At what angle should a ray of light be incident on the face of a prism of refracting angle 60° so that it just suffers total internal reflection at the other face? The refractive index of the material of the prism is 1.524.
The ray should be incident at an angle of approximately 30° on the face of the prism.
Explanation
This problem involves the condition for total internal reflection at the second face of a prism. The key concept is that for total internal reflection to just occur, the angle of incidence at the second face must equal the critical angle. Using the prism geometry relation (r₁ + r₂ = A) and Snell's law at the first face, we can determine the required angle of incidence. The critical angle is derived from the relation sin(ic) = 1/μ, where μ is the refractive index of the prism material.
Solution Steps
Step 1: Calculate the critical angle for total internal reflection. sin(ic) = 1/μ = 1/1.524 = 0.6562. Therefore, ic = sin⁻¹(0.6562) 41.04°
Step 2: For just total internal reflection at the second face, the angle of incidence at that face equals the critical angle, i.e., r₂ = ic = 41.04°
Step 3: Using the prism geometry relation r₁ + r₂ = A, we get r₁ = A - r₂ = 60° - 41.04° = 18.96°
Step 4: Apply Snell's law at the first face: sin(i) = μ × sin(r₁) = 1.524 × sin(18.96°) = = 0.4952
Step 5: Calculate the angle of incidence: i = sin⁻¹(0.4952) 29.7° 30°