Fermat’s Principle

2. Fermat’s Principle • A derivation of “Snell’s Law of Refraction”

3. Fermat’s Principle A light ray travels through space and passes through an unknown substance with an index of refraction greater than one. Medium “a” Medium “b”

4. Fermat’s Principle Snell’s Law of Refraction states that: “when a light ray travels between two points, its path is the one that requires the least time, or constant time”. Medium “a” Medium “b”

5. Fermat’s Principle Time therefore must be an extremum with respect to small variations in path. (a minimum extrema) Medium “a” Medium “b” For additional information on finding local minimum see: http://mathworld.wolfram.com/LocalExtremum.html

6. Fermat’s PrincipleREFRACTION Medium “a” Medium “b”

7. Fermat’s PrincipleREFRACTION Time (t) is equal to the distance traveled (r) at a particular velocity (v). Or: t = r / v Medium “a” Medium “b”

8. Fermat’s PrincipleREFRACTION Traveling through two different mediums with different velocities, the total time the ray travels from an arbitrary point “P” to another arbitrary point “Q” is: T = r1/v1 + r2/v2 t = r / v “P” “Q” Medium “a” Medium “b”

9. Fermat’s PrincipleREFRACTION Given, velocity is: v = c / n t = r1 / v1 + r2 / v2 “P” n1 n2 “Q” Medium “a” Medium “b”

10. Fermat’s PrincipleREFRACTION Then our equation becomes: t = r1 / (c/n1) + r2 / (c/n2) t = r1 / v1 + r2 / v2 “P” v = c / n n1 n2 “Q”

11. Fermat’s PrincipleREFRACTION Then our equation becomes: t = r1 / (c/n1) + r2 / (c/n2) This can be rewritten as: t = (n1/c) * r1 + (n2/c) * r2 t = r1 / v1 + r2 / v2 v = c / n “P” n1 n2 “Q”

12. Fermat’s PrincipleREFRACTION t = (n1/c) * r1 + (n2/c) * r2 The distances r1 and r2can be found by simple trigonometry. d “P” a d - x x n1 n2 b “Q”

13. Fermat’s PrincipleREFRACTION t = (n1/c) * r1 + (n2/c) * r2 The distance the ray travels is therefore the hypotenuse of two triangles. d “P” a d - x x n1 n2 b a2 + x2 + b2 + (d – x )2 “Q”

14. Fermat’s PrincipleREFRACTION t = (n1/c) * r1 + (n2/c) * r2 We assign “theta’s” for the angles between the rays and the normals to the surface. d “P” θ1 a d - x x n1 n2 θ2 b a2 + x2 + b2 + (d – x )2 “Q”

15. Fermat’s PrincipleREFRACTION t = (n1/c) * r1 + (n2/c) * r2 Putting the two equations together, and differentiating it with respects to time yields: d “P” a d - x x n1 n2 b a2 + x2 + b2 + (d – x )2 “Q”

16. Fermat’s PrincipleREFRACTION t = (n1/c) * r1 + (n2/c) * r2 dt n1 d n2 d --- = --- --- a2 + x2 + --- --- b2 + (d – x )2 dx c dx c dx a2 + x2 + b2 + (d – x )2

17. Fermat’s PrincipleREFRACTION n1 1 2x n2 1 2(d – x)(-1) = --- * --- * ------------- + --- * --- * -------------------- c 2 (a2 + x2)1/2 c 2 [b2 + (d – x )2]1/2 Deriving the equation gives:

18. Fermat’s PrincipleREFRACTION n1x n2(d – x) = ----------------- ------------------------- = 0 c(a2 + x2)1/2 c[b2 + (d – x )2]1/2 Simplifying and setting the equation equal to “0” yields:

19. Fermat’s PrincipleREFRACTION n1x n2(d – x) = ----------------- ------------------------- = 0 c(a2 + x2)1/2 c[b2 + (d – x )2]1/2 Recognizing the Trigonometric function of sines:

20. Fermat’s PrincipleREFRACTION n1 x n2 (d – x) -------------------- & ------------------------- c (a2 + x2)1/2 c [b2 + (d – x )2]1/2 d opposite hypotenuse “P” θ1 a d - x Sin θ1 Sin θ2 x n1 n2 θ2 b “Q”

21. Fermat’s PrincipleREFRACTION n1 Sin θ2n2 Sin θ2= 0 Simplifying the equations yields Snell’s Equation for the Law of Refraction • or - n1 Sin θ2 = n2 Sin θ2

22. Fermat’s Principle REFLECTION Your assignment is to derive Snell’s Law of Reflection the same way as I did here. This is an individual effort – Not a group effort Use the rest of this period to accomplish this. It is worth 10 points. Spread out and get to work

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24. Fermat’s PrincipleREFRACTION “P” Medium “a” Medium “b” “Q”