Proving Fermat's Principle Of Least Time For Reflection At A Plane Surface

 

                                                Sketch for mathematical analysis of situation

Fermat's principle is an intriguing one in physics, first pointed out by Feynman in his lectures.  That is, in all possible paths that light might take in its propagation the one it actually takes requires the shortest time, say as referenced in the above graphic.  Of course, if one simply demands the path for shortest time from point  x 1,a  to  x1, b the astute reader will just say, "Well just go directly across!" But we instead demand the path of light strike the mirror first - and this ends up being a bit more complicated to prove.

Our purpose here is to demonstrate that for a ray going from a to b that the angle of incidence will equal the angle of reflection and in accordance with the principle of least time.  Consider then, from the diagram, a path such that the time from a to b is a minimum, then we have;

t₁ =  ò _{x1, a} ^xo,0     ds/  C     (1st integral for path 1) 

where:   ds   =   Ö (dx² + dy ² )  =    dx Ö [ 1  +  (dy/dx) ² ]

Let:   f  =   1  +  (dy/dx) ²     

è

f  -  y'   ¶ f / ¶ y'    =  const.

t₁ =   1/ c ò _{x1, a} ^xo,0     dx f

Therefore, 

Ö  1  -  y ²    =   y'   (½ )   [y' / Ö (1  +  y'  ²  ) ]  =  const.  = A

è

1  +  y'  ²    - 2 y' ²  =   A ((1  +  y'  ²  )  (½ )

1  +  y' ²    =   1/  A²  =  B

But:   y'   =  dy/dx  =   Ö B  -1 

è

y   =   (Ö B  -1 )  x  + C      (straight line)

Then:   

 For x 1,a  to   x 0,0  to  x1, b  :

t₁ +   t₂    =   

1/ C  [(x 1   - x 0 )²  +   a ²] ^1/2  +  [(x 0   - x 2 )²  +   b ²] - ^1/2  (2 (x 0   - x 2 )

For minimum:  

¶ ( t₁ + t₂ ) / ¶ x 0    =  0

Or:

 ½ [(x 1   - x 0 )²  +   a ²] ^-1/2   (2 (x 1   - x 0) (-1) 

+ ½ [(x 0   - x 2 )²  +   b ²] ^{- 1/2}  (2 (x 0   - x 2 )   =  0   

Or:

(x 1   - x 0 )²/ (x 1   - x 0 )²  +   a ²   =  (x 0   - x 2 )² / (x 0   - x 2 )²  +   b ²

Therefore:   cos² q 1   =    cos² q 2 

q 1   =    q 2      I.e.

q i   =    q r 

The statement of the law of reflection that the angle of incidence ( q i) is equal to the angle of reflection (q r ) is equivalent to the statement that light travels (via the mirror)  from point  x 1,a  to  x1, b    in  the least possible time.   Hence, the principle of least time or Fermat's principle.