snellslaw

[tiffeney48]

what is snells law?

:help, what is snells law?

[Darryl Meister]

Snell's law is a mathematical relationship that describes how much a ray of light is refracted -- or deflected -- as it passes from medium into another (such as from air into a lens material). Specifically, it looks like this:

n1 * sin i1 = n2 * sin i2

Where n1 is the refractive index of the initial medium, n2 is the refractive index of the medium the light is passing into, i1 is the angle at which the light is striking the new medium, and i2 is the new angle of the light as it crosses the boundary between the two media.

Best regards,
Darryl

[Chris Ryser]

tiffeney48 said:
:help, what is snells law?

I found the following easy to understand explanation:

I found an article in Mathematics Teacher that gives a new way to look at the classic Boat- Ambulance problem that we have all run into at several points in our careers as math students. The problem is this:

You are captain of a ship, and one of your passengers has been injured. Your ship is 30 miles from a point that is 60 miles down shore from a hospital. You must order an ambulance to meet you ship at any point along a road that runs parallel to the shoreline. The ambulance starts at the hospital. You would like to meet the ambulance at a point that will get your passenger to the hospital in the shortest possible time. Suppose that you boat travels at a rate of 20 MPH and that the ambulance travels at a rate of 50 MPH. Describe the location of the point where the ambulance should be met to minimize the passenger¡¯s travel time to the hospital.

The article is meant to be for students who have not yet taken calculus. So it goes through the process of discovering the smallest amount of miles to travel using a table. Finally the students will see when they graph the table they created there is a minimum of the function, and that that minimum is the answer to their problem. Finding by tracing the graph they can find the coordinates that will describe the point.

The authors also add a very insightful method for discovering the answer to this problem using physics. For the purposes of this paper we will not discuss the intuitive thought process described above any more. We will be discussing the process of finding the answer using the method from physics that we will check using calculus.



Information about the Article:

The Boat-and-Ambulance Problem revisited by Michael Helfgott and P. Michael Lutz, Mathematics Teacher, April 2002, pp. 270-274.

Mathematical Concepts Presented in the Article:

The innovative mathematical concepts presented in this article are likely geared toward a calculus one student or possibly a pre-calculus student. The article uses calculus to determine the answer to the boat-ambulance problem given above.

Using the formula d=rt and the Pythagorean theorem we can come up the function:



f(t) = ¡Ì(30^2 + t^2) + (60 ¨C t)/50



Using calculus and finding the derivative of f(t) and setting it equal to zero to solve for t, we find the answer to the infamous boat-ambulance problem is (x=13.09miles, y=2.57hr).



Innovative Part:

The authors explain how to use the physics concept of Snells¡¯s Law of refraction of light:

sin(a)/sin(b) = v1/v2



where v1is the velocity of light in medium 1 and v2 is the velocity of light in medium2.

The authors explain that if Snell¡¯s law is true then a ray of light chooses the path of least time in going from a point P to a point Q. The converse of Snell¡¯s law is true. If it is accepted that whenever a ray of light goes from point P in medium 1 to point Q in medium 2, it does it in the minimum amount of time. We can relate the boat-ambulance problem to Snell¡¯s law and the law of minimum time.

In the boat ambulance-problem, ¦Â = 90 degrees.



sin(a)/sin(90) = 20/50 ¡ú sin(a) = 2/5 = 0.4 ¡ú sin(a) = x¡Ì(30^2 + x^2) = 0.4 ¡ú x^2(900 + x^2) = 0.16 ¡ú x = 13.09



We get the same x value as we did when we used our familiar calculus methods! Therefore, the innovative part of this lesson is that the authors give us an alternative approach to solving an ordinary calculus problem.



Technology Implementation:

The physics approach is a really neat way to get students to use their intuitive thought processes to reason their way through the process used. Before the authors got to the innovative part of the math content. They discussed another innovative aspect to solving this problem, but it seems less advanced. This technology would be useful for lower level math students. The authors suggested that the students use their calculators to create two lists from a table of miles (x) and time (y). Have the students graph these lists. From the graph the students can see the minimum of the function and therefore find the solution. This is a good technology to use but it seems less interesting than the physics application.

[tiffeney48]

Thank you so much for taking the time to tell me what snells law is. Now I have a better understanding!! I'm taking my ABO this next weekend. I appreiciate your help.

[Darryl Meister]

I found the following easy to understand explanation:

It's interesting to note that this example is actually based upon Fermat's principle of least time.

Best regards,
Darryl

[Optom]

Hello,

Thomas Harriot discovered laws of refraction 19 years before Willebrord Snell gave his name to it, according to Hans Baeyer. Read an article appearing on page 10 of Dispensing Optics Journal, titled "Rainbow's Bend".

Link is:http://www.abdo.org.uk/magazine/17.pdf

Regards,
Optom

[Darryl Meister]

Thomas Harriot discovered laws of refraction 19 years before Willebrord Snell gave his name to it, according to Hans Baeyer.

And the French know it as Descarte's law, though the other two described the principle before Descarte published his work on the matter... ;)

Best regards,
Darryl

[Optom]

Salut!

vous avez raison, mais c'est la vie:D

a bientot,
Optom