Question

What is the critical angle? Write down its necessary conditions.

Hint:

Total internal reflection is used in optical fibers and diamond brilliance. For water–air, \( C \approx 48.6^\circ \); for glass–air, \( C \approx 42^\circ \).

Answer 1

Step 1: According to Snell’s law, when light travels from one medium to another, the relationship between the angle of incidence \( i \) and the angle of refraction \( r \), as well as the refractive indices \( \mu_1 \) (of the first medium) and \( \mu_2 \) (of the second medium), is given by: \[ \mu_1 \sin i = \mu_2 \sin r. \] This law governs how the direction of a light ray changes as it passes from one optical medium into another. Here, \( \mu_1 \) and \( \mu_2 \) are the absolute refractive indices of the first and second medium respectively.
Step 2: From Snell’s law, it can be observed that as the angle of incidence \( i \) increases (with \( \mu_1>\mu_2 \)), the angle of refraction \( r \) also increases. This is because the sine function increases with the angle in the range \( 0^\circ \) to \( 90^\circ \), and to maintain the equality in Snell's law, \( \sin r \) must increase when \( \sin i \) increases. Therefore, the refracted ray bends further away from the normal as \( i \) increases.
Step 3: There exists a particular angle of incidence for which the angle of refraction becomes \( 90^\circ \). At this condition, the refracted ray travels along the interface between the two media. This specific angle of incidence is known as the critical angle, denoted by \( C \). Setting \( r = 90^\circ \) in Snell’s law, we get: \[ \mu_1 \sin C = \mu_2 \sin 90^\circ = \mu_2. \] \[ \Rightarrow \sin C = \frac{\mu_2}{\mu_1}. \] This equation is valid only when \( \mu_1>\mu_2 \), i.e., when light is travelling from a denser medium to a rarer medium.
Step 4: If the angle of incidence \( i \) is increased beyond the critical angle \( C \) (i.e., \( i>C \)), Snell’s law can no longer be satisfied for any real angle of refraction \( r \), because \( \sin r \) would have to be greater than 1, which is not possible. In this case, the light does not pass into the second medium at all. Instead, it is completely reflected back into the denser medium. This phenomenon is known as total internal reflection. It is characterized by the following two conditions:

The light must travel from a denser medium to a rarer medium (i.e., \( \mu_1>\mu_2 \)).
The angle of incidence must be greater than the critical angle (\( i>C \)).
Total internal reflection is the principle behind many optical devices such as fiber optics and prisms used in binoculars and periscopes.