Explain the phenomenon of total internal reflection. What are necessary conditions to perform total internal reflection? Derive the formula for the refractive index of a medium in terms of the critical angle.

Total Internal Reflection (TIR) is a phenomenon in optics that occurs when a light wave traveling through a medium hits the boundary with another medium at an angle larger than a critical angle and is completely reflected back into the original medium. This means that no refraction occurs, and all the light is reflected.

Necessary Conditions for Total Internal Reflection:

  1. The light must travel from a medium with a higher refractive index to a medium with a lower refractive index. For instance, light traveling from water (refractive index of about 1.33) to air (refractive index of about 1.00) will exhibit TIR, but not the reverse.

  2. The angle of incidence must be greater than the critical angle. The critical angle is the minimum angle of incidence at which total internal reflection occurs. If the angle of incidence exceeds this critical angle, the light will be totally reflected.

  3. The refractive index of the first medium (where the light originates) must be greater than the refractive index of the second medium (the medium into which the light would refract).

Derivation of the Formula for the Refractive Index in Terms of the Critical Angle:

To derive the relationship between the refractive index of the two media and the critical angle, let’s start with Snell’s Law, which relates the angles of incidence and refraction to the refractive indices of the two media:

n1sin⁡θ1=n2sin⁡θ2

Where:

  • n1 is the refractive index of the first medium (where the light is coming from),
  • n2 is the refractive index of the second medium (where the light would normally refract),
  • θ1 is the angle of incidence,
  • θ2 is the angle of refraction.

When TIR occurs:

At the critical angle θc, the light is refracted along the boundary between the two media, meaning the angle of refraction is 90∘. So:

θ2=90∘

At this point, Snell’s Law becomes:

n1sin⁡θc=n2sin⁡90∘

Since sin⁡90∘=1, we can simplify the equation:

n1sin⁡θc=n2

Now, solving for the refractive index of the first medium in terms of the critical angle:

n1=n2/sin⁡θc

Final Formula:

Thus, the refractive index of the medium where the light is coming from (medium 1) is related to the refractive index of the second medium and the critical angle by the following formula:

n1=n2/sin⁡θc

This equation gives the refractive index of the first medium in terms of the critical angle and the refractive index of the second medium.

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