How Can Fermat's Principle Prove the Second Law of Reflection?

[deltafee]

Hi, I am trying to prove the second law of reflection using fermat's principle and I am not entirely sure how to start it.
By the way the second law of reflection is: The incident ray, reflect ray and normal ray all lie in a single plane.

 

[Simon Bridge]

Fermat's principle - light follows path of least time?
You do it pretty much the same way as you would for the first rule and for Snell's law... fix a point that the incedent ray passes through, and another that the reflected ray passes through, but vary the point of reflection (constrained by the first law).

 

[deltafee]

Yeah I used the three variable Pythagorean Theorem and than took the derivative and than placed values for x and y so I could graph it.

Here's the typed worksheet: https://dl.dropbox.com/u/77575413/F.pdf

on the second page I have the graphs of Time and the derivative of Time and as you can see I don't get a minimum in the derivative of time graph, but I get a minimum on the time graph. So I am really not sure what I did wrong.

Oh by the way just to make it easier to see the graph I left the value of c out from the equation.

 

[Simon Bridge]

It looks like at least the derivative is wrong.
You realize you can check your calculations against the actual answer because you know it already right?

I don't get a minimum in the derivative of time graph, but I get a minimum on the time graph.

example: y=x^2 has a minimum, but the derivative function y'=2x does not have a minimum.

I don't follow what you have done though - i.e.
The diagram at the top of the first page has no labels.

That 1/2c looks a little suspect. Comes from the 2d in the first line - but since there are no labels on the diagram I have no idea if it is OK or not.

I see you have written:$$\frac{1}{2c}\left [ \frac{10+z}{\sqrt{58}+z^2}+\frac{z-6}{\sqrt{106}+(20-z)^2} \right ]$$ for both $T$ and $T^\prime$.
(Last equation page 1, and top pf page 3).

I'm surprised you didn't try for a simpler geometry.

 

[sardar]

Hello there,

The second law of reflection states that the incident ray, reflected ray, and normal ray all lie in a single plane. This means that when a ray of light hits a reflective surface, the angle of incidence (the angle between the incident ray and the normal) will be equal to the angle of reflection (the angle between the reflected ray and the normal), and all three will lie in the same plane.

To prove this using Fermat's principle, we can consider the path of the light ray as it travels from the source to the reflective surface and then to the observer. According to Fermat's principle, light takes the path that minimizes the travel time. In the case of reflection, this means that the path taken by the light ray must be the shortest possible path.

Since the shortest path between two points is a straight line, we can conclude that the incident ray, reflected ray, and normal ray must all lie in a single plane. This is because if they did not lie in the same plane, the path of the light ray would not be the shortest possible path.

In other words, if the incident ray, reflected ray, and normal ray did not lie in the same plane, there would be a shorter path for the light ray to take, which would contradict Fermat's principle.

I hope this helps. Good luck with your proof!