Derive the laws of refraction from the concept (Huygen's principle) of the wavefront.

Explanation

According to Huygen.s principle of wavefront is as follow.

Let PP. represent the surface separating medium-1 and medium-2, as shown in figure.
And `v_(1)`, and `v_(2)` represent the speed of light in medium-l and medium-2 respectively and `v_(2)ltv_(1)`.
And a plane wavefront AB propagating in the direction AA. incident on the interface of two medium at an angle i.
Let `tau` be the time taken by the wavefront to travel the distance BC.
`:.BC=v_(1)tau` In order to determine the shape of the refracted wavefront, draw a sphere of radius `v_(2)tau` from the point A in the second medium (the speed of the wave in the second medium is `v_(2)` and the distance covered in time `tau` is `v_(2)t`.)
Let CE represent a tangent plane drawn from the point C on the sphere. Then `AE=v_(2)tau` and CE would represent the refracted wavefront.
In `DeltaABCandDeltaAEC,`
`sini=(BC)/(AC)=(v_(1)tau)/(AC)".........(1)`
and `sini=(AE)/(AC)=(v_(2)tau)/(AC)".........(2)`
where i and r are the angles of incidence and refraction respectively.
Taking ratio of equation (1) and (2),
`(sini)/(sinr)=(v_(1))/(v_(2))" "......(3)`
From this equation, we get the important result that if `rlti` (i.e., if the ray bends toward the normally), then
`(sini)/(sinr)gt`
[`:.i` is increasing function in first quadrant]
`:.(v_(1))/(v_(2))gt1impliesv_(1)gtv_(2)`
the medium-1. This predication is opposite to the prediction from the corpuscular model of light but predication is as according to wave theory.
Suppose the speed of light in vacuum is c, absolute refractive index `n_(1)=(c)/(v_(1))` where the speed in medium-1 is `v_(1)` and `n_(2)=(c)/(v_(2))` where the speed in medium-2 is `v_(2)`.
`:.(n_(2))/(n_(1))=(v_(1))/(v_(2))" "......(4)`
From equation (3) and (4),
`(sini)/(sinr)=(v_(1))/(v_(2))=(n_(2))/(n_(1))`
`:.n_(1)sini=n_(2)sinr`
This is the Snell.s law of refraction.
For more information : Speed,
`v=(lamda)/(t)`
`:.vt=lamda`
`:.v_(1)tau=lamda_(1)andv_(2)tau=lamda_(2)`
`:.(lamda_(1))/(lamda_(2))=(v_(1))/(v_(2))`
or `(v_(1))/(lamda_(1))=(v_(2))/(lamda_(2))`
This relationship shows that when the wave refracted in denser medium `(v_(1)gtv_(2))`, its wavelength and speed decreases but the its wavelength `lamda` and speed decreases but the frequency remains constant.