A prism of refractive index n₁ and another prism of refractive index n₂ are stuck together (as shown in the figure). n₁ and n₂ depend on λ, the wavelength of light, according to the relation
n₁ = 1.2 + (10.8 × 10⁻¹⁴)/(λ ²) and n₂ = 1.45 + (1.8 × 10⁻¹⁴)/(λ ²)
The wavelength for which rays incident at any angle on the interface BC pass through without bending at that interface will be _____________ nm.

Solution

Physics sir · 4 steps

No bending at BC for every angle means the two glasses must have the same refractive index there.

1. Condition at BC. Snell's law at BC gives n₁ sin θ₁ = n₂ sin θ₂. For θ₂ = θ₁ at every angle of incidence, we need n₁ = n₂ at that wavelength.

   1.2 + (10.8 × 10⁻¹⁴)/λ² = 1.45 + (1.8 × 10⁻¹⁴)/λ²

2. Solve for lambda. Collect terms: (9 × 10⁻¹⁴)/λ² = 0.25, so λ² = 36 × 10⁻¹⁴ m² and λ = 6 × 10⁻⁷ m.
3. Quick check. At this wavelength 10⁻¹⁴/λ² = 1/36. That gives n₁ = 1.2 + 0.3 = 1.5 and n₂ = 1.45 + 0.05 = 1.5, so the two indices match.
4. So. λ = 6 × 10⁻⁷ m = 600 nm.

The idea to keep. Light crosses a boundary without bending at every angle only when both sides have the same refractive index. The prism angles do not matter at all here.

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