Show the refraction of light wave at a plane interface using Huygens' principle and prove Snell's law.

Consider the triangles ABC and AEC: $\displaystyle \sin i=\frac{B C}{A C}=\frac{v_{1} \tau}{A C}$ ................(i) $\displaystyle \sin r=\frac{A E}{A C}=\frac{v_{2} \tau}{A C}$ ..............(ii) Divide eq. (i) & (ii): $\displaystyle \frac{\sin i}{\sin r}=\frac{v_{1}}{v_{2}}$ ........................(iii) Refractive index of medium $\displaystyle 1$, $\displaystyle n_{1}=\frac{c}{v_{1}}$ ................(iv) Refractive index of medium $\displaystyle 2$, $\displaystyle n_{2}=\frac{c}{v_{2}}$ ................(v) From eq. (iii), (iv) & (v): $\displaystyle \frac{n_{2}}{n_{1}}=\frac{\sin i}{\sin r}$