued on unimpeded by the reflective medium. The blue line and the orange line are equal and so by basic geometry the reflected line and the red line are equal in length. Since we know that a straight line is the shortest distance between two points we can say that the angle that the light makes at the reflective medium that minimizes the distance between the two points (if it has to first travel to the reflective medium) is the angle that the light beam makes with the surface of the reflective medium. Since the light is traveling at a constant speed the minimum distance also corresponds to the minimum time. So we see that the minimum time proposition explains the law of the reflection of light. We can now explore the more complicated scenario of light traveling from one medium to another. Below is a diagram showing the path that the light travels. Here we know that the light is traveling from the initial point to the final but we are trying to find the x value (where it crosses the two mediums) that minimizes the time taken. If we look at this diagram it is clear that in general the total time that the beam travels to get from point x1,y1to point x2,y2 is: Tx=x-x12+y12v1+x2-x2+y12v2 If we consider Fermat’s principle of least time to be true then the true time taken is that for which the function is a minimum. To find this we take the derivative of T(x) and set it equal to zero. The derivative is: T'x=x-x12v1x-x12+y12-x2-x2v2x2-x2+y12 Setting this equal to zero we get the following result. x-x12v1x-x12+y12=x2-x2v2x2-x2+y12 Looking

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