Fermat's Principle
Before we embark upon an analysis of reflection and refraction from the point of view of
scattering light waves, it is worth exploring an alternative explanation for the propagation of
light. Fermat's principle is a variational principle which states that:
The path taken by light going between any two points is the one that is traversed in the least
time.
Indeed, by considering all possible paths for a light ray and choosing one which takes the least
time, it is possible to determine how a light ray will move. Consider a situation where a particle
moves from one medium to another.
Figure %: Fermat's principle applied to refraction.
If the point at which the light crosses the boundary is a distance
x
from the origin, and the speeds
in the media are
v
A
and
v
B
respectively, then the time taken by the light is:
t
=
+
Minimizing time with respect to
x
:
=
+
= 0
Rearranging this we find:
=
which is the
law of refraction
. In general, paths of minimum time are those paths which deviate
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 Fall '10
 DavidJudd
 Physics, Light, Reflection And Refraction, Geometrical optics, Fermat's principle, adjacent paths

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