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Refraction in a sphere

  1. Jan 15, 2014 #1
    [Mentor's note: this thread does not use the normal homework forum template because it was originally posted in a non-homework forum, then moved here.]

    A light beam is incident on a sphere with refractive index n=√3 at an angle i from air and emerges parallel to the horizontal axis passing through the center of the sphere. Find i?

    Here I made a line from the center of the sphere 25.jpg to the exit point of the ray after which it becomes parallel to the axis. I attached a rough diagram I made in paint for it.

    Now the triangle formed inside the circle is isosceles and hence two sides and corresponding angles are equal so we have two sides as r and I named the last side as d and the angle at the center is unknown. Here I only have one numerical value (n=√3) and I have to find the angle of incidence. I know there is some geometry involved so I used the cosine rule to solve for the refracted angle but even combined with snells law and other properties of triangles (S and Δ) I just don't see how to proceed, can you guys please help me out?
     
    Last edited by a moderator: Jan 15, 2014
  2. jcsd
  3. Jan 15, 2014 #2
    There you go, this image should help.

    note that θ=Θ-θ because it's and isosceles triangle. so Θ=2θ (θ is the angle of refraction from air to the sphere)

    so √3 sin(2θ-θ) = sin(2θ)

    expand the sin on the RHS you get: √3 sin(θ) = 2sin(θ)cos(θ)

    solve that you get θ = 30°

    now it's easy to solve this equation sin(i)=√3 sin(θ) to get i.
     

    Attached Files:

  4. Jan 15, 2014 #3
    i=600
    Thanks a gazillion. Just extending one line makes a hell of a difference, well, I guess that's geometry for you. Anyhow, thanks again. World is simpler again till I find another brainteaser from Euclid.
     
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