Refraction - Proof of Snell's Law

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Proving the following:
First a few definitions:
(x1,y1) = point a.  It is in medium A
(x2,y2) = point c.  It is in medium C
(x ,y0) = point b.  It is where Medium A intersects medium C
* Medium A is not necessarily different than medium C
q1 = Angle between line (a,b) relative to normal
q2 = Angle between line (b,c) relative to normal
Now to Prove:
To prove this, we have to first show that to travel from a to b is a straight line, however simple this might seem, it is quite a lengthy formal proof, so lets just assume that the quickest way to get from point a to b is a straight line.  Then we will show that the quickest way to get from a to c via b is Snell's Law.
1) The Total Time to get to c is:
2) Which does not help much, so lets expand this into an integral to get something more exact.
3) Where S is the distance and V the velocity, and obviously, distance over velocity is time.  Then:
4) Now lets get the change in distance in terms of the points a, b and c:
5) Since we want the minimum time, we have to take the derivative of t as a function of x and set it equal to 0:
6) Then we bring the term multiplied by 1 over V2 to the other side to make it easier to work with:
7) This may look complicated, but with some simple trigonometry, the big mess in the brackets simplify:
8) Very nice, lets plug this back into part 6 and see what happens:
9) How convenient, now lets multiply both sides by the speed of light, it will not change the result since both side are being multiplied by the same thing, we are using the speed of light constant for ease of use:
10)Lets see what we get with speed of light over velocity.  It is the indices of refraction:
Now plug this back into 9:
Voila, Snell's Law!

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Patrick Blochle - Canisius College, Buffalo, NY