Download Fermat_s_Principle_Honors_Project_

Fermat’s Principle
MAC 2312/ German Farias
Roberto Frota
1
Historical Form
• The actual path between two
points taken by a beam of light
is the one which is traversed in
the least time.(Fermat’s
Principle)
• It can be used to describe the
properties of light-rays
reflected off mirrors, refracted
through different media, or
undergoing total internal
reflection.
• It can also be used to derive
Snell's law of refraction and
the law of reflection.
MAC 2312/ German Farias
Roberto Frota
2
Modern Form
The historical form is
incomplete. The modern, full
version of Fermat's Principle
states that the optical path
length must be extremal, which
means that it can be either
minimal, maximal or a point of
inflection (a saddle point).
Minima occur most often, for
instance the angle of refraction
a wave takes when passing
into a different medium or the
path light has when reflected
off of a planar mirror. Maxima
occur in gravitational lensing. A
point of inflection describes the
path light takes when it is
reflected off of an elliptical
mirrored surface.
MAC 2312/ German Farias
Roberto Frota
3
Important Facts
• This principle was first
stated in a letter dated
January 1st, 1662, to
Cureau de la Chambre by
Pierre de Fermat.
• Fermat's principle can be
derived from Quantum
Electrodynamics and thus is
a consequence of Quantum
physics.
• Fermat's principle can be
considered as a
mathematical consequence
of Marpertuis principle.
MAC 2312/ German Farias
Roberto Frota
4
Derivation
How do light rays propagate in the
presence of material bodies?
It does so by reflection and
refraction of rays of light going
through a change of medium.
MAC 2312/ German Farias
Roberto Frota
5
The Equations
MAC 2312/ German Farias
Roberto Frota
6
The Graph
MAC 2312/ German Farias
Roberto Frota
7
Use of Trig Substitution
MAC 2312/ German Farias
Roberto Frota
8
MAC 2312/ German Farias
Roberto Frota
9