Question

Question related to the angle of refraction?

Answer 1

There is no exit ray from the second piece or associated refractive angle because the Critical Angle for the interface has been exceeded.

Explanation:

Here we have a case of serial refractions. We need to calculate the angle of refraction in the top piece to find the angle of incidence for the second piece. Then we can calculate the angle of refraction from the second piece. Snells’ Law states that `n_i * sin(Theta_i) = n_r * sin(Theta_r)`

where Θi ("theta i") = angle of incidence = 55
Θr ("theta r") = angle of refraction = Need both `r_1 and r_2` here.
`n_i` = index of refraction of the incident medium = 1.91 for first piece, 1.00 for air - top
`n_r` = index of refraction of the refractive medium = 1.91 for the first piece, 1.50 for the second piece, 1.00 for air - bottom

1. FIRST PIECE
   `1.00 * sin(55) = 1.91 * sin(Theta_(r1))`
   `sin(Theta_(r1)) = (sin(55)) /1.91`
   `sin(Theta_(r1)) = 0.429`
   `Theta_(r1) = 25.4`

2. SECOND PIECE
   `Theta_i = 90 – Theta_(r1) = 64.5`
   `1.91 * sin(64.5) = 1.50 * sin(Theta_(r2))`
   `sin(Theta_(r2)) = sin(64.5)*1.91/1.50`
   `sin(Theta_(r2)) = 0.903*1.91/1.50`
   `sin(Theta_(r2)) = 1.15`

In this case we have an internally reflected ray
`sin(Theta_(r2)) > 1`

So, there is no exit ray from the second piece or associated refractive angle.

The critical angle is the first angle for which the incident ray does not leave the first region, namely when the "refracted" angle is `90^o`. Any incident angle greater than the critical angle will consequently be reflected from the boundary instead of being refracted.
In this example to find the critical angle, we set it at `90^o`.

Using Snell's Law, `n_i/n_r * sin(Theta_i) = sin(90)` we see that any incident angle greater than about `51^o` will not leave the first piece into the second piece.
`n_i/n_r * sin(Theta_i) = sin(90)` ; `1.91/1.50 * sin(Theta_i) = 1` ; `sin(Theta_i) = 1.50/1.91 = 0.785`
`(Theta_i) = 51.2o`
https://www.math.ubc.ca/~cass/courses/m309-01a/chu/Fundamentals/snell.htm