Question

Derive Snell's law from Fermat's Principle. Assume that the interface between the two materials is a planer surface.

Answer 1

From figure , which shows that the points A and B are in ywo optically different medium with their refractive index $n_1$ and $n_2$ respectively. Lets the velocity at the two side of the moundary are $v_1$ and $v_2$ respectively. From figure, total time of travel is,

$t=\frac{\sqrt{a^2+x^2}}{v_1}+\frac{\sqrt{b^2+(d-x)^2}}{v_2}=\frac{l_1}{v_1}+\frac{l_2}{v_2}$

By using relation

$t=\frac{n_1l_1+n_2l_2}{c}=\frac{l}{c}$

Where is optical path length. According to fermat principle ,

$\frac{dl}{dx}=0$

hence

$\frac{dl}{dx}=n_1\frac{x}{\sqrt{a^2+x^2}}-n_2\frac{d-x}{\sqrt{b^2+(d-x)^2}}=0\\ \Rightarrow n_1\frac{x}{\sqrt{a^2+x^2}}=n_2\frac{d-x}{\sqrt{b^2+(d-x)^2}}\\ \Rightarrow n_1\sin\theta_1=n_2\sin\theta_2$

Which is Snells law.

18 March 201711:23 91 0