Comparing the refractive index of a medium with its critical angle

## PURPOSE

This lab was conducted to prove a relationship between the critical angle and index of refraction of a ray of light passing through a denser medium to air.

## QUESTION

How does the critical angle vary across media with different refractive indices?

## BACKGROUND KNOWLEDGE

Examining the law of refraction, it was known that light rays traveling from denser to rarer media bend away from the normal because light travels faster in rarer media. On the other hand, the critical angle of a medium is the incidence angle at which the angle of refraction is 90 degrees. Considering this, to decrease the critical angle, the original medium has to be much denser than the medium the light escapes to. This is because, to achieve an angle of refraction of 90 degrees with the smallest incidence angle, the light needs to bend as much as possible. Refractive index is calculated by dividing the speed of light in a vacuum by the speed of light in that medium (n = c/v). Therefore the slower the light travels in the first media compared to air, the more the light will bend.

## HYPOTHESIS

Based on the background knowledge discussed, it was hypothesized that as the index of refraction increases, the critical angle decreases. In other words, the two variables are inversely proportional.

## Materials

-Thin Clear Containers

-Vegetable Oil

-80% Alcohol Solution (Hand Sanitizer)

-50% Sugar Solution

-Water

-Acrylic

-Lexan (Polycarbonate)

-Light Ray Box

-Protractor (printed on paper)

## Procedure

First, the different media being examined were prepared by pouring them into thin clear containers. It was ensured that the volume of liquid in the container was large enough for the light rays to evenly pass through, without disruptions.

Next, the light ray box was plugged into the wall outlet, was powered on, and set up with the light blocker that only allows one light ray to pass.

Following, the first medium was placed on the flat line of the protractor on the protractor paper so that the straight edge touched the line.

Then, the ray box was placed 5-10 centimeters so the light shined through the curved side of the container, passed through the center on the protractor paper, and was perpendicular to the straight edge of the containers.

Next, the refracted ray and its angle from the normal was observed

Until the observed angle wasn’t 90 the incidence angle was increased.

When this critical angle point was obtained, the angle of the incidence ray was measured with the protractor paper, and was recorded under the critical angle column for the appropriate medium.

Steps 3-7 were repeated for each media until all observations were made.

Once the data was collected the light ray box was turned off and packed up, the containers were emptied and washed and everything was returned the way it was received.

## ANALYSIS

Equation of Trendline:  y = -52.2x + 120

y represents the dependent variable, Critical Angle, in degrees

x represents the independent variable, Index of Refraction

Therefore, there is an inversely proportional relationship between the refractive index and the critical angle of a medium. In other words, the lower the refractive index of a medium, the higher the critical angle.

## DISCUSSION

Main Discoveries:

As a result of conducting this experiment, an inversely proportional relationship between the refractive index and critical angle was proved. This makes sense considering the equation given by Snell’s Law as isolating for n1 gives you the following equation:

An important idea to consider is the factor by which increases as n1 decreases because even the relation formed from the data in this experiment showed a factor of -52.2. This is because sine 1 is inversely proportional to n1. To test the accuracy of the relationship drawn from this experiment, refractive indices of other media were substituted into both this isolated equation and the formula obtained from the data:

Upon examining the accuracy of the obtained results, it can be seen that for interpolated points (data within the range of tested refractive index values; 1.33-1.58), the results are quite accurate with an absolute, average experimental error of just 2.65%. For extrapolated points like the refractive index of diamonds, the results were way off with an experimental error of -126.24%. Therefore, this experiment can be considered successful in proving the inverse relationship between refractive indices and critical angles to an extent (for any refractive index within the range of the values tested in this lab).

Real Life Applications:

Through obtaining this relationship, many opportunities have opened for the use of this idea to make life in optics easier. For example, part of our eye includes something called the cornea (refracts light entering the lens) which normally has a refractive index of 1.38. However, through diseases like Keraoconus and conditions like astigmatism, refractive errors can occur, making the refractive index change. By understanding this inverse relationship, it can make it easier to correct refractive errors. For example, corneas with a lower than normal refractive index can bend light less therefore, an optical instrument such as contact lenses need to be used to correct this.

Side Discoveries:

Apart from the quantitative results discussed in the previous paragraph, some qualitative observations were also made during this experiment.

The phenomenon of light being divided into its individual colors or wavelengths as it travels through a material is known as dispersion. The difference in the medium’s refractive index with respect to the wavelength of light is what causes this effect. Dispersion may be involved in the experiment in several ways. First off, if the medium under investigation has a high level of dispersion, the various hues of light flowing through it might refract at various angles. Instead of a single, concentrated area, this would cause the pattern of refracted light to be more evenly distributed.

Secondly, the refractive index of a medium for various hues of light may have been calculated based on the presence of dispersion. By examining the angle of refraction for different wavelengths of light, it was possible to determine how the material’s refractive index varies with respect to wavelength.  The medium’s dispersion can be calculated using this knowledge, which would be useful in areas like optics and materials research.

The presence of dispersion, in this experiment, is a factor to consider because this experiment involves refraction of light and can have a significant impact on the observed results. The dispersion of light was also observed during the experiment as mentioned in the qualitative observations, and varied depending on the medium that was investigated.

Experimental Errors & Limitations:

Although this lab was successful in proving the relationship between critical angles and indices of refraction, it did have a small margin of error and some limitations that must be considered. First, the refractive indices range was quite small as only values between 1.33 and 1.58 were tested.

This is in large part due to the limitations of available materials as media with higher refractive indices are either rare/expensive to obtain or are dangerous to use in a school setting (e.g. diamonds, sapphires, strontium titanate, titanium dioxide, etc.) Additionally, working with these materials would also be difficult as with the current equipment available, it would be difficult to obtain the appropriate shape to be able to observe refraction and total internal reflection. Perhaps in the future a more sophisticated setup with safety features for other materials could be used to extend this study.

Aside from material limitations, the equipment used during this experiment also caused some inaccuracy. The source of light in the experiment was a light ray box with a single slit attachment to produce a singular straight ray of light. However, as discussed in the qualitative observations, this setup was unable to produce a constant straight light beam as a lot of light escaped through the sides making it difficult to identify a distinct straight line.

The slit also was not thin enough, producing a very thick light ray which made measuring the incidence and refracted angles very difficult and introduced human error in identifying the correct angle. A possible solution to this is using a thinner slit cover and a more intense light source like a laser.

Finally, the last relevant experimental error identified in this study was measuring the media’s refractive indices. Measuring the refractive index requires the use of a refractometer which was unavailable at the time of this experiment. Therefore, common materials like water, oil, and hand sanitizer were used so that their refractive indices can be obtained from online sources. However, even within common materials, there can be some variations such as temperature, which affects density and how fast light moves through it, or mixing with other media. To prevent slight inaccuracies caused by this, a refractometer can be used to accurately measure the exact media’s refractive index.

## CONCLUSION

Upon completion of this experiment, the relationship between critical angles and refractive indices was proven to be an inverse relationship. This confirmed the hypothesis as the critical angle did decrease as the refractive index increased. However, this study is restricted to the range of materials with refractive indices between 1.33-1.58 and no accurate predictions can be made for values outside this using the data obtained from this experiment. Another flaw in the hypothesis was identified as the relationship is not directly between the critical angles and refractive indices but rather the sine of the critical angles. In the future, this lab can be extended to cover a wider range of refractive indices and obtain more accurate observations.

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William Anderson (Schoolworkhelper Editorial Team)
William completed his Bachelor of Science and Master of Arts in 2013. He current serves as a lecturer, tutor and freelance writer. In his spare time, he enjoys reading, walking his dog and parasailing. Article last reviewed: 2022 | St. Rosemary Institution © 2010-2024 | Creative Commons 4.0