Question

Using Huygen's principle for refraction of a plane wave, arrive at Snell's law of refraction.

Answer 1

Let $XY$ represent the surface separating the medium I and medium II.
Let $v_1$ and $v_2$ be the speed of light in medium I and medium II respectively.
Consider a plane wave front $AB$ incident in medium I at angle i.
Let the secondary wavelet from $A$ will travel a distance $v_{2}t$ as radius draw an arc in the medium II in the same time $t$. Therefore with $A$ as centre and $v_{2}t$ as radius draw an arc in the medium II. The tangent from $C$ touches the arc at $D$.
Let $r$ be the angle of refraction
From ${△}^{le}ABC,\sin i=\frac{BC}{AC}=\frac{v_{1}t}{AC}$
${△}^{le}ACD,\sin r=\frac{AD}{AC}=\frac{v_{2}t}{AC}$
$\frac{\sin i}{\sin r}=\frac{v_{1}t}{AC}\times \frac{AC}{v_{2}t}=\frac{v_1}{v_2}$
If $c$ represents the speed of light in vacuum then,
$n_1=\frac{c}{v_1}$ and $n_2=\frac{c}{v_2}$
$\frac{n_2}{n_1}=\frac{c}{v_2}\times c\frac{v_1}{c}=\frac{v_1}{v_2}$
$\therefore \frac{\sin i}{\sin r}=\frac{n_2}{n_1}$

$n_1\sin i=n_2\sin r$
This is the Snell's law of refraction