Mathematics in the sugarcontent measuring project with laser

[Flame9991]

Hello! I'm actually an 8th grade student, but I chose this project to get ahead of my friends and get a higher mark. In this project, you have a laser pointer shot at a hollow prism with a liquid, and the refraction indicates the liquid that is inside. But to find the index of refraction of the liquid (or the number the ratio of light in a vacuum and in the substance) you need to do an equation. I do understand Snell's law, but when it is entered twice, it gets way more confusing. I'm the kind of guy that must know everything before he does it, and I do not understand the first equation
n = nair × sin[0.5(θmd + θp)] / sin(0.5θp)
Why are we dividing by half in the little brackets? Why are we dividing the sines? It would be greatly appreciated if you explained everything. And in the second equation, the index of refraction for air is doubled? Why?
n = 2.00056 × sin[0.5(θmd + 60°)]
Thanks again

[deleted-2131]

HI Flame9991,

Kudos to you for taking a challenge and seeking to understand the underlying physics.

To get the equation for the angle of minimum deviation (equation 2 in the project idea background) from Snell's law, you apply Snell's law two times, just like you said. Then, to get the equation in the form shown in equation 2, you use the geometry of the prism to relate the various angles you calculated using Snell's law. Dividing by two and dividing the sines are the result of applying this geometry. The following two websites go through this derivation, so check them out for details:

http://www.thorlabs.com/newgrouppage9.c ... 20Tutorial
http://www.a-levelphysicstutor.com/optics-prisms.php

In equation 5, the index of refraction is doubled because we are dividing the equation by the sin of 30 degrees (see equation 3). The value of sin(30 degrees) is 1/2 (shown in equation 4). Dividing by 1/2 is the same as multiplying by 2 (remember that dividing fractions involves inverting and multiplying). So, you end up with twice the index of refraction of air in equation 5.

Let me know if you have any other questions, or if something is still confusing - I'm happy to help!

[Flame9991]

Thanks for the info and links. Just one more question
Why are we using a laser instead of an ordinary light?

[deleted-2131]

Hi Flame9991,

Great question! Lasers produce collimated light, which is different from ordinary flashlights. "Collimated" means that all of the photons in the laser travel in together in a single laser "beam". That is why a laser pointer makes a small dot on a wall, while a flashlight makes a big circle--the laser is collimated, the flashlight is not. Second, lasers are only composed of one wavelength of light, while ordinary lights generate multiple wavelengths. The index of refraction depends on wavelength, so different wavelengths of light bend by different amounts when they interact with the same medium. This is why a prism "splits" white lights into all of its color--the different wavelengths of light refract by different amounts when they travel through the prism, which spreads the colors apart. If you didn't use a laser for this project, then (1) you wouldn't have a nice tight beam of light and (2) the index of refraction would not be just a single number.

This article has some more discussion about the differences between lasers and regular light sources (like a flashlight):

http://physics.yoexpert.com/light-and-o ... -2621.html

[Flame9991]

Wow, I cannot thank you enough for your help. School wide science fair is in a few days, and I'm pretty confident in my project. But i'm stressed over this one question: How does this relate to real life situations and/or how does it benefit society?
Thanks for answering all my questions, you've been a great help
Flame9991

[deleted-2131]

Hi Flame 9991,

It's exciting that your science fair is so close! A LOT (seriously, a lot) of analytical techniques involve passing light through a liquid or suspension (or even a solid). For example, a technique called spectrophotometry can determine the composition and abundances of dissolved species in a liquid (or transparent solid) by measuring how much light of different wavelengths passes through the sample. In my own research, I used a technique called Fourier transform transmission spectroscopy to measure the water content of glasses. In this technique, you again pass light through a sample (the glass, in my case) and measure how much light is passed through the sample, as a function of wavelength. The "Fourier transform" part just describes some mathematical procedures involved in processing the data. You can also measure the oxygen isotopic composition of glacial meltwater using a laser technique. People then use that data to reconstruct changes in global climate. So, the take away here is that millions (if not billions) of dollars are spent every year developing, building, and using analytical techniques that involve passing light through a sample. It's a technique that can be used for a wealth of applications in biology, chemistry, environmental science, geology, physics, materials science . . . the list just keeps on going. Doing this project, and learning the physics involved, is like opening a door to this HUGE area of science.

This webpage describes spectrophotometry in more detail, if you are interested. It might be a little advanced, depending on your chemistry background.

http://chemwiki.ucdavis.edu/Physical_Ch ... photometry

Another key features of this project is that is uses a "non-destructive" method to determine sugar content. In other words, you don't use up the sugar solution (or soda, fruit juice, gatorade, etc.) during the analysis, and you don't do anything nasty to it that would make it undrinkable (like adding some toxic chemical). If you are running a soda manufacturing plant, you want to make sure that your soda has the right amount of sugar in it. So, you might periodically test samples of the soda to ensure you've got everything in the correct proportions. (In industry this is called "quality control testing".) You want the test to be quick, easy, non-destructive, and accurate. A technique similar to the one you explore in this project could meet those criteria.

[Flame9991]

Thanks. I want to say this is the last question, but I've said that for all my other questions and that wasn't true. Anyway: Why do we have to do this project with a prism?
Reply soon
Flame9991

[deleted-2131]

Hi Flame9991,

Sorry for not replying sooner, I didn't see your post (eek!). This project uses a prism because a prism refracts light in a very specific way. The geometry of the refraction would be different if you didn't use a prism for this project.