Group 4
Edith Reshef
Ian Tracy
Mark Yen
Nick Sisler
Michael Aponte
Caroline Hane-Weijman
Purpose
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Hypothesis
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Materials
Procedure
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3.003 Lab 1 Report
Due: Feb 14th, 2008
To determine the specific relationship between the angle of incidence (theta 1)
and the angle of refraction (theta 2).
To quantify the refraction of light through air and water
The index of refraction (n2) of water will be slightly more than the index of
refraction of air (n1) because water is a denser medium than air.
2.2 L Water
Clear rectangular container (8 * 16 * 24.5 cm)
Butter knife
Ruler
Fill the container with water at 2 cm away from the top of the container
Place the knife in the container, with the tip of the knife on the bottom end of
the container and its handle resting on the top of the edge of the container such
that the knife is perpendicular to the edge of the container
Measure the vertical distance from the water’s surface to the point where the
knife meets the lip of the container. This will be x.
Measure the length of the knife between the surface of the water to the point
where the knife meets the lip of the container, this will be y.
y and x form a hypotenuse and leg of a right triangle respectively, use inverse
sine to find the angle θ1.
Next hold the ruler to the side of the container so your view of the measurement
will not be bent by the light and eyeball a measurement of the perceived depth
of the tip of the knife in the water. This will be a.
Then hold the ruler above the water and measure a horizontal length from the
end of the ruler in the water to where it meets the water. Call this b.
a and b form two legs of a right triangle use inverse tangent to find θ2.
Data
The experiment was conducted by each group member one time and measurements were recorded and rounded as
indicated on the table below. θ represents theta, so θ1 = theta 1, etc.
To define variables:
x1 represents the horizontal (along the surface of the water) distance between where the height of the rules was measured
and the point at which it entered the water;
x2 represents the horizontal distance between the point where the knife broke the surface of the water and where its end
rested at the base of the container;
y represents the hypotenuse of the triangle created by the knife above the surface of the water, aka the distance between
its highest end and the point at which it breaks the water’s surface;
z was the perceived depth of the end of the knife (vertical direction only) below the surface of the water;
θ1 was the angle of reflection, formed by the slant of the knife above the water and the water’s surface (<90degrees);
θ2 was the angle of refraction, formed by the slanted, submerged knife just below the surface of the water and the
surface of the water (<90 degrees).
Nick
Ian
Edith
Mark
Caroline
n1
1
1
1
1
1
x1
(cm)
1.9
1.6
1.8
1.6
1.8
y
(cm)
4.7
4.4
4.6
4.6
4.5
θ1 (degrees)
23.84
21.3
23
20.35
23.6
θ1 (radians)
0.416008
0.371685
0.40135
0.3551075
0.41182
sin(θ1)
0.4041
0.3632
0.3907
0.3477
0.4003
Average
1
1.74
4.56
22.418
0.3911941
0.3812
x2
3.7
2.9
2.7
2.9
3.2
3.0
8
Z
(cm)
8.9
9.8
9.8
9.2
8.9
θ2 (deg.)
15.6
16.5
15.4
17.5
19.8
θ2 (rad.)
0.27222
0.28793
0.26873
0.30538
0.34551
sin(θ2)
0.2689
0.284
0.2655
0.3007
0.3387
n2
1.503
1.279
1.471
1.156
1.182
9.3
16.96
0.29595
0.2915
1.308
Arithmetic Mean of refraction index of water = Average n2 = 1.308
Percent Error from theoretical value (1.33) = | 1.308-1.33 | /(1.33) * 100 = 1.65 percent experimental error
Deviation from theoretical value = (1.308-1.33)/(1.33) * 100 = -1.65
Here, the graph of
a linear function
passing through
the origin of sine
of theta 1 versus
the sine of theta 2
of our
experimental
values yields a
line with a slope
of approximately
our experimental
arithmetic mean
for n2: ~1.3,
which has ~2.3
percent
experimental
error.
continuation of Data
For the critical angle θcritical, or critical angle with respect to the normal of the surface (water): what must be realized is
that when light travels from a given medium into a more dense medium, the angle of refraction cannot be calculated by
Snell’s law when the incident angle’s (θ2) sine value is greater than 1. At the point where the angle θ2 attains this value
θcritical, θ1 will be 90 degrees (normal to the surface of the water) and light is reflected only within the denser medium,
and is not reflected into the less dense medium. This phenomenon is known as internal reflection. So theoretically, at
exactly θcritical, the ray of light does not yet internally reflect; it instead travels directly along the barrier between the two
concerned mediums (the surface of the water in this experiment).
sin(θ1)*n1 = sin(θ2)*n2
In order to calculate the critical angle for air and water, let θ1 = 90o and use Snell’s law to solve for θcrit:
θcritical = sin-1(n1/n2)
Which (theoretically) gives sin^-1 (1.0003/1.33) ~ 48.8 o
When θ1 > θcritical, the light/incident ray is reflected only internally and thus no refraction is observed.
Thus, if light is launched into the denser medium at angle θ1 exceeding 48.8 degrees, there should be no reflection from
the surface of the denser medium.
This can be confirmed logically in this case as follows: when θ2 = θcritical, simple algebra or inspection will prove that θ1
will be 90 degrees, and the light will be shined into the denser medium from a point normal to the surface of that denser
medium (water).
Experimentally, θcritical can be determined using this group’s set-up by adjusting the angle θ1 formed by the portion of the
knife outside of the water and the surface of the water until it appears that the knife looks straight, as this marks the point
when no light is reflected to cause a difference between a perceived and actual behavior of an object (the knife) in the
denser medium.
Conclusion
This lab explored the relationship between the angle of incidence and the angle of
reflection of an object placed in a tub of water. The goal of this was to determine the
index of refraction for water, based on the known refraction for air (n = 1.003~1.0) and
our perception of the angle of reflection. We placed a knife in a Tupperware container
of water and measured (in centimeters) various heights and lengths of triangles that the
knife formed with the container and the water (see figure 1) and then used trigonometry
to figure out θ1 and θ2. With these two values, Snell’s law was then used to calculate
the unknown index of refraction of water.
Looking at the resulting data, the experimental value of the index of refraction of water
was 1.308, a value higher than that of air. This experiment was controlled in that each
observer measured every element of data in their trial. The measurement of the
perceived height of the knife (a) was the experimental factor, differing slightly
depending on the angle that each individual was looking at the tub from. This was
accounted for by maintaining the same experiment conditions for all trials, placing the
ruler in the same location and attempting to place each individuals eye level at the
same location (same chair and relatively the same height). However, this accounted for
most of the error made. When measuring a, the perceived depth, the ruler was held
outside of the container so that the ruler measured accurately the perceived height as
opposed to placing the ruler in the water which would have measured the accurate
height. Again, this provided for error as this measurement was more difficult to take
since the object that was being measured was at a distance away.
The goal of this experiment was to quantify the refraction of light through air and
water. Our resulting data supported our hypothesis that the index of refraction of water
will be slightly greater than that of air since water is a denser medium, as we calculated
the index of refraction of water to be 1.308, a value higher than the known index of
1.003 of air.
Furthermore, it was determined that of θ1 exceeds θcritical~ 48.8 o total internal
reflection occurs in n2, the denser medium. This means that no refracted ray appears,
and no light is reflected from the denser medium back into n1 if θ1 attains or surpasses
this angle. Consequently, as exemplified by optical fibers, all of the incident light will
become trapped inside n2.