fermat’s principle
DESCRIPTION
Fermat’s Principle. A derivation of “Snell’s Law of Refraction”. 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”. Fermat’s Principle. - PowerPoint PPT PresentationTRANSCRIPT
Fermat’s Principle
• A derivation of “Snell’s Law of Refraction”
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”
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”
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
Fermat’s PrincipleREFRACTION
Medium “a”
Medium “b”
Fermat’s PrincipleREFRACTION
Time (t) is equal to the distance traveled (r) at a particular velocity (v).
Medium “a”Medium “b”
Or: t = r / v
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/v2Medium “a”
Medium “b”
“P”
“Q”
t = r / v
Fermat’s PrincipleREFRACTION
Given, velocity is:
v = c / n
Medium “a”Medium “b”
“P”
“Q”
t = r1 / v1 + r2 / v2
n1 n2
Fermat’s PrincipleREFRACTION
Then our equation becomes:
t = r1 / (c/n1) + r2 / (c/n2)
“P”
“Q”
t = r1 / v1 + r2 / v2
n1 n2
v = c / n
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
“P”
“Q”
t = r1 / v1 + r2 / v2
n1 n2
v = c / n
Fermat’s PrincipleREFRACTION
The distances r1 and r2 can be found by simple trigonometry.
“P”
“Q”
t = (n1/c) * r1 + (n2/c) * r2 d
x
d - x
n1 n2
a
b
Fermat’s PrincipleREFRACTION
The distance the ray travels is therefore the hypotenuse of two triangles.
“P”
“Q”
t = (n1/c) * r1 + (n2/c) * r2 d
x
d - x
n1 n2
a
b
a2 + x2 + b2 + (d – x )2
Fermat’s PrincipleREFRACTION
We assign “theta’s” for the angles between the rays and the normals to the surface.
“P”
“Q”
t = (n1/c) * r1 + (n2/c) * r2 d
x
d - x
n1 n2
a
b
a2 + x2 + b2 + (d – x )2
θ2
θ1
Fermat’s PrincipleREFRACTION
Putting the two equations together, and differentiating it with respects to time yields:
“P”
“Q”
t = (n1/c) * r1 + (n2/c) * r2 d
x
d - x
n1 n2
a
b
a2 + x2 + b2 + (d – x )2
Fermat’s PrincipleREFRACTION
dt n1 d n2 d --- = --- --- a2 + x2 + --- --- b2 + (d – x )2
dx c dx c dx
t = (n1/c) * r1 + (n2/c) * r2
a2 + x2 + b2 + (d – x )2
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:
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:
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:
Fermat’s PrincipleREFRACTION
“P”
“Q”
d
x
d - x
n1 n2
a
b
n1 x n2 (d – x) -------------------- & ------------------------- c (a2 + x2)1/2 c [b2 + (d – x )2]1/2
θ1
θ2
Sin θ1
oppositehypotenuse
Sin θ2
Fermat’s PrincipleREFRACTION
n1 Sin θ2 n2 Sin θ2 = 0
Simplifying the equations yields
Snell’s Equation for the Law of Refraction
-or -
n1 Sin θ2 = n2 Sin θ2
Fermat’s Principle
Your assignment is to derive Snell’s Law of Reflection the same way as I did here.
REFLECTION
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
• E N D
Fermat’s PrincipleREFRACTION
Medium “a” Medium “b”
“P”
“Q”