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Theory:
When light passes from one transparent medium to another,
it bends according to Snell's law which states:
Ni * Sin(Ai) = Nr * Sin(Ar),
where:
Ni is the refractive index of the medium the light is leaving,
Ai is the incident angle between the light ray and the
normal to the meduim to medium interface,
Nr is the refractive index of the medium the light is
entering,
Ar is the refractive angle between the light ray and the
normal to the meduim to medium interface.
So, here at the critical angle, Ar becomes 90 degrees, and Ai is theta
Ni = 1.543
Nr=1.508
Sin(theta) = 1.508 / 1.543
=.977 Here angle is in radians. You have to take Sin inverse.
Criticale angle Theta = 1.49 radians
To convert to degrees, please use, 180 degrees = pi
In the case of air and core, see the picture attached.
With simple geometry, you can see, reflective angle to the core= 90-theta
Therefore,
1 x Sin(alpha) = 1.543 x Sin(90-theta)
=1.543 x cos(theta)
=1.543 x Cos(1.49 rad)
=1.543 x 0.081
= 0.125
alpha = Sin inverse .125
=0.125 radians