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The second law of reflection

  1. Feb 10, 2013 #1
    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.
    Last edited: Feb 10, 2013
  2. jcsd
  3. Feb 10, 2013 #2

    Simon Bridge

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    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).
  4. Feb 13, 2013 #3
    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 [Broken]

    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.
    Last edited by a moderator: May 6, 2017
  5. Feb 15, 2013 #4

    Simon Bridge

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    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.
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