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The Path of Light Using Calculus of Variations

  1. Mar 12, 2012 #1
    1. The problem statement, all variables and given/known data

    A rectangular solid of height h
    increases in density as its height
    increases, so the index of refraction of
    the solid increases with height
    according to:
    n(y) = 1.20(2y + 1)
    where y is the distance, in meters,
    from the origin (see diagram). A beam
    of light traveling in air (n = 1.00) in the
    x-y plane strikes the bottom of the
    tank at the origin, making an angle of
    incidence with the normal of θ1. Assume:
    • n varies only with y, not with x or z.
    • The light travels in the x-y plane.
    • The block is wide, so the light leaves through the top and not through a side.

    a) Use Fermat’s principle and Euler’s equation to find the equation for the path of the
    light y(x) in the tank. There will be two undetermined constants.
    HINT: If the equations become too complicated (and they will), remember we learned
    some alternative ways to solve calculus of variations problems.
    HINT: The answer should come out as a hyperbolic function.

    2. Relevant equations

    Euler's equations for extremum: f - y'(∂f/∂y') = constant
    n=c/v

    3. The attempt at a solution

    I'm looking to use Fermat's least time principle, I've tried to find the minimum of an integrated functional equal to time t.
    Currently I have it as:

    t = ∫ ds/v = ∫(1.20/c)(2y+1)√(1+y'^2)dx

    which would make my functional:

    f=(2y+1)√(1+y'^2)

    and using the Euler equations above I have from that:

    (2y+1)(1+y'^2)^.5 - y'^2(2y+1)(1+y'^2)^-.5 = (2y+1)(1+y'^2)^-.5 = constant

    and that's where I'm stuck...

    (2y+1)(1+y'^2)^-.5 = constant seems like it should be easy enough to integrate with dy and dx somehow, but try as I might I'm totally stumped.

    Somehow I'm supposed to get a hyperbolic function for y(x), which leads me to believe I'm approaching this whole problem the wrong way.

    Please help?
     
  2. jcsd
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