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.

proof of snell's law

Suppose the speed in medium 1 is  
  and the speed in medium 2 is  
  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  
\[\frac{dt}{dx} = 0\]
\[\frac{dt}{dx}= \frac{x}{v_1 \sqrt{a^2+x^2}} - \frac{c-x}{v_2 \sqrt{b^2 + (c-x)^2}}= \frac{sin \theta_1}{v_1}- \frac{sin \theta_2}{v_2}=0\]
\[\frac{sin \theta_1}{v_1}= \frac{sin \theta_2}{v_2}\]
This is one version of Snell's law. Becuase the number of waves passing each point per second - the frequency  
  - cannot change, and because  
\[v= f \lambda\]
, where  
  is the wavelength, the frequency is proportional to the wavelength so we can write Snell's Law as  
\[\frac{sin \theta_1}{\lambda_1}= \frac{sin \theta_2}{\lambda_2}\]

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