Animated Solution for Physics - Optics: A green light is incident from the water to the air-water interface at the critical angle ($\theta$). Select the correct statement.

The phenomenon of Total Internal Reflection (TIR) is one of the most beautiful consequences of Snell's Law. But what happens when we introduce the concept of dispersion into the mix? Let's dive into this fascinating problem where a green light ray is perfectly balanced at the edge of two worlds.

Green Light at the Edge

Imagine a beam of green light traveling through water and striking the boundary with air. We are told that it hits the interface exactly at its critical angle ($\theta =C_g$). At this precise angle, the refracted ray grazes the surface of the water, making an angle of $90^{\circ }$ with the normal.

The critical angle is governed by the refractive index of the medium, given by the relation:

This means that the critical angle is inversely proportional to the refractive index. But here is the catch: the refractive index is not a universal constant for all colors!

To understand what happens to other colors, we must invoke Cauchy's formula, which beautifully connects the refractive index of a material to the wavelength of light passing through it:

This equation tells us that light with a longer wavelength ($\lambda$) experiences a lower refractive index ($\mu$).

Now, let's think about the spectrum of visible light. The frequency of light ($ u$) is inversely proportional to its wavelength ($\lambda = \frac{c}{ u}$). Therefore, colors with a frequency less than that of green light—such as yellow, orange, and red—have a longer wavelength.

Because these lower-frequency colors have a longer wavelength ($\lambda >{\lambda }_g$), the water offers them a lower refractive index ($\mu <{\mu }_g$).

According to our critical angle formula, a lower refractive index mathematically results in a larger critical angle ($C>C_g$).

Now, let's bring it all together. The incident angle $\theta$ is fixed for all colors in this beam, and it is exactly equal to the critical angle of green light ($\theta =C_g$). For the lower-frequency colors (like red), their critical angle is now greater than the incident angle ($\theta <C_{\text{red}}$).

Since the incident angle is no longer large enough to cause Total Internal Reflection for these colors, they will cross the boundary and refract out into the air medium.

Conversely, if we look at higher-frequency colors like blue or violet, their shorter wavelengths mean a higher refractive index and a smaller critical angle. For them, the incident angle exceeds their critical angle ($\theta >C_{\text{blue}}$), and they will be trapped inside the water due to Total Internal Reflection!