## Proof of Snell's Law

When light passes from a medium to an optically denser medium the path of the light is refrated towards the normal. Suppose the speed in medium 1 is
$v_1$
and the speed in medium 2 is
$v_2$
then the time taken to pass from point A to B above is
$t= \frac{d_1}{v_1}+ \frac{d_2}{v_2}= \frac{\sqrt{a^2+x^2}}{v_1} + \frac{\sqrt{b^2+(c-x)^2}}{v_2} = \frac{sin \theta_1}{v_1} + \frac{sin \theta_2}{v_2}$
.
The time is minimised when
$\frac{dt}{dx} = 0$
.
$\frac{dt}{dx}= \frac{x}{v_1 \sqrt{a^2+x^2}} - \frac{c-x}{v_2 \sqrt{b^2 + (c-x)^2}}= \frac{sin \theta_1}{v_1}- \frac{sin \theta_2}{v_2}=0$
.
Hence
$\frac{sin \theta_1}{v_1}= \frac{sin \theta_2}{v_2}$
.
This is one version of Snell's law. Becuase the number of waves passing each point per second - the frequency
$f$
- cannot change, and because
$v= f \lambda$
, where
$\lambda$
is the wavelength, the frequency is proportional to the wavelength so we can write Snell's Law as
$\frac{sin \theta_1}{\lambda_1}= \frac{sin \theta_2}{\lambda_2}$
. 