DETERMINING THE CRITICAL ANGLE

Refer again to Fig. 1. The light passes from a medium having a relatively higher index of refraction r into a medium having a relatively lower index s. Therefore, r  s. As angle x increases, angle y approaches 90°, and ray S gets closer to the boundary plane B. When x, the angle of incidence, gets large enough (somewhere between 0° and 90°), angle y reaches 90°, and ray S lies exactly in plane B. If angle x increases even more, ray R undergoes total internal reflection at the boundary plane B. Then the boundary acts like a mirror.
The critical angle is the largest angle of incidence that ray R can subtend relative to the normal N without being reflected internally. Let’s call this angle x_c. The measure of the critical angle is the arcsine of the ratio of the indices of refraction:

x_c = sin^-1 (s/r)

Fig. 1. A ray passing from a medium with a relatively higher refractive index to a medium with a relatively lower refractive index.