A ray of light incident at an angle in a rectangular prism grazes the surface of the prism on emerging from…

Analyzing the question

A ray enters one face of a rectangular (right-angled) prism at angle θ, travels inside, and meets an adjacent face at 90° to the first. "Grazing the surface on emerging" means the emergent ray travels along the surface, so the angle of refraction there is 90°. That means the angle of incidence at the second face is the critical angle, c.

Concept

• Critical angle: sin c = 1/n = 1/1.5 = 0.6667, so c = 41.8°
• Snell's law at the first face: sin θ = n sin r
• In a rectangular block, the two faces are perpendicular, so the refraction angle r at the first face and the incidence angle at the second face add up to 90°: r + c = 90°.

Solution

1. c = sin⁻¹(0.6667) = 41.8°
2. r = 90° − 41.8° = 48.2°
3. sin θ = 1.5 × sin 48.2° = 1.5 × 0.7454 = 1.118

Since sin θ cannot exceed 1, no real angle θ satisfies this. A ray cannot graze the second face if it enters the first face from air at any real angle. So, taken strictly, the question is flawed.

What the question intends

The question asks for "the angle" at which the ray grazes the surface on emerging. The key quantity is the angle of incidence at the emerging face, which is the critical angle, 41.8°.

My working agrees with the key only in that sense: the angle at the emerging face is 41.8°. If θ is literally the angle of incidence at the first face from air, no valid value exists, so the question is poorly worded. Option A is still the intended answer.

Tip

"Grazes the surface on emerging" means the angle of refraction is 90°, so the angle of incidence inside is the critical angle. Use sin c = 1/n. For n = 1.5, remember c ≈ 41.8°.