Question

A transparent container contains layers of three immiscible transparent liquids A, B and C of refractive indices n, `3/4` n and `2/3` n, respectively. A laser beam is incident at the interface between A and B at an angle θ as shown in figure. Prove that the beam does not enter region C at all for sin θ > ≥ `2/3`.

Answer 1

Using Snell’s law:

n_A sin θ = n_B sin r

n sin θ = `(3 n)/4` sin r

sin r = `4/3` sin θ

For the beam to enter C:

sin r ≤ sin i_c    ...(i)

Critical angle for B-C:

sin i_c = `n_C/n_B`

= `((2n//3))/((3n//4))`

= `8/9`

Total internal reflection at B-C requires:

sin r ≥ `8/9`    ...(ii)

Substitute sin r = `4/3 sin theta` in equation (ii),

`4/3 sin theta ge 8/9`

sin θ ≥ `2/3`

If sin θ ≥ `2/3`, the beam undergoes total internal reflection at the B-C interface and never enters region C.