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?
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.