Question

Prove that ₁n₂ × ₂n₃ × ₃n₁ = 1.

Answer 1

The proof that ₁n₂ × ₂n₃ × ₃n₁ = 1 is given by considering the refractive indices between three media: 1, 2, and 3.

Using Snell’s law:

`""_1n_2 = (sin i_1)/(sin r_1), ""_2n_3 = (sin r_1)/(sin r_2), ""_3n_1 = (sin r_2)/(sin i_1)`

Multiplying these three indices:

`""_1n_2 xx ""_2n_3 xx ""_3n_1 = (sin i_1)/(sin r_1) xx (sin r_1)/(sin r_2) xx (sin r_2)/(sin i_1) = 1`

Hence, ₁n₂ × ₂n₃ × ₃n₁ = 1

This proves that the product of refractive indices in a closed loop equals one.