fermat’s principle

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”