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Question
Using Huygen’s wave theory of light, prove Snell’s law of refraction of light.
Solution
Consider a plane wavefront AB incident on a surface PQ separating two medium 1 and 2. The medium 1 is a rarer medium of refractive index n1 in which light travels with a velocity c1. The medium 2 is a denser medium of refractive index n2 in which light travels with a velocity c2.
The angle between the incident ray FA and the normal NA at the point of incidence A is equal to i.
The angle is also equal to the angle between the incident plane wavefront AB.
Similarly, the angle between the refracted wavefront and the surface of separation PQ is equal to the angle of refraction r.
i.e., ∠ADC = r
Consider the triangles ΔBAD, ΔACD figure below.
sin i = sin ∠BAD = `"BD"/"AD" = ("c"_1"t")/"AD"`
sin r = sin ∠ADC = `"AC"/"AD" = ("c"_2"t")/"AD"`
`(sin "i")/(sin "r") = ("c"_1 "t")/("c"_2 "t") = "c"_1/"c"_2` = constant
This constant is called the refractive index of the second medium (2) with respect to the first medium (1).
`"c"_1/"c"_2 = "n"_2/"n"_1` = 1n2
This equation proves the Snell's law.
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