Lab O3:
Snell's Law and the Index of
Refraction
Introduction.
The bending of a light ray as it passes from air to water is determined by Snell's law. This law also applies to the bending of light by lenses and to the guiding of light by the fiber optic cables that carry modern communications signals. In part 1 of this lab you will investigate the bending of light as it travels from air to plastic and plastic to air. Your data will illustrate Snell's law. In part 2, you will find that the speed of light in plastic varies slightly with the wavelength.
The speed of light in a vacuum is c
= 3.00 x 10^{8} m/s. Because of
a complex interaction between an electromagnetic wave (light) and the charges
in matter(electrons and protons), a light signal travels more slowly in
transparent materials, such as glass or water, than in vacuum. The ratio of c, the speed of light in a
vacuum, to v, the speed of light in a medium, is called the index of
refraction, n, of the medium
medium 
n 
vacuum 
1 
air 
1.000293 
water 
1.333 
CF_{4} 
1.461 
diamond 
2.419 
(1) _{}.
Note that n ³ 1,
always. In most applications, the index
of refraction of air can be taken to be one, with negligible error.
A ray of light passing from one
medium to another (say, from air to water)
is bent or refracted from its original path according to Snell's Law,
which states

(2) _{},
where n_{1}
and n_{2} are the indices of refraction in medium 1 and medium 2, and
the angles are measured with respect to the normal (the perpendicular) to the
interface. The light is always bent more toward the normal in the material with
the larger n, and, in the diagram to the left, we have drawn the angles for the
case _{}. Note that if the ray strikes the interface
perpendicularly, at an angle of q_{1}
= 0°, then it is
not refracted and exits perpendicularly with q_{2}
= 0°.
Although Snell's Law can be proven
from fundamental principles, we will consider it to be an experimental fact and
use it to determine the index of refraction of Lucite (a transparent plastic).
Consider a ray of light passing from
air (n_{air} = 1) into another medium such as Lucite with an index of
refraction n (n >1). In this case,
the ray is incoming (incident) on the air side and outgoing (refracted) on the
Lucite side of the interface, and we have
(3) _{} (air
® Lucite)
where q_{i} and q_{r}
are the angles of the incident and refracted rays, respectively.
Suppose now a ray of light passes
from a medium (Lucite) with index of refraction n > 1 into air . We then have
(4) _{} (Lucite
® air)
In the two cases just considered, if
instead of labeling the angles as q_{i}
and q_{r},
we label the angles q_{a}
(q in air) and q_{p}
(q in Lucite
plastic), then both formulas (3) and (4) become
(5) _{}.

The index of refraction is, in
general, a function of the wavelength of the light. It turns out that n is smaller for long
wavelength (red) light and larger for short wavelength (blue) light. That is, blue light is bent more by glass or
plastic lenses and red light is bent less.
This is the cause of chromatic aberration in simple lenses. Multielement lenses which correct for
chromatic aberration are called achromats and are rather expensive.
This spread in the values of n, or dispersion, is the cause of the rainbow
of colors produced by a prism. This
effect was first studied quantitatively by Isaac Newton when he was still a
student.
The colors of the
spectrum, in order from longest to shortest wavelength are: red, orange,
yellow, green, blue, violet.
Procedure
In this lab, using a semicircular
Lucite lens, we will measure the angles q_{a}
and q_{p}
as a ray of light passes between the flat side of the Lucite and air. The angles are easily measured because the lens sits at the center of a
circular platform with an angle scale along its perimeter. A ray of light, defined by a slit on the side
of the platform, passes through the lens and the angles _{}and _{} are read by noting
where the shaft of light touches the angle scale. The apparatus is shown
below. The lamp and the slit are fixed
in position, while the lens and the platform with the angle scale can be
rotated with a handle under the platform.
Note that refraction
of the light ray takes place only on the flat side of the semicircular
lens. On the circular side, the ray
strikes the interface exactly normal to the surface and there is no refraction.
Recall that _{} is the angle on the air side of the flat interface and _{} is the angle of the plastic side of the interface. So if the flat side is facing the slit, _{} is the incident angle,
but if the curved side is facing the slit, then _{} is the incident angle.
Before taking measurements, it is
crucial that the lamp and the lens be carefully positioned with respect to the
angle scale platform. Unless the
alignment is perfect, your data will have systematic errors. The alignment procedure is as follows:
1. Begin by adjusting the lamp to give a
good parallel beam. Point the lamp at a
distant wall (at least 30 feet away) and observe the image of the light bulb
filament. Slide the bulb back and forth
slightly, with the knob on the underside of the lamp, to get a sharp image of
the filament on the distant wall. Adjust
the orientation of the lamp to make the image of the filament vertical (so that
maximum light passes through the vertical slit on the platform).
2.
Remove the Lucite lens from the platform and then position the lamp and
rotate the platform so that a fine, sharp shaft of light is seen entering at 0^{o},
passing over the exact center of the platform, and exiting at 0^{o}. You have to get the light slightly higher
than the platform and angle it down slightly to get a bright shaft of light
showing on the platform.
3. Now place the Lucite lens on the
platform with the flat edge centered on the platform and perpendicular to the 0^{o}
line. Adjust the position of the lens
slightly until the ray of light exits at 0^{o} when it enters at 0^{o}. Check that this occurs when both the flat
side is toward the slit and when the platform is rotated 180^{o} so
that the rounded side is toward the slit.
4. As a further check of the alignment,
rotate the platform to the 45^{o} position and observe the reflected
ray, which should also be at 45^{o}.
Part 1. Index of Refraction of Lucite
At last, you are ready to take
data. With the flat side toward the
slit, take several measurements of _{} and _{} for various _{}’s from 5^{o} up to about 70^{o}. Estimate the angles to the nearest 0.1^{o}. For each value of _{}, take two
measurements of _{} for both possible
orientations of the lens, as shown below, and average your two values for _{}. If your alignment is good, the two values should be very
close, within a few tenths of a degree. Record both _{}'s in your handwritten notes, but enter only the average _{} into your Mathcad
document.
Now repeat all these measurements,
with the circular side of the lens toward the slit. Take data for values of _{} from 5^{o} to
about 40^{o}, every 5^{o} or so. When _{} is near 75^{o}
, which occurs when _{} is near 40^{o},
you will notice that the refracted beam is no longer a narrow beam of white
light, but has broadened into a rainbow of colors. Use a white piece of paper as a screen to see
this clearly. Use the position of the
middle of the broadened light beam to
determine _{}.
Now combine all your data to make a
single table of _{} and _{}.
To make the new arrays with the
combined data, simply enter the data again. You can avoid much tedious typing
by using copy and paste. Be sure that
all the air side angles are in one column and all the plastic side angles are
in the other column.
Using
all the data make a plot of _{} vs. _{}, with both angles in degrees. Now convert
the angles from degrees to radians, and then make a plot of _{}. [The default mode of
Mathcad assumes that angles are in radians when computing sine and
cosine.] The plot should be a straight
line.
For each _{} /_{} pair, compute _{}, and graph n vs._{}. The index of
refraction n should be a constant, independent of _{}; however, for small q,
the fractional errors are large, and you will probably observe large deviations
of n from the average value n_{avg} when q is small. [The curve of n vs._{} should be flat, except for random fluctuations; if not,
there is some systematic error, probably due to misalignment.]
By examining the graph of n vs._{}, decide which smallq
values of n to throw out, and then use your “good” data to compute the mean of
n, standard deviation, the standard
deviation of the mean.
_{}
Part 2. Dispersion of Lucite
In this part, you will measure n_{red}
and n_{violet}, the values of n at the extreme red and violet ends of the visible spectrum. To do this, orient the platform so the
circular side of the lens is facing the slit, and then adjust the angle so that
the refracted beam is close enough to 90^{o } to give a good spectrum of colors, but not so
close that some of the colors are lost to internal reflection. Make measurements of the angles of the
refracted beam at the red and violet edges of the spectrum, _{} and _{}, as well as _{}. Compute
_{}.
Make measurements at 2
or 3 slightly different incident angles and repeat each measurement for the two
orientations of the lens.
You now have three values of n for
three different wavelengths: n_{avg} in the middle of the visible
spectrum (yellow), n_{red}, and n_{violet}. Look at the spectrum chart on the wall to
estimate the wavelengths of the red, yellow, and violet light. Make a plot of n vs. wavelength with your
three points.
PreLab Questions:
1. What is Snell’s
Law? Your answer should include a diagram.
2. A ray of light in air
strikes the surface of a pool of water at an angle of 30^{o} from the
normal. What is the angle (measured from
the normal) of the refracted ray in the water?
3. In this experiment,
why is there no refraction of the light ray on the circular side of the Lucite
lens?
4. When a light ray
passes from a material of higher n to a material of lower n (for instance, a
light ray passing from glass to air), the refracted angle is larger than the
incident angle. In this case, the angle
of the incident ray can exceed a critical
angle, at which the angle of the refracted ray is 90^{o}. For
incident angles greater than the critical angle, there is no refracted ray,
only a reflected ray. This phenomenon is
called total internal reflection. What is the critical angle for water with n =
1.33? What is the critical angle for
glass n = 1.50?
5. Show with a sketch
what the graph of sin(q_{a})
vs. sin(q_{p})
for Lucite should look like (for a given fixed wavelength l). Your sketch should show the shape of the
curve (linear, parabolic, etc.) as well as indicate whether the curve goes
through the origin. What can you say
about the slope of this curve?
6. Suppose you measured n
vs. q_{a}
using a red light source and a blue light source. On the same graph, sketch both n_{red}
vs. q_{a} and n_{blue} vs. q_{a},
where n_{red} is the index measured with red light and n_{blue } is the index measured with blue light. (Qualitative
sketch only! No numbers.)
7. What does a graph of n
vs. wavelength look like for glass or plastic?
8. When the curved side
of the lens is toward the slit, which angle is _{}? Which angle is _{}? Indicate your
answers with a diagram.
9. If a piece of clear
glass with an index of refraction n is placed in a clear liquid with the same
index of refraction n, the glass becomes completely invisible. Why?
10. When a light ray passes from air through the Lucite semicircular lens and back into air the ray path appears as shown below, with q_{a} > q_{p}. Suppose that a Lucite lens with n = 1.55 is submerged in a clear liquid with n_{liq}=1.70, so the ray path is now liquid ® Lucite ® liquid, and the angle q_{a} should now be labeled q_{liq}. Is q_{p} greater than, less than, or equal to q_{liq}? Suppose q_{liq }= 20^{o}, what is q_{p}?