Download Chapter 4 Experiment 2: Snell`s Law of Refraction

Chapter 4
Experiment 2:
Snell’s Law of Refraction
4.1
Introduction
In this and the following lab the light is viewed as a ray. A ray is a line that has an
origin but does not have an end. Light is an electromagnetic disturbance and, as such, is
described using Maxwell’s equations, which expresses the relationship between the electric
and magnetic fields in an oscillating wave.
Light propagates as a wave; yet, many optical phenomena can be explained by describing
light in terms of rays. In the model for light, rays in a homogeneous medium travel in
straight lines. This model is referred to as Geometric Optics and is a very elementary
theory. In this theory light travels from its origin at a source in a straight line, unless it
encounters a boundary to the medium. Beyond this boundary may be another medium
which is distinguished by having a speed of light different from the original medium. In
addition, light may be reflected at the boundary back into the original medium. A light ray
that returns to the original medium is said to be “reflected”. A ray that passes into the other
medium is said to be “refracted”. In most interactions between light and a boundary, both
reflection and refraction occur.
In order to frame laws that govern these phenomena we must define some terms. The
boundary between two media is defined as a surface. The orientation of a surface at any
specific point is characterized by a line perpendicular to the surface that we call the normal.
A ray may encounter a boundary at any arbitrary incidence angle. The angle of incidence
is measured with respect to the normal line. A reflected ray will have an angle of reflection
that is also measured with respect to the normal. The refracted ray will be oriented by the
angle of refraction measured between the ray and the normal to the surface.
Checkpoint
For geometric Optics what assumption is made about the nature of light?
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CHAPTER 4: EXPERIMENT 2
What distinguishes the two media is that the speed of light is different from one medium
to the other. We define the index of refraction n to be the measure of how much different
the speed of light is in a certain medium from that of light through a vacuum. Light travels
through a vacuum at 299,792,458 m/s. This speed is thought to be a universal constant and
the highest speed allowed in nature as postulated in Einstein’s theory of Special Relativity.
We use the symbol c to represent this speed.
The index of refraction is a characteristic of the medium. It is the only thing that
distinguishes one medium from another in geometric optics. It is defined as the ratio of the
speed of light in a vacuum to the speed in a particular medium of interest,
n=
c
c
or v = .
v
n
(4.1)
Therefore, the value of the index of refraction is always greater than unity. Gasses have an
index of refraction close to 1 (nair = 1.00028), while for water the index is about 1.33 and
for plastic it is approximately 1.4. Depending on the type of glass the index of refraction of
glass can vary from 1.5 to 1.7.
Normally we might think that the index of refraction is a constant that is the same for
all light. The index of refraction actually depends on the frequency (color) of the light wave
to a small degree across the visible part of the spectrum and as such is different for different
colors of light. The rules for reflection of light are:
1) The angle of incidence is equal to the angle of reflection,
θ1 = θ10
(4.2)
where θ1 is the angle of incidence and θ10 is the angle of the reflected ray that
propagates in the same medium. (This is the commonly known rule, but this next
rule is rarely stated though equally important)
2) The incident ray, the reflected ray, and the normal to the surface, all lie in the
same plane.
Checkpoint
What is the law of reflection?
We will not formally investigate these rules in this lab although you will be able to observe
the phenomena of reflection as a side issue while performing this lab experiment. The rules
for refraction are not so obvious although they where well known to the ancients.
1) The first rule is often cited as Snell’s Law; it is:
sin θ1
n2
=
or n1 sin θ1 = n2 sin θ2 .
sin θ2
n1
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(4.3)
CHAPTER 4: EXPERIMENT 2
where θ2 is the angle of refraction of the ray that is transmitted into the second
medium.
2) The incident ray, the refracted ray and the normal to the surface, all lie in the
same plane.
Checkpoint
What is Snell’s Law? What phenomenon does Snell’s Law describe?
In general the path of a light ray is reversible in that if a light ray were to be reversed it
would follow the same path. A ray traveling from a low index of refraction to a high index of
refraction will experience a bending toward the normal. However a ray passing from a high
index of refraction to a lower index will experience a bending away from the normal. The
angle of refraction will be larger than the angle of incidence. So, what happens when the
angle of refraction is greater than 90◦ for a given incidence angle. In this case light cannot
be transmitted through the interface and as such it is reflected totally. The efficiency for this
reflection is 99.99% (as compared to 95% for a typical silvered surface mirror). The largest
angle for which a ray will be transmitted is the critical angle. One can show that the sine
of this angle is the inverse of the ratio of the index of refraction of the first medium to the
index of the second medium. If the second medium is air (n = 1.00028), the sine of the angle
is effectively the reciprocal of the index of refraction of the first material.
Checkpoint
In optics angles are always measured with respect to what?
4.2
The Experiment
The experiment consists of a single thin bundle of light rays exiting a light box. This ray will
be incident upon various objects so that we may deduce whether the laws of reflection and
of refraction are obeyed by the interaction between the light and the objects. A picture of
the ray box is shown in Figure 4.1.
4.2.1
Part 1: Reflections
In this experiment you will use the Light Ray Box shown in Figure 4.1. It consists of a light
source and a Multi-slit Slide. The light source housing is mounted on a colored plastic base
in which it can slide back and forth. The utility of this feature will be explained in a future
experiment. The Multi-slit Slide is a flat square piece of plastic with notches cut into each
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CHAPTER 4: EXPERIMENT 2
of the four edges. The Multi-slit Slide slips into a slot on the end of the ray box to create
rays of light. Choose the side with just one narrow notch and place that side down as you
slip the a Multi-slit Slide into place. A single narrow beam should be observed emerging
from the ray box. Place the ray box on the end of a sheet of paper provided, such that the
ray is projected onto the paper.
Draw a long line for the surface of a mirror
using the ruler. Construct the surface normal
90◦ from this surface line using the protractor
and the ruler. Position the flat side of the shiny
triangle exactly on the surface line. Carefully
slide the paper (and shiny triangle) so that the
ray from the ray box hits the shiny triangle at
Figure 4.1: A photograph of the light ray
the position of the surface normal. Carefully
box and the multi-slit slide that we will use
rotate the paper and mirror about a vertical axis
to investigate ray optics.
passing through the surface-normal intersection
and note the relation between the incident and reflected rays. If the mirror moves on the
paper, carefully reposition it before marking the ray positions. Now mark the positions of
one incident ray and its reflected ray. One point far from the mirror and the point where
the ray hits the mirror (and the surface normal) define each of the two rays. Remove the
triangle, draw the incident and reflected rays, and measure the incident and reflected angles.
Be sure to record your units and to estimate your experimental errors. How accurately
can you construct the surface normal? How accurately can you position the mirror on the
surface line? How accurately can you point the ray at the intersection between the surface
and the surface normal? How accurately can you mark the incident and reflected rays? How
accurately can you construct the rays from the three marks? How accurately can you position
the protractor? How accurately can you read the protractor? All of these contribute to your
measurement errors and must be considered while you contemplate your error estimates. Is
your angle of reflection equal to your angle of incidence? Are your error estimates large
enough to cover the difference? If not, repeat the experiment more carefully.
4.3
Part 2: Index of Refraction
In this experiment you will also use the Light Ray Box shown in Figure 4.1. Locate the oddly
shaped Lucite block. Place the block on the paper with the frosted side down. Aim the ray
at one of the two parallel sides of the block. Note that the ray emerges from the opposite
side of the block in a direction parallel to the original ray but displaced slightly due to the
refraction of the ray inside the block. One option is to print the template in Figure 4.4 and
to use it instead of constructing everything on a blank page.
You will now draw lines representing the incident and refracted rays so that you can
measure the angle of incidence and refraction for several angles of incidence. Arrange the
block for a specific angle of incidence. You will repeat the following steps through a series
of five angles from a shallow angle to a steep angle of incidence.
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CHAPTER 4: EXPERIMENT 2
Use a pencil or pen and a ruler to
draw a long line to represent the surface of
incidence. Use the protractor to construct
a surface normal perpendicular to the first
line. (Alternately, the worksheet following
these instructions may be photocopied or
removed and used directly.) Place the Lucite
block’s surface exactly on the first line and
carefully slide the paper and block (friction
Figure 4.2: A sketch of light refraction at the
will move the Lucite if the paper is moved
surface of a Lucite block. The ray enters the
without sudden jerks) so that the light ray
block at angle θ1 from the surface normal and
enters the Lucite above the intersection of
is refracted to angle θ2 from the same normal.
the surface and the normal constructed earBy noting the point the ray enters the block
lier. It might be possible to improve the
and exits the block, we have two points on the
length of the rays or their brightness by
line of the refracted ray.
elevating the front or back of the light ray
box with the ruler and/or protractor. Mark the location of the incident ray at another point
along the ray, such that its direction can be recreated later. It is advisable that this point
be near the ray box. Lastly, draw a two points on the opposite side of the block where
the ray emerges. These points should be separated as far as possible to make reproducing
the ray’s angle as accurate as possible. These points should be sufficient to recreate the
geometry of the refraction. Label these points “1” and carefully drag the paper and Lucite
block to set another incident angle with the ray once again entering the Lucite block at
the location of the normal line. Mark the incident and refracted rays as before and label
them “2”. Continue to mark three more rays for a total of five or more. If the Lucite block
should move off of the surface line, carefully reposition it before marking the refracted ray.
Remove the block and draw lines representative of the incident and refracted rays. On the
refracted side, construct the line between each pair of points and extend it back until it
intersects the Lucite block’s back surface. This is the point where the ray emerged from the
Lucite block, so inside the block the ray proceeded from O in a straight line to this point
of emergence. Connect O to this point of emergence with the ruler and extend the line to a
length that will allow you to use the protractor to measure its angle from the normal. Use
a protractor to measure the incidence and refraction angles; keep in mind that these angles
must be measured from the normal line and not from the surface. Record these measured
angles in a properly formatted table. Be sure to include the units and error estimates for
these measured angles. How accurately can you position the Lucite block along the surface
line? How accurately can you point the ray at the surface normal? How accurately can you
mark the incident and refracted rays? How accurately can you construct the incident and
refracted rays through the points that mark the rays? How accurately can you position the
protractor? How accurately can you read the protractor? All of these questions should be
considered while you are considering the sizes of your angle errors.
Order the results in a table in your lab book from smallest to largest angle. Plot the index
of refraction as a function of increasing incident angle using Vernier Software’s Graphical
Analysis 3.4 (Ga3) program. Your lab instructor will show you how to use the program if
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CHAPTER 4: EXPERIMENT 2
you haven’t already used it in previous labs. GAX can also calculate your index of refraction
in a “calculated” column.
Observe if there are any systematic changes in the index of refraction. To do this,
one must have an idea of how much deviation in the index can be attributed to random
variations. One way to assess this is to find an average value for the index, and from that to
determine a standard deviation. The average and standard deviation can be obtained from
the Analyze/Statistics menu item in Ga3. You will need to draw a box around your data
points before the Statistics will be active. Note the result in your Data. Use the std. dev.
(sn ) to find the standard error or the deviation of the mean using
sn
sn̄ = √ .
N
N is the number of values averaged (probably 5. . . ).
refraction as n = n̄ ± sn̄ .
4.3.1
(4.4)
Specify your measured index of
Part 3: Total Internal Reflection
In this experiment you will also use the Light
Ray Box shown in Figure 4.1. One option
is to print the template in Figure 4.5 and
to use it instead of constructing everything
on a blank page. Draw a long line for the
slanted surface and another at 90◦ for the
surface normal. Position the slanted end of
the Lucite block along the surface line. Slide
the paper to arrange the ray to enter the
Lucite block at the squared off end of the
block. This is the end opposite the slanted Figure 4.3: A sketch illustrating internal
end. Have the ray cross through the block reflection at the surface of a Lucite block. We
and exit the slanted side. Observe the ray can see and note where the ray enters the block,
that exits this side as you turn the paper and where the ray hits the surface, and where the
block to vary the angle that the ray strikes reflected ray exits the block.
the inside surface of the slanted side; this is the incident angle θ1 . You should also observe
that there is a significant amount of intensity in the ray reflected off the inside surface.
Record your observations in your Data. As you increase the incident angle the angle of
refraction, θ2 , will also increase. Since θ2 will always be larger than θ1 , there will come an
angle of θ1 for which θ2 should equal the maximum of 90◦ . At all angles of incidence larger
than this critical angle, total reflection occurs and you should observe a noticeable increase
in the intensity of the reflected ray. Total internal reflections are the most efficient optical
reflections. Record your observations in your Data.
Adjust the block so that the refracted ray just skims the slanted surface of the block
and the incident ray hits the surface at the surface normal line. Note any observations of
interest as you do this operation. Decide on what position best represents the critical angle
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CHAPTER 4: EXPERIMENT 2
for the incident ray. Now mark the point where the incident ray enters the block and where
the reflected ray emerges. If the block moves on the paper, reposition it on the surface line
before marking the rays. Now, remove the block, use the points (and the point of incidence)
to recreate the incident and reflected rays as they appeared inside the block. Measure the
angle of incidence. Check this with the angle of reflection since they are equal. Be sure to
record your units and to estimate your measurement errors. Consider all of the steps needed
to construct the rays and to measure the angles. Use this critical angle to calculate the index
of refraction for the Lucite block,
n=
1
nair
≈
,
sin θc
sin θc
and compare with the value you got in part one. Does it agree within the uncertainty
of the first measurement as determined from the standard error? Might the error in your
measurement of critical angle be large enough to explain any disagreement? Comment on
this in your Analysis.
Checkpoint
If two adjacent media have the same index of refraction, n, can you observe the
phenomena of reflection or refraction?
Checkpoint
What is a critical angle? What are the two conditions that allow total internal
reflection to take place?
4.3.2
Part 4: Optical Fiber
One of the more recent and ubiquitous applications of total internal reflection is the optical
fiber. Fiber optics has revolutionized the communications industry. A fiber consists of a
long thin thread of glass called the core surrounded by an outer shell of a material with a
lower index of refraction called the cladding. Light entering the end of the core of an optical
fiber is transmitted or piped to the other end with very little loss in intensity even though
the fiber may be bent in a circular shape. The thinness of the fiber and the lower index
of refraction in the cladding ensures that light will always strike the fiber side at an angle
greater than the critical angle for total internal reflection and be totally reflected back into
the fiber. Thus the light bounces off the inside as it caroms down the length of the fiber
following every bend or twist.
Find the optical fiber among the items on your lab table. Be careful in handling the
fiber. It is, after all, a piece of glass and will break if bent too sharply. Align one end of the
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CHAPTER 4: EXPERIMENT 2
fiber with the ray emerging from the ray box. Observe the light emerging from the other end
of the fiber. It will look like a bright pinpoint of light. Point this end of the fiber vertically
down onto a sheet of paper on the table with the end of the fiber held about five millimeters
above the table. You should see a bright spot on the paper where the light rays coming from
the fiber hit the paper. The light has been channeled through the fiber even around the
bends as the ray bounces off the inside surface of the fiber, possibly many times. The light
persists with little loss of intensity because of the great efficiency of these reflections that are
at such steep angles to the surface normal as to always be in the realm of total reflection.
Now, carefully rotate the incident end of the fiber to introduce and angle between the
incident ray and the fiber end’s normal. Observe that the spot on the paper has expanded
to form a ring of light. Note these observations and try to explain them in your Data.
Checkpoint
In the case of fiber optics do you expect the core or the cladding to have a greater
value for the index of refraction? Why are optical fibers immune to electrical noise?
4.4
Analysis
Which is larger between your ability to construct the light rays on paper and to measure the
angles and the difference between your incident and reflected angles? Between the accepted
index of refraction (1.495) of Lucite and those measured in Part 2 and/or Part 3? Between
the two measurements of Lucite’s index of refraction from Part 2 vs. Part 3? This last
question suggests the self-consistency among your measurements.
How accurately have you constructed and measured your ray diagrams? What other
sources of error have you noticed and how has each affected your data? Might these errors
be large enough to explain your differences?
4.5
Conclusions
What physical relations do your data support? Contradict? Which are not satisfactorily
tested? Communicate with complete sentences and define all symbols.
What have you measured that you might need to use in the future?
46
1
2
3
4
5
O
3
3
3
CHAPTER 4: EXPERIMENT 2
Figure 4.4: A template to aid the several measurements of index of refraction.
47
2
2
O
1
1
CHAPTER 4: EXPERIMENT 2
Figure 4.5: A template to aid the observation of total internal reflection and the critical
angle.
48