My fishtank that I had used in a previous project had been sitting still for a number of weeks. I had created it in the beginning of May and over a month later, the tank had developed some interesting characteristics. The tank had always had various indexes of refraction. So when pointing a laser through it you would always see very nice curves through it.

This was related to the topic of Gradient Indexes of Refraction. This is also known as GRIN medias. They can be easily identified because the optical rays always follow a curved trajectory. This was determined to be a catenary.

The aim of the project was now set out. It is described that the curve of a light ray when passing through various indexes of refraction should be in the shape of a catenary. My plan is to develop methods for analyzing the shape of the curve.

Topics such as these can be related to real-life ideas. There is refraction within the Earth's atmosphere. Atmospheric refraction is greatest for signals near the horizon where they enter at the lowest angle. The apparent altitude of the signal appears to be about 1/2 degree higher than its true height. When the Earth rotates, the object gains altitude and it reduces. At the altitude it is zero. This has an effect on the sun. This means that at equatorial latitudes there are about five extra minutes of daylight. For more information about this go to the website: (http://www.jpl.nasa.gov/basics/bsf6-6.html)

There are certain factors that affect the index of refraction. Some of these changes can also be brough on by a change in temperature, salinity or pressure.

When observing a light ray passing through a medium and seeing this bend for yourself, the question will arise why is this such? Why should the light bend? Refraction occurs at the boundary of two different indexes. The speed of the light is changed. However if the light ray strikes the normal there will not be any bending.

In a prism with non-parallel sides, the displacement is described by the angle of deviation between the ray incident to the prism and the ray emerging from it. When a light ray is incident on a prism it will undergoe refrction at the boundary of face 1. Travelling through it, the ray refracts again exiting the prism at face 2. The angle of deviation is a fuction of the angle of incidence. So to find the angle of deviation a manipulation of the angle of incidence through moving the prism about the original light and looking at the reflection of the light ray through the second face you can mark the point. Through these lengths and the use of the tan function you can calculate this angle of deviation.

The angle of deviation will be a minimum when the angle of incidence for both faces is the same. This is saying that the incident ray and the emerging ray will be equal. The refracted ray will then be parallel to the base.

This can show how only a simple laser can be needed to make measurements instead of a precision spectrometer table and vernier scales. The values have been proven comparable or maybe even better.

1. Fill the prism first with water.

2. At the end of the table set up a wall with graph paper on it for the emerging ray to strike.

3. Using a stable base set up a laser at the far end of the table. It is important that the laser does not move during the experiment.

4. Be sure to mark the initial point of the laser so that the amount of movement of the laser can be identified.

5. Move the prism so that it interacts with the laser. Rotate the prism around the laser until the angle of deviation can be found.

6. Mark the new point of the laser on the wall.

7. Measure out the change of the distances. Using the new point of the wall mark out the path of the incident ray. Then draw out the point where the incident ray and the emerging ray meet. This is the apex angle. For ease, I set up a sheet of graph paper underneath the prism and marked out the points where the laser interacted with the prism on the graph paper and then transformed onto this sheet. Then simply using a ruler I could measure out the different lengths and calculate the angle of deviation. With the angle of deviation, the index of refraction for the solution inside the prism can be found. For this trial, water was used.

8. The formula to use:

n = 1.00028 * sin 1/2(Angle of Deviation)(Angle of Prism)

/ sin 1/2 (Angle of Prism)

For the most accurate value I took into account the index of refraction of air which is 1.00028 (almost 1).

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My Measured Values:

Distance from the wall to the critical angle (denoted as L): 52 cm + 32.1cm = 84.1cm.

Distance on the wall between the original laser point and through the angle of deviation (denoted as X): 24.9cm

Using the tangent law the angle of deviation is calculated to be 0.2859236 radians. Plugging that into the given formula you will get an index of refraction of:

n = 1.33414

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Error Analysis:

All of these measurements were made by hand. So I took into account that I could have made a +/- 1mm variation. Solving through the formulas with this error there is a 0.00137 difference in the index. This shows that this is a very easy and accurate method.

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Comments:

So once you feel confident using this method. It is very easy to analyze changes in a solution. Different substances can now be added to the water to see how much of a change is induced. The substance can also be heated or cooled. There are many different observations that can be made and now speedily calculated.

In the first measurement I already knew what the value of the index of refraction for the water was supposed to be. If I did not calculate a value close to 1.33 I knew that it was an error on my part. So it was through this that I had tried various different methods until I was able to devise the one described earlier. That was the procedure that gave me the most accurate values. Now I was ready to try measuring substances that had an unknown index of refraction. I wanted to create substances of my own and compare how much the index of refraction would change. To create a change in index of refraction, I added slowly different amounts of sugar in two instances. I compared the difference when a measured amount of 100ml of sugar were added and then 325ml of sugar in a final trial.

1. 100ml of sugar added.

X distance = 24.3cm

(The original distance measured for the pure water was 24.9cm)

L distance = 79.9cm.

(The original L distance measured for the pure water was 84.1cm )

Using these two distances the angle of deviation can now be determined with the tan function.

Theta is equal to 0.295342radians. This is compared to the original value of theta which was 0.2859236radians.

This is a difference of .0094184radians. This is a very small difference so already I know that the index of refraction will not be much different from that of the water.

n = 1.00028 * sin (0.295342 + pi/4)/2 / sin(pi/8)

n = 1.3446

So the difference from the original pure water solution is: 1.3446 - 1.33414 = 0.01046.

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2. 325ml of sugar added.

X distance = 25.05cm.

L distance = 78.35cm.

The value for theta is: 0.309448radians.

This has more of a significant change in angle than the first trial. This is because 3.5 times more sugar has now been added. When the final calculations for the index of refraction are made, it is measured to be: 1.36391.

The difference here is 0.02977 in index of refraction.

Overall I had hoped for a more significant change in the index. The value was change was probably small because the sugar was never able to thoroughly mix. It was combined with room temperature water. Had the water been intially heated it might have dissolved better and mixed more homogenously throughout the water. To counter this I tried to stir the sugar into the water as thoroughly possible and then the measurements were taken right afterwards while the sugar was still suspended throughout the tank. So another possible measurement that can now be taken will be to see if the prism filled with this substance, after sitting still for almost a week, has any kind of change within it. Other ideas that I thought of trying were to add cold or hot water. Also increasing the salinity of water will have an effect on its index of refraction. So seeing the effects of adding salt could also be an interesting idea.

My goals for all of this is to try to be as accurate possible with measuring the index of refractions. For now I am just measuring out homogenous liquids. Liquids where I assume that at any height they will have the same index of refraction. Later on I will practice measuring with a method known as the tank method. Once I am skilled with the tank measurement I will have a better understanding of the tank I had created a few weeks ago and be able to measure out many of the interesting properties that it displays.