To determine the refractive index of the prism, let's analyze the given conditions using Snell's law and the geometry of the prism. The prism has an apex angle of
30∘ and the incident angle on one of its faces is
60∘. The emergent ray is normal to the other surface.
First, let's consider the refraction at the first surface of the prism where light enters. According to Snell's law:
nisin‌(θi)=nrsin‌(θr)where
ni is the refractive index of the medium the light is coming from (which we assume to be air, so
ni=1),θi is the angle of incidence
(60∘),nr is the refractive index of the prism, and
θr is the angle of refraction.
We have:
‌sin‌(60∘)=nsin‌(θr)‌‌=nsin‌(θr) Next, we need to find the relationship between
θr and the internal angles of the prism. Since the prism angle is
30∘ and the emergent ray is normal to the surface (meaning the angle of emergence is
0∘ ), we can use the fact that the sum of angles inside the prism should be equal to
90∘ (since the light travels through the prism and the final deviation angle should cancel out the prism angle). Therefore:
θr+φ=30∘ Where
φ is the internal angle of deviation. Given that
φ+0∘=30∘, it means
θr should equal
30∘.
Therefore we have:
sin‌(30∘)=‌ Substituting
θr=30∘ into Snell's law equation:
‌=n⋅‌Thus, solving for
n, we find:
n=√3Hence, the refractive index of the prism is:
Option A:
√3