A diamond (n=2.42) is immersed in a transparent liquid. A light ray inside the diamond is incident at the critical angle at the diamond-liquid interface. If the refractive index of the liquid increases, what happens to the critical angle?
The critical angle increases as the refractive index of the liquid increases
The critical angle (θc) is the angle of incidence in the optically denser medium at which the refracted ray just grazes along the boundary (refraction angle = 90°); for any angle of incidence greater than θc, the light undergoes total internal reflection (TIR) instead of refracting out. Using Snell's law at the diamond–liquid interface, with light travelling from the diamond (refractive index n₁ = 2.42) into the surrounding liquid (refractive index n₂):
n₁ sin θc = n₂ sin 90° = n₂
sin θc = n₂ / n₁
Since n₁ (the diamond's refractive index) is fixed at 2.42, the critical angle depends entirely on n₂, the refractive index of the surrounding liquid. If the liquid's refractive index n₂ increases (i.e., the liquid becomes optically denser and closer in optical density to the diamond), the ratio n₂/n₁ increases, so sin θc increases, and therefore θc itself increases.
This makes physical sense: total internal reflection relies on there being a significant "optical mismatch" between the denser medium (diamond) and the rarer medium (liquid). As the liquid's refractive index approaches that of diamond, this mismatch shrinks, so light can escape (refract out) more easily and only undergoes TIR at larger and larger angles of incidence. In the limiting case where n₂ approaches n₁ (the liquid becomes almost as optically dense as diamond), sin θc approaches 1, so θc approaches 90° — meaning total internal reflection becomes possible only at grazing incidence, and diamonds lose much of their brilliant "sparkle" when immersed in such a liquid (this is actually used as a simple way to demonstrate that a diamond's sparkle depends on TIR, since immersing a diamond in a liquid of similar refractive index makes it appear dull).
The other options can be dismissed accordingly: the critical angle clearly does not remain unchanged, since it explicitly depends on the ratio n₂/n₁; it does not decrease with increasing n₂, since the ratio n₂/n₁ moves in the same direction as n₂; and TIR does not occur "at all angles" as n₂ increases — in fact the opposite happens, since a larger θc means TIR occurs over a narrower range of angles (only for angles greater than the now-larger θc), not a wider one. Hence, increasing the liquid's refractive index correctly causes the critical angle to increase.
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