Find the Angle of Incidence for Total Internal Reflection in a Glass Sphere

AI Thread Summary
The problem involves determining the angle of incidence for total internal reflection in a glass sphere with a refractive index of sqrt 3. The relationship between the angle of incidence (i) and the angle of refraction (r) is expressed through the equation μ = sin i/sin r, leading to the conclusion that i = 60° and r = 30°. A key point of confusion was the identification of angle ABO as equal to r, which is clarified by recognizing that triangle ΔABO is isosceles due to the equal lengths of the radii. The discussion highlights the importance of accurately constructing diagrams for better understanding. Overall, the solution to the problem is confirmed as correct.
betaleonis
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Homework Statement



A ray of light incident on a glass sphere (refractive index sqrt 3) suffers total internal reflection before emerging out exactly parallel to the incident ray. What was the angle of incidence?

μ = refractive index of the glass sphere, i = angle of incidence, r = angle of refraction

Homework Equations



μ = sin i/sinr

The Attempt at a Solution



From the figure, it turns out that < AOE = i (since L1 is parallel to L2, the corresponding angles are equal)

i = 2r, which implies that sin i/ sinr = μ, or 2cosr = μ, or r = 30° and i = 60°, which is the answer.

I do not understand why < ABO = r. Why is it so? It'd be helpful if I someone could come up with a different way of approaching the problem.
 

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betaleonis said:
I do not understand why < ABO = r. Why is it so?
Isosceles triangle (two radii).
 
I do not understand why < ABO = r. Why is it so? It'd be helpful if I someone could come up with a different way of approaching the problem.
ΔABO is an isosceles triangle: |OA|=|OB|

[edit] beaten to it :)
 
Oops! I ought to have thought a bit more before posting that question. Thank you. :)
 
No worries. Everyone does it sometimes :)
You'd probably have noticed right away if the diagram was constructed rather than sketched, even though you noticed about the equal base-angles.
 
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