Total Internal Reflection

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Light inside a diamond of refractive index 2.42 tries to escape into air, and a gemologist needs the critical angle beyond which total internal reflection occurs.

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Total internal reflection, discussed in NCERT, happens when light in a denser medium meets the boundary with a rarer medium at an angle exceeding the critical angle, defined by $\sin	heta_c=\frac{n_{rarer}}{n_{denser}}$. For diamond to air, $n_{denser}=2.42$ and $n_{rarer}=1$, so $\sin	heta_c=\frac{1}{2.42}=0.413$. Taking the inverse sine yields $	heta_c=\sin^{-1}(0.413)\approx24.4^\circ$. The unusually small critical angle explains why diamonds sparkle: light striking most internal facets exceeds this angle and is trapped, bouncing repeatedly before exiting. The value $41.8^\circ$ is wrong because it corresponds to water ($n=1.5$ region) not diamond, using a lower index. The value $48.6^\circ$ is wrong as it belongs to glass of index 1.33-type distractors, again the wrong medium. The value $65.6^\circ$ inverts the ratio and takes the arcsine of 2.42, which is impossible since sine cannot exceed 1. As stated in NCERT Class 12, Chapter 9, a higher refractive index produces a smaller critical angle. A magnitude check confirms the very high index of diamond gives a strikingly small critical angle, consistent with its brilliant appearance.

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