Sample Problems for The Critical Angle

Sample Problem for The Critical Angle

Important Information

The critical angle is reached when the angle of refraction just equals 90^o. Thus,

sin θ_c=(n₂/n₁) sin 90^o = (n₂/n₁)

where n_i represents the index of refraction in medium i, and θ_c represents the critical angle made by the light in medium 1.

Sample Problem 1:

A man on the street offers you a diamond.

You are approached by a man on the street who offers to sell you a rather impressive “gen-u-ine diamond.” He produces a certificate of authenticity while telling you of the hardships forcing him to part with his wife’s engagement stone (a family heirloom). Being skeptical, you decide to verify the composition of the stone with the laser pointer attached to your key ring. (a) What is the critical angle for a diamond/air interface? For a glass/air interface? (b) How could you use this information to determine the composition of the stone?

Solution:

(a)

To observe total internal reflection, light must travel from the denser medium of diamond or glass into the rarer medium of air. From the Table of indices of refraction, we see that n_1d = n_diamond = 2.4, while n_1g = n_glass = 1.5. n₂ = n_air = 1.0. Using the definition of the critical angle provided above, we have θ_{c,diamond/air}= sin^-1 (n_air/n_diamond) = sin^-1 (0.417) = 24.6^o.
θ_c,glass/air= sin^-1 (n_air/n_glass) = sin^-1 (0.667) = 41.8^o.

(b)

We can observe the path of laser light as it travels through the stone. By rotating the stone until TIR is just reached for the light exiting the stone, we can observe the critical angle. If the critical angle for light incident on the back face of the stone is fairly close to 45^o (halfway between face and normal to face so "easy" to reference), the stone is glass. If, on the other hand, the critical angle for the light incident on the back face is significantly closer to the normal than halfway, the stone might actually be a diamond. Of course observing TIR on the back face of a cut stone might be difficult unless the stone is very large and/or has few facets.

Practice Exercises

On-Line Reading

At-Home Assignments

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