Nick
Doty, Alexandra Saewetwong: 2001
The refractive index of water and other liquids changes with temperature. The effect is similar to the dependence of the physical density on temperature.
Question
Can we confirm the relationship between temperature and the refractive index of water?
Hypothesis
Although the change in refractive index (according to published
figures) is small, if the distance between a water filled 60°
prism and a wall is large the deviation of a laser spot will be
seen to change with temperature.
The values of D, the minimum deviation, for very small changes
in the refractive index can be calculated using a graphing calculator.
n = sin ((A+D)/2) / sin (A/2)
Using the intersection function of the calculator, the minimum deviation angles are found to be.....
The difference of 1.09 degrees is large enough to see a significant difference on a wall -- to create a difference of 10 cm on the wall, for example, a distance between the prism and the wall must be only 4.4 meters. If the full length of the classroom is used the change in the position of a laser spot will be easily measured with a mm scale.
Procedure
A hollow triangular 60° glass prism was placed on a table with a laser shining through it.
The prism was filled with water in order to maintain a constant temperature as much as possible. By rotating the prism, the angle of minimum deviation was found, (the point at which the laser spot at the other end of the room began to move back to the left). The minimum deviation of the laser spot was marked on a meter stick at the back of the room and the temperature of the water was measured using a thermometer. The water was then replaced with higher temperature water and the new position of the laser spot was marked. As the water cooled, several more readings were taken of both the temperature and laser position.
To measure the minimum deviation at room temperature, we marked the path of the laser on a piece of paper - both with and without water. A triangle is created from the center of the prism, and the endpoints of the two laser paths. Using a protractor we measured the two outer angles and were thus able to find the angle of minimum deviation.
Data [see Table 1]
Analysis
The change in angle of minimum deviation was found by finding the inverse tangent of the change in position over the distance from the prism to the ruler.
The distance from the prism to the meter rule was 1320 cm.
The changes in angle are negative because when the water is heated and the refractive index decreases, the laser point moves closer to where it would be if there was no water in the prism and the angle becomes smaller.
The refractive index, n, was found using Graphical Analysis and the equation:
... where n is the refractive index, A is the angle of the prism (in this case 60°) and D is the angle of minimum deviation. The values are plotted in ...
The graph is a straight within errors. if the line is drawn using only the last six points.
The initial point was clearly not part of this line. It is suspected that one piece of the apparatus (either the laser, prism or ruler) was moved and created this anomalous point. The remaining points give a clear and accurate description of the relationship.
The slope of the line from the published refractive indices is expected to be - 0.00019 [assuming an approximately linear relationship]. The slope of the measured line is -0.00019(5) which is the same within errors.
Discussion
These measurements have confirmed the published variation of
refractive index with temperature but different absolute values
were obtained. The refractive index at 45 °C was expected
to be 1.3290 but the measured value was 1.3336.
Although errors in measuring absolute distances in the experiment
are fairly large, and there may be some small difference in the
calculated refractive index due to a prism angle of not quite
60° we are able to measure very precisely the change in position
and hence the change in refractive index.
This situation is common in Physics - a simple thermometer measures temperature to within only ±1°, but with a microscope one is able to measure differences in temperature over a range of a few degrees to within ± 0.01°. The same situation occurs in measuring the mass of individual atoms. The relative mass of an atom in atomic units is known much more accurately than the mass of the atomic unit in kilograms.
Suggestions for further work
To obtain more accurate measurements, a more closely temperature controlled environment would help in maintaining a constant temperature throughout the water. Using a perfectly semicircular hollow prism would reduce the complexity of the experiment. With a semicircular prism, minimum deviation would not be involved, only the calculation of n as sin i / sin r. A perfectly semi-circular prism would also remove any inaccuracies involved in the non perfect equilateral shape of the prism used in this experiment. Unfortunately, we were unable to obtain a semicircular glass container.
The measurements could be repeated with other substances, such as alcohol or another clear liquid -- to again determine differences in optical density at different temperatures.