The critical angle for a denser-rarer interface is 45°. The speed of light in rarer medium is 3 × 10⁸ m / s. The speed of light in the denser medium is:

Solution

Physics sir · 4 steps

At the critical angle, sin C is the ratio of the two speeds, so one line gives the answer.

1. Critical angle condition. At the critical angle the refracted ray grazes the surface, so the refraction angle is 90°. Snell's law then gives sin C = nᵣₐᵣₑᵣ/n_(denser).
2. Switch to speeds. Since n = c/v, the ratio of refractive indices is the inverse ratio of speeds. Light is slower in the denser medium, so sin C is less than 1, as it should be.

   sin C = nᵣₐᵣₑᵣ/n_(denser) = v_(denser)/vᵣₐᵣₑᵣ

3. Put in numbers. v_(denser) = vᵣₐᵣₑᵣ sin 45° = (3 × 10⁸)/√2 = 2.12 × 10⁸ m s⁻¹.
4. So. The speed of light in the denser medium is 2.12 × 10⁸ m s⁻¹, which is option (a).

The idea to keep. For any denser-rarer pair, sin C = v_(denser)/vᵣₐᵣₑᵣ. The slower medium is on top, so the ratio stays below 1.

Answer from the NTA final answer key. Solution by PrepWiser.