26
Refraction of Light
Refraction of Light
Finding the Index of Refraction and the Critical Angle
OBJECTIVE
Students will verify the law of refraction for light passing from water into air. Measurements of the
angle of incidence and the angle of refraction, along with the critical angle will be utilized to determine
the index of refraction of water.
T E A C H E R
P A G E S
LEVEL
Physics
NATIONAL STANDARDS
UCP.3, A.1, A.2, B.6
TEKS
2(A), 2(B), 2(C), 2(D), 2(E), 2(F), 3(A), 7(B), 8(A)
CONNECTIONS TO AP
IV. Waves and optics, C. Geometrical optics, 1. Reflection and refraction
TIME FRAME
45 minutes
MATERIALS
(For a class of 28 working in pairs)
14 laser pointers
water
14 protractors or sheets of polar graph paper
14 metric rulers
14 semicircular plastic dishes
paper
14 viewing screens or backing papers
powdered non-dairy creamer
Jello® (optional)
TEACHER NOTES
A nice opening demonstration is to place a meter stick into a small aquarium filled with water.
Demonstrate the apparent bending of the meter stick by inserting it at a 45˚ angle. Point out that the
apparent bending is due to the way light interacts with matter. Further reinforcement of light refraction
can be done with a demonstration using a penny placed at the bottom of an opaque coffee cup and
slowly filling the cup with water allowing the penny to come into view to a student standing far enough
away that he or she could not see the coin initially.
Laser pens or pointers may be purchased at any discount store or scientific catalogue, and are much less
expensive than a decade ago. The semicircular plastic dishes can be purchased from any of several
scientific catalogues, such as Sargent Welch www.sargentwelch.com. Jello® may be substituted as the
medium for water. It is actually much easier to see the angles of reflection and refraction with Jello® as
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Refraction of Light
26
the medium. You may want to have the students perform the experiment with water first, and then a
more dense medium such as Jello®.
Light and matter appear quite different, but there must be an underlying connection at some level
because they interact with each other. Interaction implies some fundamental relationship between them.
To observe and verify this interaction between light and matter, we will determine the index of
refraction of water and the critical angle of water.
In any homogeneous material, light travels in straight lines. When light encounters a boundary (a change
in optical medium) some of the light reflects back obeying the law of reflection and some of the light is
transmitted into the new medium. The transmitted light does not travel in the same direction as the
original light. Instead it is bent (refracted) at the boundary and travels in a different direction. This
phenomenon is called refraction.
T E A C H E R
Reflected
light ray
Incident
light ray
Refracted
light ray
The refraction of light at the interface between two materials is described mathematically by Snell’s
Law. In Figure 1 above, the long dashed line represents the normal, a line perpendicular to the surface.
The angle θ measures the angle of incidence relative to the normal. The angle φ measures the angle of
refraction relative to the normal. Snell’s Law states that
ni sin θ = nr sin φ
The quantity ni is the index of refraction for the medium in which the light was incident. The quantity
nr is the index of refraction for the medium in which the light was refracted. The index of refraction n of
a material is a measure of the speed of light in that medium. It is defined as the ratio of the speed of light
in vacuum c to the speed of light v in the medium.
n =
c
v
The index of refraction of the vacuum is 1. The index of refraction of air, which depends somewhat on
the temperature and density of the air, is very nearly 1 as well.
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P A G E S
Figure 1
26
Refraction of Light
POSSIBLE ANSWERS TO THE CONCLUSION QUESTIONS AND SAMPLE DATA
DATA AND OBSERVATIONS
P A G E S
Index of Refraction Worksheet
90°
T E A C H E R
80°
70°
60
°
50
°
°
40
°
30
20°
10°
0°
614
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Refraction of Light
sin θwater
sin θair
nwater
5˚
5˚
0.087
0.087
1.00
10˚
12˚
0.174
0.208
1.19
15˚
18˚
0.259
0.309
1.19
20˚
26˚
0.342
0.438
1.28
25˚
33˚
0.423
0.545
1.29
30˚
41˚
0.500
0.656
1.31
35˚
50˚
0.574
0.766
1.34
40˚
60˚
0.643
0.866
1.35
45˚
69˚
0.707
0.934
1.32
50˚
___
___
___
___
55˚
___
___
___
___
60˚
___
___
___
___
65˚
___
___
___
___
70˚
___
___
___
___
75˚
___
___
___
___
80˚
___
___
___
___
85˚
___
___
___
___
P A G E S
θair
T E A C H E R
θwater
26
Critical angle = 49.7˚
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Refraction of Light
ANALYSIS
1. Calculate the index of refraction of water for each incident angle using Snell’s Law. Record your
values for nwater in the data table. Average all of your measurements. Calculate your percent error.
The accepted value for the index of refraction for water is 1.33.
• nw sin θ w = na sin θ a
•
•
Average of all your measurements: nwater =
1.25
% Error =
5.85%
2. On the axes below, plot a graph of sin θair (y-axis) vs. sin θwater (x-axis). Be sure to sure to use proper
graphing techniques, including a title, scaling, labeling the axes, and drawing the best-fit curve that
represents the average of the data.
T E A C H E R
P A G E S
•
1.00 (sin 45°)
= 1.32
sin 69°
1.00 + 1.19 + 1.19 + 1.28 + 1.29 + 1.31 + 1.34 + 1.35 + 1.32
= 1.25
naveragewater =
9
1.33 − 1.25
× 100 = 5.85 %
% error =
1.33
nw =
Angle of Incidence (º)
Linear Fit For: Angle of Incidence vs. Angle of Refraction: Angle of Incidence
y = mx+b
m(Slope): 1.39 º/º
b(Y-Intercept): –0.0373º
Correlation: 1.00
Angle of Refraction (º)
616
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Refraction of Light
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3. Determine the index of refraction from the slope of the graph.
• Slope = 1.39
Graphical Estimate of nwater =
1.39
% Error =
4.5%
4. In the space provided, calculate the index of refraction for water and your percent error.
n
• sin θ c = air
nwater
•
nwater =
nair
sin θ c
Critical angle =
=
1.00
= 1.31
sin(49.7°)
49.7˚
nwater(from the critical angle) =
1.31
% Error =
1.41%
T E A C H E R
5. Using nwater = 1.33, determine the velocity of light in water.
m
3.0 × 108
s
• 1.33 =
vwater
•
vwater = 2.26 × 108 m/s
•
The light ray will bend toward the normal in a medium with a large index of refraction.
7. What happens when light travels to a medium of lower refractive index?
• The light speeds up in the new medium and bends away from the medium.
8. Will light be refracted more while passing from air into water or while passing from water into glass
(n = 1.50)? Explain.
• Light will be refracted more while passing from air into water since the difference in the indexes
of refraction for the two media is greater than for light going from water into glass.
9. Will light traveling from air into water undergo total internal reflection? Explain.
• No, total internal reflection occurs only when light goes from a more optically dense medium to
a less optically dense medium.
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P A G E S
6. If a medium has a large index of refraction, what does that say about the speed of light in that
medium? What can you say about the way the light ray bends in relation to the perpendicular (or the
normal) to the surface to the media?
• A large index of refraction indicates that light will slow down more than in a medium with a
smaller index of refraction.
26
Refraction of Light
CONCLUSION QUESTIONS
P A G E S
1. A diligent physics student is given the following equipment: a transparent acrylic cube, a visiblespectrum laser, a metric ruler, a protractor, and a viewing screen. Her instructor asks her to devise a
method to measure the index of refraction of the transparent solid. After much reflection, the lights
come on and she readily measures the index of refraction of the transparent solid. Describe in detail
how she determined the index of refraction. The results of her illumination are shown in the two
diagrams below. P2 is the path of the light from the laser through the air and P1 represents the path of
the light through the transparent cube.
s
P1
T E A C H E R
•
P2
The student measures the length s of one side of the cube. She determines the angle of refraction
by examining the distance between the undeflected laser beam and the exit point P1 of the beam
in the cube. Taking the arctangent of the ratio of l1 to side s gives the refraction angle. Likewise,
the distance l2 between the undeflected laser beam and the exit point P2 yields the angle of
incidence.
θ22
θ11
s
s
l2
l1
P2
P1
•
•
•
618
θ1 = tan −1
l1
θ 2 = tan −1
l2
s
s
Finally by applying Snell’s Law the index of refraction of the transparent solid is readily
determined.
n1 sin θ1 = n2 sin θ 2
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Refraction of Light
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2. Use Snell’s Law to determine the path of the light through this rectangular sheet of glass ( n = 1.50).
Draw a normal, perform the appropriate measurements and calculations for the entry point and draw
the refracted ray for the light entering the glass. Continue the ray to the opposite side, draw a normal,
perform the appropriate measurements and calculations for the exit point and draw the refracted ray
for the light exiting the glass. Show all your calculations in the space below.
T E A C H E R
n1 sin θ1 = n2 sin θ 2
•
•
•
•
(1.00) sin 44° = 1.50 sin θ
θ = 27.6°
(1.50) sin 27.6° = (1.00) sin θ
θ = 44°
P A G E S
•
3. Use Snell’s Law to determine the path of the light through the triangular glass ( n = 1.50 ). Draw a
normal, perform the appropriate measurements and calculations for the entry point and draw the
refracted ray for the light entering the glass. Continue the ray to the opposite side, draw a normal,
perform the appropriate measurements and calculations for the exit point and draw the refracted ray
for the light exiting the glass. Show all you calculations in the space below.
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T E A C H E R
P A G E S
26
Refraction of Light
•
n1 sin θ1 = n2 sin θ 2
•
•
•
•
(1.00) sin 46° = (1.50) sin θ
θ = 28.7°
(1.50) sin 28.7° = (1.00) sin θ
θ = 46°
4. In our experiment, the beam of light actually passed through three different media (air, plastic, and
water). We assumed that the interaction of the light with the plastic could be ignored. Is that
assumption reasonable or is it an additional source of error? We will examine this question by
imagining three media in layers as shown in the diagram below. The beam passes from air into
medium X and then from medium X into the water. There are four angles to measure. Use your
knowledge of geometry and Snell’s Law to find the relationship between angles 1 and 4. Given that
relationship, how important is it to find the index of refraction of X, assuming we are interested in
knowing the index of refraction of the water?
1
Air
3
X
2
Water
4
•
620
Angles 2 and 3 are congruent since they are alternate interior angles. Hence angle 1 and angle 4
must also be congruent by applying Snell’s Law. The angle of incidence from the air into
material X is equal to the angle of refraction from material X into the air. Thus, medium X can
be ignored, just as we ignored the plastic dish in determining the index of refraction of water.
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Refraction of Light
Finding the Index of Refraction and the Critical Angle
Light and matter appear quite different, however, there must be an underlying connection at some level
because they interact with each other. Interaction implies some fundamental relationship between them.
To observe and verify this interaction between light and matter, you will determine the index of
refraction of water and the critical angle of water.
In any homogeneous material light travels in straight lines. When light encounters a boundary (a change
in optical medium) some of the light reflects back obeying the law of reflection and some of the light is
transmitted into the new medium. The transmitted light does not travel in the same direction as the
original light. Instead it is bent (refracted) at the boundary and travels in a different direction. This
phenomenon is called refraction.
Reflected
light ray
Incident
light ray
Refracted
light ray
Figure 1
The refraction of light at the interface between two materials is described mathematically by
Snell’s Law. In Figure 1 above, the long dashed line represents the normal, a line perpendicular to the
surface. The angle θ measures the angle of incidence relative to the normal. The angle φ measures the
angle of refraction relative to the normal. Snell’s Law avers:
ni sin θ = nr sin φ
The quantity ni is the index of refraction for the medium in which the light was incident. The quantity
nr is the index of refraction for the medium in which the light was refracted. The index of refraction n of
a material is a measure of the speed of light in that medium. It is defined as the ratio of the speed of light
in vacuum c to the speed of light v in the medium.
n =
c
v
The index of refraction of vacuum is 1. The index of refraction of air, which depends somewhat on the
temperature and density of the air, is very nearly 1 as well. We will use this approximation for the index
of refraction of air.
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Refraction of Light
The phenomenon of total internal reflection occurs when the light travels from a medium with a higher
index of refraction to a medium with a lower index of refraction. When ni > nr , the refracted ray bends
away from the normal. If the angle of incidence is large enough, the angle of refraction will be 90˚ and
the light travels parallel to the interface between the two media. The angle of incidence for which this
occurs is called the critical angle. If the angle of incidence is increased further, then the calculated value
of the angle of refraction is greater than one which is mathematically impossible! At the critical incident
angle, the light does not pass through the surface, it reflects off the surface, such that the surface
becomes a mirror and obeys the law of reflection. Therefore, the critical angle for light passing from a
more dense medium of n1 to a less dense medium of n2 , where n1 > n2 , is
sin θ c =
n2
n1
The phenomenon of total internal reflection is important in numerous fiber optic technologies, from
communication to surgical procedures. Total internal reflection explains how light (and information) can
be transmitted great distances with little loss of energy.
PURPOSE
In this activity you will investigate the refraction of light as it passes from water into air. Measurements
of the angle of incidence and the angle of refraction, along with the critical angle will be utilized to
determine the index of refraction of water.
MATERIALS
laser pointer
water
protractor or polar graph paper
metric ruler
semicircular plastic dish
paper
viewing screen or backing paper
powdered non-dairy creamer
Safety Alert
Caution – Do NOT look into the laser and do NOT direct the laser at others.
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Refraction of Light
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PROCEDURE
1. Place the Index of Refraction Worksheet on a flat surface or table. Fill a semicircular dish with water
and center the semicircular dish on the outline of the dish.
2. Sprinkle a small amount of non-dairy creamer on the water. This will make the laser beam visible in
the water.
3. The dish has an etch mark on its flat side at the center of the semicircle. Shine the laser into the dish
through the curved wall of the dish. Aim the beam so that it hits the etch mark on the flat wall of the
semicircle. Vary the angle beginning with an incidence angle of 5˚ and approaching 90˚, by moving
the laser pen around the curve of the dish. Always shine the laser perpendicular to the curved wall so
that the beam strikes the etch mark (midpoint) of the flat wall. Place a viewing screen or some
backing paper opposite the flat wall of the dish and perpendicular to the flat surface or table. The
purpose of the viewing screen or backing paper is to help you locate the exit point of the laser beam
and measure the angle of refraction for each angle of incidence.
4. Draw a ray on the worksheet from the center of the semicircle (the etch mark on the flat surface)
through each of the angles of refraction and extend it to the margin of the paper. Draw an arrowhead
on each incident ray and all the refracted rays you used showing the path of the light. Measure the
angle that the refracted rays make with the normal and record them in the data table. Fill in as much
of the table as possible. Use the data and Snell’s Law to determine the index of refraction.
5. Move the laser pen around the curved surface of the semi-circular dish and observe the phenomenon
of total internal reflection. At some position, the light ray exiting the flat side will reach an angle of
90˚ and then reflect back out the curved side of the dish. When the refracted angle reaches 90˚ draw
a line along the flat side of the dish (parallel to the interface between the air and the water). Draw a
line perpendicular to and through the center of the flat side of the dish. Measure the incident and
reflected angles. The angle of incidence (which should also be the angle of reflection) is the critical
angle. Determine the index of refraction for the water using the relationship:
sin θ c =
n2
n1
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Refraction of Light
Name _____________________________________
Period _____________________________________
Refraction of Light
Finding the Index of Refraction and the Critical Angle
DATA AND OBSERVATIONS
Index of Refraction Worksheet
90°
80°
70°
60
°
50
°
°
40
°
30
20°
10°
0°
624
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Refraction of Light
θwater
θair
sin θwater
sin θair
26
nwater
5˚
10˚
15˚
20˚
25˚
30˚
35˚
40˚
45˚
50˚
55˚
60˚
65˚
70˚
75˚
80˚
85˚
Critical angle = _____________˚
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Refraction of Light
ANALYSIS
1. Calculate the index of refraction of water for each incident angle using Snell’s Law. Record your
values for nwater in the data table. Average all of your measurements. Calculate your percent error.
The accepted value for the index of refraction for water is 1.33.
Average of all your measurements: nwater = _____________
% Error = _____________
2. On the axes below, plot a graph of sin θair (y-axis) vs. sin θwater (x-axis). Be sure to sure to use proper
graphing techniques, including a title, scaling, and labeling the axes, and drawing the best-fit curve
that represents the average of the data.
626
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Refraction of Light
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3. Determine the index of refraction from the slope of the graph.
Graphical Estimate of nwater = _____________ % Error = ____________
4. In the space provided calculate the index of refraction for water and your percent error.
Critical angle = _______˚ nwater (from the critical angle) = ____________
% Error = _____%
5. Using nwater = 1.33, determine the velocity of light in water.
6. If a medium has a large index of refraction, what does that say about the speed of light in that
medium? What can you say about the way the light ray bends in relation to the perpendicular (or the
normal) to the surface to the media?
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26
Refraction of Light
7. What happens when light travels to a medium of lower refractive index?
8. Will light be refracted more while passing from air into water or while passing from water into glass
(n = 1.50)? Explain.
9. Will light traveling from air into water undergo total internal reflection? Explain.
CONCLUSION QUESTIONS
1. A diligent physics student is given the following equipment: a transparent acrylic cube, a visiblespectrum laser, a metric ruler, a protractor, and a viewing screen. Her instructor asks her to devise a
method to measure the index of refraction of the transparent solid. After much reflection, the lights
come on and she readily measures the index of refraction of the transparent solid. Describe in detail
how she determined the index of refraction. The results of her illumination are shown in the two
diagrams below. P2 is the path of the light from the laser through the air and P1 represents the path of
the light through the transparent cube.
s
P1
628
P2
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Refraction of Light
26
2. Use Snell’s Law to determine the path of the light through this rectangular sheet of glass ( n = 1.50).
Draw a normal, perform the appropriate measurements and calculations for the entry point and draw
the refracted ray for the light entering the glass. Continue the ray to the opposite side, draw a normal,
perform the appropriate measurements and calculations for the exit point and draw the refracted ray
for the light exiting the glass. Show all your calculations in the space below.
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26
Refraction of Light
3. Use Snell’s Law to determine the path of the light through the triangular glass ( n = 1.50 ). Draw a
normal, perform the appropriate measurements and calculations for the entry point and draw the
refracted ray for the light entering the glass. Continue the ray to the opposite side, draw a normal,
perform the appropriate measurements and calculations for the exit point and draw the refracted ray
for the light exiting the glass. Show all your calculations in the space below.
4. In our experiment, the beam of light actually passed through three different media (air, plastic, and
water). We assumed that the interaction of the light with the plastic could be ignored. Is that
assumption reasonable or is it an additional source of error? We will examine this question by
imagining three media in layers as shown in the diagram below. The beam passes from air into
medium X and then from medium X into the water. There are four angles to measure. Use your
knowledge of geometry and Snell’s Law to find the relationship between angles 1 and 4. Given that
relationship, how important is it to find the index of refraction of X, assuming we are interested in
knowing the index of refraction of the water?
1
Air
3
X
2
Water
4
630
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Refraction of Light
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