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Optics - Refraction and a transparent sphere

  1. Feb 16, 2013 #1
    Optics -- Refraction and a transparent sphere

    1. The problem statement, all variables and given/known data
    A spider is hanging by means of its own silk thread directly above a transparent fixed sphere of radius R=20 cm as shown in the figure. The refractive index of the material of the sphere is equal to ##\sqrt{2}## and the height of the spider from the centre of the sphere is 2R. An insect, initially sitting at the bottom, starts crawling on the sphere along a vertical circular path with the constant speed of ##\frac{\pi}{4}## cm/s. For how long time the insect will be invisible for the spider, assume that it crawls once round the vertical circle.
    (see attachment)

    2. Relevant equations



    3. The attempt at a solution
    Honestly, I have no idea. I don't have a clue about which equations I have to use here.

    Any help is appreciated. Thanks!
     

    Attached Files:

  2. jcsd
  3. Feb 16, 2013 #2

    SammyS

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    Re: Optics

    This question has to do with total internal reflection. Start there.
     
  4. Feb 16, 2013 #3
    Re: Optics

    I did think of this before but the insect keeps on moving which is confusing me.
     
  5. Feb 16, 2013 #4

    SammyS

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    Re: Optics

    Find out the location of the insect just as a ray from the insect to the spider reaches the point of criticality.

    I actually found this easier to do by tracing the ray from the spider, back to the insect.
     
  6. Feb 17, 2013 #5
    Re: Optics

    I too think this would be easier.
    If a ray from the spider is tangent to the sphere, then by Snell's law, it deviates towards the normal by pi/4. (see attachment)

    I am facing problems with geometry. Look at the triangle where the angles come out to be 150 and 45 degrees, the sum goes over 180 degrees! I can't find where I went wrong.

    EDIT: Woops, please ignore this reply. I think I have found out my mistake. One more question, would the rays meet, after refraction, at the point where insect was initially placed?
     

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  7. Feb 17, 2013 #6

    SammyS

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    Re: Optics

    They don't meet there, if my solution is correct.

    attachment.php?attachmentid=55855&d=1361161755.png

    The angle you have labelled as 60° is 120° .
     
  8. Feb 17, 2013 #7
    Re: Optics

    Yes, I figured that out already, that's why I said to ignore the reply. I will be back with a solution. :smile:
     
  9. Feb 17, 2013 #8

    SammyS

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    Re: Optics

    After being refracted, the ray from the spider then intercepts the circle again. A radius from each place the ray intercepts the circle together with the ray itself, form an isosceles right triangle, with the ray falling along the hypotenuse.
     
  10. Feb 18, 2013 #9
    Re: Optics -- Refraction and a transparent sphere

    I am still not getting the right answer.
    (see attachment)
    The angular displacement of the insect when it is invisible to the spider is ##2\cdot \frac{13\pi}{24}##.
    The time taken is ##t=\frac{\theta}{\omega}## where ##\omega## is the angular velocity of insect.
    Substituting the values, ##t=\frac{260}{3} sec## which is wrong. :(
     

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  11. Feb 18, 2013 #10

    SammyS

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    Re: Optics -- Refraction and a transparent sphere

    Your drawings are misleading.

    The refracted ray makes an angle of 45° (π/2) with respect to the normal .

    In both drawings, the refracted ray appears to be drawn at an angle of 45° with respect to the horizontal.
     
  12. Feb 18, 2013 #11
    Re: Optics -- Refraction and a transparent sphere

    How do you know that the rays go like you have shown in your diagram? :confused:
     
  13. Feb 18, 2013 #12

    ehild

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    Re: Optics -- Refraction and a transparent sphere

    The bug is visible along the red arc. The refractive index is 1/√2 so the angle of refraction is 45°. The radii make 90°angle between the point of incidence and point where the refracted ray arrives to the circle again.

    ehild
     

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  14. Feb 18, 2013 #13
    Re: Optics -- Refraction and a transparent sphere

    Thank you both for your inputs, I have got the right answer! :smile:
     
  15. Feb 18, 2013 #14

    SammyS

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    Re: Optics -- Refraction and a transparent sphere

    You, yourself, said the angle of refraction is π/4 .
     
  16. Feb 18, 2013 #15
    Re: Optics -- Refraction and a transparent sphere

    Yes, I did but I was confused by the path the rays would take after refraction. I messed up with simple geometry. :redface:
     
  17. Feb 18, 2013 #16

    ehild

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    Re: Optics -- Refraction and a transparent sphere

    The rays travel along straight lines :tongue2:
     
  18. Feb 18, 2013 #17
    Re: Optics -- Refraction and a transparent sphere

    :biggrin:
    Yes but see my previous diagrams, they are completely wrong.
     
  19. Feb 18, 2013 #18

    ehild

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    Re: Optics -- Refraction and a transparent sphere

    Yes, the upper central angle was wrong, and also, you did not draw the refracted light according to the angle you got.

    ehild.
     
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