Prove laws of refraction using Hugyen's principle.

Explanation

Let us consider a parallel beam of light is incident on a refracting plane surface XY such as a glass surface. The incident wavefront AB is in rarer medium (1) and the refracted wavefront A.B. is in denser medium (2). These wavefront are perpendicular to the incident rays L, M and refracted rays L.,M. respectively. By the time the point A of the incident wavefront touches the refracting surface, the point B is yet to travel a distance BB. to touch the refracting surface B..
The two normals A and N. are considered at the points where the rays L and M fall on the refracting surface. As refraction happens from rarer medium (1) to denser medium (2), the speed of light is `v_(1) and v_(2)` before and after refraction and `v_(1)` is greater than `v_(2) (v_(1) gt v_(2))`. But 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))=(A A.)/(v_(2))(or)(BB.)/(A A.)=(v_(1))/(v_(2))`
(i) The incident rays, the refracted rays and the normal are in the same plane.
(ii) Angle of incidence,
`i = angle NAL = 90^(@) - angle NAB = angle BAB.`
Angle of refraction,
`r = angle N.B.M. = 90^(@) - angle N.B.A = angle A.B.A.`
For the two right angle triangles `Delta ABB. and Delta B.A.A`.
`(sini)/(sinr)=(BB.//AB.)/(A A.//AB.)=(BB.)/(A A.) = (v_(1))/(v_(2))=(c//v_(2))/(c//v_(1))`
Here, c is speed of light in vacuum. The ratio c/v is the constant, called refractive index the medium. The refractive index of medium (1) is, `c//v_(1) = n_(1)` and the of medium (2) `c//v_(2) = n_(2)`.
`(sini)/(sinr) = (n_(2))/(n_(1))`
In product form,
`n_(1) sin i = n_(2) sinr`
Hence, the laws of refraction are proved.