Rainbow Formation: Dispersion and Total Internal Reflection

Introduction
Rainbows are beautiful natural phenomena that occur when sunlight is refracted, dispersed, and reflected within raindrops. This leads to the formation of a circular spectrum of colors in the sky. The process of rainbow formation involves two main optical phenomena: dispersion and total internal reflection.

Dispersion
Dispersion is the phenomenon in which light is separated into its component colors when it passes through a prism or a medium with varying refractive index. In the case of a rainbow, dispersion occurs when sunlight enters a raindrop and is refracted. The different wavelengths (colors) of light bend at slightly different angles due to their varying speeds in the water droplet.

Refractive Index and Dispersion
The refractive index of water is wavelength-dependent, which means that different colors of light have different speeds and angles of refraction. This causes the white light from the sun to split into its constituent colors as it enters the raindrop. The shortest wavelength (violet) is refracted the most, while the longest wavelength (red) is refracted the least.

Formation of Primary Rainbow
1. Refraction: When sunlight enters a raindrop, it undergoes refraction, bending as it passes from air to water and then back to air.
2. Dispersion: As the light refracts, it is dispersed into its component colors due to the varying refractive index of water.
3. Reflection: Once inside the raindrop, the dispersed light undergoes multiple internal reflections from the inner surface of the raindrop.
4. Total Internal Reflection: For a specific range of angles, called the critical angle, the light undergoes total internal reflection and exits the raindrop. This is because the angle of incidence exceeds the critical angle, causing the light to reflect back into the water droplet.

Circular Rainbow
The combination of dispersion and total internal reflection results in the formation of a circular rainbow. Each raindrop acts as a tiny prism, dispersing and reflecting light back towards the observer. The observer sees the dispersed light in the form of a circular arc, with red on the outer edge and violet on the inner edge.

Conclusion
In conclusion, rainbow formation is a result of the combined optical phenomena of dispersion and total internal reflection. When sunlight enters raindrops, it is refracted, dispersed into its component colors, and then reflected internally before exiting the droplets. This process creates the beautiful circular spectrum of colors that we observe as a rainbow.