What are the necessary conditions for total internal reflection to take place?

Total internal reflection occurs when light traveling through a medium is completely reflected back into the same medium, rather than being transmitted into another medium. The necessary conditions for total internal reflection to take place are as follows:

Refractive Index Difference: Total internal reflection occurs at the interface between two different media when the refractive index of the medium where the light is coming from is greater than the refractive index of the medium it is approaching. In other words, the incident medium should have a higher refractive index than the refracted medium.

Angle of Incidence: The angle of incidence of the light ray should be greater than the critical angle. The critical angle is defined as the angle of incidence that produces an angle of refraction of 90 degrees (i.e., the refracted ray travels along the boundary between the two media). For total internal reflection to occur, the angle of incidence must be greater than the critical angle.

Mathematically, the critical angle (θc) can be calculated using Snell's Law:

θc = sin^(-1)(n2/n1),

where n1 is the refractive index of the incident medium and n2 is the refractive index of the refracted medium.

Light Propagation: The light must travel from a medium with a higher refractive index to a medium with a lower refractive index. If the light travels from a medium with a lower refractive index to a medium with a higher refractive index, total internal reflection cannot occur.
When these conditions are met, total internal reflection takes place, and all of the incident light is reflected back into the same medium, resulting in no transmission of light across the interface. Total internal reflection is commonly observed in phenomena such as fiber optics, mirages, and prism applications.