Laws of Optics

Why study the Laws of Optics

Extra Practice with Pythagorean Theorem Practice with d=rt relationship Study Domain and Range Mathematical Modeling

Even a dog could do it…

Tim Collins is a professor of mathematics at Hope College in Michigan. This is his dog, Elvis. Tim and Elvis study Calculus together.

Elvis studied Calculus?

Tim and Elvis worked together to find the Optimal path from A to B. The optimal path required the least amount of time to travel from point A to point B.

Least Time is not Necessarily the Least Distance

Your parents may have driven from New Port Richey all the way to I 75 to get to Sarasota. A more direct route would be to take US 19. Why drive all the way to I75?

http://www.metas2010.com.ar/img/autovia.jpg

What did Tim do?

Tim stands at the waters edge and throws the ball into the water. Elvis, eager to please, finds the quickest way to get the ball back to his beloved master.

How does Elvis do it?

If Tim stands at point A and throws the ball to point B, Elvis could go from A to B. Or he could go along the legs of the right triangle, from A to C to B.

How does Elvis do it?

Instead, Elvis goes from A to D to B. Elvis knows that this is the quickest path. If Elvis’ faster speed on land is combined with his slower speed in the water, the quickest way for him to get the ball is to start swimming at point D. What a dog!!

Check out Elvis’s DataX represents the distance from the ball to the shore.

Y represents the shoreline distance from were the doge entered the water, to the ball.

A dog that knows Calculus?

Tim should have named his dog Einstein instead of Elvis.

(It has not been confirmed that the real Elvis knew Calculus)

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html

Pierre de Fermat,Another Mathematician

Fermat was a French mathematician who was given a small amount of credit for calculus. Once, he was struck by the plague and was reported dead!! Later, this information was corrected and he continued his love for mathematics

http://www-gap.dcs.st-and.ac.uk/~history/PictDisplay/Fermat.html

Fermat’s Contribution to Optics

Fermat’s Principle states:

The actual path between two points taken by a beam of light is the one which is traversed in the least time.

Look at this example

Fermat’s Principle

Using the formula d=rt, we can solve for the time and compare these two paths.

Look at the straight path marked in red.

tr

t

Fermat’s Principle

There are two regions for the red path. One in the water and one on land. We need to know the intersection on the x axis. Slope of that line is 11/6 so the equation is

y x 11

63

Fermat’s Principle

y x 11

63

Given this equation, we know that there is an x intercept when y=0. So when we solve for x, we find that there is an x intercept at 18/11.

Fermat’s Principle

Now, you can find the distance of each portion of the red path. And given the velocities in the water and on land, you can calculate the time on each part. Add these values and you will have the total straight distance time.

Use the Pythagorean Theorem

Calculate the distance from the point (0,-3) to the x axis.

Did you get 3.41? Now calculate the time using the formula. Did you get 0.43 seconds?

Use the Pythagorean Theorem

Next calculate the time for the trip to the point (6,8)

Do you get 9.11 for the distance? For the time, you should have gotten 1.82

seconds. The total time for the red path is 0.43+1.82

is 2.25 seconds.

Use the Pythagorean Theorem

Now try on your own to calculate the total time needed to take the green path.

You should have found that the time for the green path is 1.52 seconds, total.

This is much faster than the direct path that you recall was 2.25 seconds.

Find a Generic Formula for the green path

Use the same mathematical principles that you did in the numerical example

tx a

Vt

b x c

V

tx a

V

b x c

V

landland

waterwater

totalland water

2 2 2 2

2 2 2 2

( )

( )

Calculus students: Take the derivative of this time function. What does the derivative tell you?

Willebrord van Roijen Snell

This Swiss Mathematican was also known as Snellius. He was a lawyer, but discovered the sin law and calculated an approximation for

to seven decimal places.

Snell’s Law

Snell’s Law can be derived directly from Fermat’s Principle.

Remember that Snell was responsible for the law of sines.

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/refr.html

Snell’s Law

Snell’s Law

n n1 1 2 2sin sin

Snell’s Law is also known as the Law of Refraction.

Speed of Light