1. Definition of Critical Angle ($i_c$): When light travels from a denser medium (refractive index $n_1$) to a rarer medium (refractive index $n_2$), it bends away from the normal. As the angle of incidence ($i$) increases, the angle of refraction ($r$) also increases.
2. The Condition at $i_c$: The critical angle is defined as the angle of incidence for which the refracted ray grazes the boundary surface between the two media. This means the refracted ray is perpendicular to the normal.
3. Mathematical Proof using Snell's Law: $$n_1 \sin i_c = n_2 \sin r$$
At the critical angle, the light is at the limit of escaping the denser medium. For this to happen, $\sin r$ reaches its maximum value of $1$.
$$\sin r = 1 \implies r = 90^\circ$$
If the angle of incidence exceeds the critical angle, refraction is no longer possible, and the light undergoes total internal reflection.