Question

A plane wavefront propagating in a medium of refractive index 'μ₁' is incident on a plane surface making the angle of incidence 'i' as shown in the figure. It enters into a medium of refractive index 'μ₂' (μ₂ > μ₁). Use Huygens' construction of secondary wavelets to trace the propagation of the refracted wavefront. Hence verify Snell's law of refraction.

Answer 1

Let the plane wavefront AB be incident on the plane surface XY at an angle of incidence i.

Then, from the figure, we have

∠BAA' = ∠i

According to Huygens's principle, every point on AB is a source of secondary wavelets. 

Let the secondary wavelet from B strike XY at A' in t seconds. Then,

B​A' = c₁t     .....(i)

Here, c₁ is the velocity of light in the rarer medium.

The secondary wavelet from A travels in the denser medium and covers a distance AB'.

Then,

AB' = ​c₂t     .....(ii)

Thus, A'B' will be the secondary wavefront after t seconds. This would later advance to rays 1', 2', and 3', which are the corresponding refracted rays perpendicular to A'B​'.

We can also show that the secondary wavelets starting from any point D on the incident wavefront AB after refraction at P will reach point D' on A'B' at the same time in which the secondary wavelets from B reach A'. Therefore, A'B' is the true refracted wavefront with the angle of refraction r.

So, we have

∠A A'B' = ∠r

Now in  ΔA A'B'

`sini = (BA^')/(A A^') = (c_1t)/(A A^')`

In ΔAA'B

`sinr = (BA^')/(A A^') = (c_2t)/(A A^')`

`therefore (sini)/(sinr) = (c"/"mu_1)/(c"/"mu_2) = mu_2/mu_1`

⇒ `mu_1 sini = mu_2 sinr`

Hence, snell's law is verified.