Question

Prove laws of refraction using Huygens’ principle.

Answer 1

Law of refraction

1. Let us consider a parallel beam of light is incident on a refracting plane surface XY such as a glass surface.
2. The incident wavefront AB is in rarer medium (1) and the refracted wavefront ” AB is in denser medium (2).
3. These wavefronts are perpendicular to the incident rays L, M and refracted rays L’, M’ respectively.
4. By the time point A of the incident wavefront touches the refracting surface, point B is yet to travel a distance BB’ to touch the refracting surface at B’.
5. When point B falls on the refracting surface at, B, point A would have reached A’ in the other medium.
6. The time taken t for the ray to travel from B to B is the same as the time taken for the ray to travel from A to A’.
   t = `"BB'"/"v"_1 = "AA'"/"v"_2` (or)
   t = `"BB'"/"AA'" = "v"_1/"v"_2`
   1. The incident rays, the refracted rays and the normal are in the same plane.
   2. Angle of incidence,
      i = ∠ NAL = 90° – ∠NAB = ∠ BAB’
      Angle of refraction,
      r = ∠N’B’M’= 90° – ∠N’B’A’ = ∠ A’B’A
7. For the two right angle triangle ABB’ and B’A’A,
   `(sin "i")/(sin "r") = ("BB'"//"AB'")/("AA'"//"AB'") = "BB'"/"AA'" = "v"_1/"v"_2 = ("c"//"v"_2)/("c"//"v"_1)`
8. Here, c is speed of light in vacuum. The ratio c/υ is the constant, called refractive index of the medium.
   `(sin "i")/(sin "r") = "n"_2/"n"_1`
   In product form,
   n₁ sin i = n₂ sin r
   Hence, the laws of refraction are proved.