Please show all work/answers legibly. A simple form of an optical fibre is shown

Question

Please show all work/answers legibly.

A simple form of an optical fibre is shown in the diagram. A high-refractive core is surrounded by air. Light enters the end of the optical fibre with an incident angle theta, and then stays inside the fibre by total internal reflection. The index of refraction of the core material is n, and outside the index is 1. Find the maximum angle theta for which the light ray will stay contained to the core. When n is above a critical value, ANY value of theta will work, from zero to 90 degrees. Find this critical value of n.

Explanation / Answer

n*sin() = constant

=> 1*sin() = n*sin(r)

=> sin(r) = sin() / n

sin(c) = 1/n

c + r = 900

=> r = 900 - c

=> sin(r) = sin(900 - c)

=> sin(r) = cos(c)

=> sin() / n = [1- (1/n2)]

=> max = sin-1((n2-1))

For any value of to work,

max = 900

=> sin(max) = 1

=> (n2-1) = 1

=> n = 2

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