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Non-constant index of refraction due to layered material.

  1. Aug 22, 2010 #1
    A ray of light travels through a medium with an index of refraction [tex]n_{1}[/tex] and strikes an layered medium such that the index of refraction is [tex]n_{2}=ky+1[/tex] where [tex]y[/tex] is the depth of the medium and [tex]k[/tex] is a constant. If it hits at an angle of [tex]\theta_{1}[/tex] with respect to the normal, find the angle [tex]\theta_{2}[/tex] at which the light ray refracts as a function of time.

    Source: A post that I made on the Art Of Problem Solving forum.
     
    Last edited: Aug 22, 2010
  2. jcsd
  3. Aug 22, 2010 #2

    Integral

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    As with all homework like questions you must show some work before getting help.
     
  4. Aug 22, 2010 #3
    My work:
    So far I know that [tex]v=\frac{c}{n_{2}}[/tex] is the speed of the light beam, which is also equal to [tex]v=\frac{dy}{dt}[/tex]. So a differential equation to solve would be [tex]\frac{dy}{dt}=\frac{c}{n_{2}}[/tex]
     
  5. Aug 23, 2010 #4

    ehild

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    The light ray does not travel along y but at an angle θ2 with respect to it. θ2 itself is a function of y.

    ehild
     
  6. Aug 23, 2010 #5
    APhO 2004 problem 2. It is similar to this one. Look at the solution there.
     
  7. Aug 23, 2010 #6
    But because [tex]y[/tex] is a function of time, that also makes [tex]\theta_{2}[/tex] a function of time.
     
  8. Aug 23, 2010 #7

    ehild

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    And how are y and θ2 related?


    ehild
     
  9. Aug 23, 2010 #8
    If the material is layered infintesimally so that the index of refraction is proportional to the y, which I stated in the problem, then y is related to [tex]\theta[/tex]2 because the index of refraction is related to [tex]\theta[/tex]2
     
  10. Aug 23, 2010 #9

    ehild

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    What is the relation between the refractive index and θ2?

    ehild
     
  11. Aug 23, 2010 #10
    The refractive index and [tex]\theta[/tex]2 are related through Snell's Law.
     
  12. Aug 24, 2010 #11

    ehild

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    Well. At depth y, the light ray encloses the angle θ2(y) with the y axis. The light travels along a curved path s and ds/dt = c/n2(y). At depth y, θ2 is obtained from Snell's law. Now you can set up the differential equation for θ2 as function of t.

    ehild
     
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