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Homework Help: Refractive index question - light beam around the world

  1. Jan 3, 2012 #1
    1. The refractive index of the Earth’s atmosphere is
    n = 1.01 + α(R − r),
    where α is a constant, r is the radial distance from the Earth’s centre and
    R = 6.4 × 10^6 m is the Earth’s radius. By considering a path comprising a series of
    total internal reflections or otherwise, find a value of α for which a light ray emitted
    horizontally close to the Earth’s surface would go around the Earth. (The effects of
    absorption may be ignored and the Earth may be taken to be a perfect sphere of radius
    R.)



    2. Relevant equations
    n1sin(x1)=n2sin(x2)

    3. The attempt at a solution
    I really have no idea what to do as I understand the refractive index is constantly changing?
    Don't know where to start, any help would be much appreciated.
     
  2. jcsd
  3. Jan 3, 2012 #2

    rude man

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    Think of a wavefront launched horizontally. Realize that the top of this wavefront will move faster than the bottom, since n is lower at higher altitudes ... yet the wavefront phasing across its front has to be the same for all heights - whar does that requirement impose on grad(n)?
     
  4. Jan 4, 2012 #3
    Thank you very much for your reply, are you suggesting the light curves around the world? I considered this but thought that only really happened at black holes, or have I misunderstood your response?
     
  5. Jan 4, 2012 #4

    rude man

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    I never heard of the herm 'light curves around the world' per se but that is what I had in mind. And I know next to zilch about black holes.

    Can you go from there?

    BTW the problem did not state that it was possible, it just asked what the n gradient had to be were it possible.
     
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