Download experiment 31 plane mirrors and index of refraction of light

EXPERIMENT 31
PLANE MIRRORS AND INDEX OF REFRACTION OF LIGHT
I. Object
The purpose of this experiment it to verify experimentally the behavior of electromagnetic
radiation at the interface of two media. We chose the visible region - light - of that radiation. In this
case the only phenomena occurring at the surface is reflection and refraction. The laws governing
these phenomena can be derived using geometric or ray optics that are based on Huygens's principle.
That is, the speeds and direction of a segment of the wavefront or the perpendicular to the wave
front, called the ray, are used to chart the progress of a wave. This procedure will yield:
A. the law governing the specular reflection (that is, from a smooth surface) of light incident
on a plane mirror and the position of the image of a source in front of the mirror;
B. the law governing the refraction of light by glass, and the index of refraction of the medium.
This last quantity is a measure of the speed of light in the medium and is defined as n = c/v
where c is the speed of light in vacuo (air) and v is the speed of light in the medium. For air,
n = 1.0;
C. The lateral displacement of a beam of light entering, traversing, and exiting a medium with
two parallel surfaces and an index of refraction greater than the surrounding media.
Reflection
In this case the speed of the wave is not changed because the wave remains in the same medium so
the speed of the reflected wave is the same as that of the incident wave. In Figure 1, the incident
wavefront is approaching the surface from the left and at an angle i to the surface. Let the medium
be air so that n=1.
Figure 1. Reflection of a plane wave at a smooth
mirror.
A point on the left of the incident wavefront touches the surface at A while a point to the right is
a distance BC = ct from the surface. As soon as the point on the left of the wavefront has reached
point A, a wavelet is generated and spreads with a speed c. After a time, t, the right end (B) reaches
point C. while the wavelet generated at A has traveled to point D, AD = ct. The triangles ABC and
ADC have a common side AC to that, BC = AD and
θ r = θi
(1)
This is the law of specular reflection and leads to the position of the image of the source.
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Image of a Source in Front of a Plane Mirror.
Now consider the spherical wavefronts emanating from a point source at S shown in Fig. 3. The
wavefronts (ray) strike the mirror and the reflected wavefronts (rays) obey Eq. 1. If the reflected rays
are continued in a backward direction, they appear to diverge from the point I which is an image of
the source. This image is called a virtual image because the source only appears to be there. The
human eye or a camera will interpret the rays as coming from I.
Fig. 2. The location of the virtual image of a point source in front of a plane mirror.
The relation of the image distance from the mirror is obtained from the geometry of Fig.2.
tan ( 90° − θi ) =
∴I = S
S
OA
and
tan ( 90° − θ r ) = tan ( 90° − θi ) =
I
OA
(2)
Refraction and index of refraction.
An optical medium is characterized by an index of refraction, n, which is defined as the ratio n = c/v
where c is the speed of light in a vacuum and v the speed of light in the medium. When the medium
on the other side of an interface is different, experience shows that the speed of the wave changes,
so the index of refraction is different. The wavelength, , of the wave also changes, Fig. 3, because
v = f and the frequency f is fixed by the source. Fig. 3. shows a wavefront in medium 1 incident
at an angle 1 and index of refraction n1, speed v1, wavelength 1. The corresponding quantities in
the second medium have the subscript 2 with n2 n1, so that v2 v1.
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Figure 3 Refraction of a plane wave traveling from a medium where its speed is v1 to a second
medium where its speed is v2 < v1
As soon as the wavefront enters the second medium, that portion of the wavefront slows down so
the point entering the second medium travels a distance AD = v2t in time t while the point at C
travels a distance CB = v1t. The angles the wavefronts make with the interface are
sin θ 2 =
AD v2t
ct
=
=
AC AC n2 AC
(2)
sin θ1 =
BC v1t
ct
=
=
AC AC n1 AC
(3)
and
Dividing (2) by (3)
n2sinθ2 = n1sinθ 1
(4)
This is Snell’s law of refraction. The index of refraction for the second medium is then easily
obtained from Eq, (4)
n2 =
sin θ1
n1
sin θ 2
(4a)
Generally, a beam of light is partially reflected and partially refracted when it is incident on a
smooth, clear surface. There is a special case of zero refraction when a ray of light traveling in a
medium of higher index of refraction strikes the interface with a medium of smaller index of
refraction, at some angle 1 such that the angle of refraction, 2 is 90 . In this case the refracted ray
travels along the interface of the two media and begins to be reflected into the first medium so that
the incident beam is totally reflected. This is called total internal reflection. This effect can be used
to determine the index of refraction of the denser medium (Eq. 4a).
Displacement of a beam traversing a medium with parallel surfaces.
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When a beam of light is incident of a slab of glass of thickness t with parallel surfaces it is refracted
at both surface, which leads to a lateral displacement, d, of the beam leaving the second interface
as shown in Fig. 4.
L
Fig. 4. The lateral displacement, d, of a beam of light traversing a medium of thickness t
with parallel sides and with an index of refraction greater than that of the surrounding
medium
The lateral displacement of the beam is obtained by noting
L=
sin (θ1 − θ 2 )
t
, and d = L sin (θ1 − θ 2 ) = t
cos θ 2
cos θ 2
(5)
II. Equipment and Procedure
Equipment
Optical bench, goniometer, line light source, plane mirror with holder, acrylic D-shaped lens, acrylic
rhomboid, prism
Procedure
This experiment requires careful alignment of the mirror, or the D-shaped lens, and the
direction of the incident beam.
A. Reflection from a plane mirror
Lay the light source on its side on the table and turn the screen in front of the sources so that there
is a single slit in front of the light source.
Place the goniometer and the light source, with a single vertical slit, shown in Figure 5, so the light
beam is along the normal line of the goniometer. Place the reflective surface of the plane mirror
with the middle point of the plane over the center of the goniometer (on the 90o -270o line) so that
it is perpendicular to the beam. The mirror (or surface) is perpendicular to the beam when the
reflected ray overlaps the incident beam.
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Fig.5. Setup to determine the angles of incident and reflection
Rotate the goniometer/mirror assembly so that the incident beam makes an angle θi , about 30E from
the normal to the mirror, and look for a reflected beam. You may have to incline the beam very
slightly to get a strong reflection along the goniometer. Measure the angles of incidence and
reflection from the normal to the mirror. Also measure the width of the incident and reflected beam.
This is a source of uncertainly. Repeat this procedure for two other angles of incidence and record
the data in a table
Position of the image of a source in front of a plane mirror
In this part of the experiment, you will to use parallax (the relative motion between two objects
when you move your eyes to-and fro in a plane perpendicular to them while looking at one of them)
to locate the position of the virtual image behind a mirror of a source in front of the mirror. Your
laboratory instructor may help you get started. Use one of the two thin rods provided as a source,
(A), and one to view the image, (B), as shown in Fig. 6
Fig 6 Position of object and image for a plane mirror
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Place the mirror about half way along the optical bench so that the silvered surface of the mirror is
at a marking on the measuring tape on the side of the bench. Record this position.
Place one of the thin rods at point A in front of the mirror, Fig. 6, and observe the image in the
mirror. For your safety, make sure that the corks are on the ends of the thin rods. Place a second
thin rod behind the mirror and adjust the position of the second rod at point B behind the mirror
until there is no parallax between it and the image of object at A in the mirror. There is no parallax
when you can move your eye from side to side while looking at the image and there is no apparent
motion of the rod at point B relative to the image of A in the mirror - the distance of the virtual
image behind the mirror coincides with rod at position B. Mark the positions of the rods at positions
at points A and B, and measure the distances from A(AM) and B(BM) to the silvered surface of the
mirror. Repeat the procedure for a different distance of the rod in front of the mirror. Record the
data in Table 2.
B. Refraction and Index of Refraction
Index of Refraction from a D-Shaped acrylic lens.
Fig 7. D-shaped lens on the goniometer and rotated through and angle θ2
Place the D-shaped acrylic lens on the outlined shape of it on the goniometer (Fig. 7) so that the flat
surface is perpendicular to the beam. Rotate the goniometer to about 30E. Record the angle on the
emerging beam to the normal. This is θ2 for use in Eq. (4.a).
Enter the data in table 3. To obtain the index of refraction of the lens using the internal reflection
of the beam, place the D-shaped lens in the outline and such that the half-way point along the curved
surface faces the beam and is perpendicular to it; the flat surface also will be perpendicular to the
beam. Rotate the goniometer table/lens; there will be a weak reflected and a refracted beam (exit)
beam; until the refracted beam is along the flat surface of the lens. A small angular increment will
yield total internal reflection. Record the angle of incidence the beam makes in the lens with the
normal to the flat surface of the lens. Calculate the index of refraction of the lens. Notice that the
angle of reflection is now equal to the angle of incidence as previously determined with the mirror.
Index of Refraction for a prism. (Optional)
Place the prism on a white sheet of paper. Shine the incident beam shine on the angular part of the
prism. Rotate the prism slightly to get a large deviation as shown in Figure 8,
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Fig. 8. Experimental setup to determine the index of refraction of a glass prism.
Mark two points along the incident beam and two points along the exiting beam. Carefully outline
the prism with a pencil, then remove the prism and the pins. Draw the solid line, as shown in Fig.8,
representing the path of the ray through the prism. Construct a normal line to the surfaces where
the ray enters and leaves the glass. The angle of incidence (θi1) of the entering ray and the angle of
refraction (θr1) of the ray in the glass are shown in Figure 8, as are angles for the existing rays, θi2
and θr2.
Measure the angle θi1 and θr2 with a protractor and calculate the index of refraction for the glass
prism. Record all information in Data Table 4.
C. Lateral Displacement of a beam traversing a medium with parallel sides.
Measure the thickness, t, of the rhomboid. Place the rhomboid on the goniometer with the short face
facing the beam and along the line marked “Component” and centered on the center of the
goniometer (where the incident beam is directed). Set the beam at 90Eto the surface (along the line
marked “Normal”). Rotate the goniometer table/rhomboid until the beam makes an angle of 30E (or
more) to the normal t the short face. A refracted beam will appear outside the second surface. It will
be parallel to, but displaced a distance d from the continuation of the incident beam. Carefully
measure and record the angle of incidence, θ1, of the beam with the first (short) face and, using a
vernier caliper, the displacement of the beam from the continuation of the incident beam to the
center of the refracted beam. To determine the angle θ1 - θ2, place the flat side of the large Dshaped acrylic piece against the long side of the rhomboid and determine the angle of the beam in
the material with respect to the continuation of the original beam. The angle θ2 also can be
determined by the angle θ1 and the index of refraction determined by the two methods in B1 and (Eq.
4a).
Calculate the lateral displacement of the beam from Eq. 5.
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III Data.
A. Angle of reflection from a plane mirror.
Angles of incidence and reflection for the assigned angles of incidence.
Table 1
Furnish a Heading
Trial
Angle of
Incidence
Estimated
Error
Angle of
Reflection
Estimated
Error
1
±
±
2
±
±
3
±
±
Position of the image of a source in front of a plane mirror
Table 2
Furnish a Heading
Trial
Object Distance AM (m)
Image Distance BM (m)
1
2
B. Index of Refraction for the D-shaped lens
Table 3 Furnish a Heading
Surface
Angle
Incidence θi
Sinθi
Angle of
Refraction θr’
Sinθr’
1
2
Incident angle that yields total internal reflection _________________
Index of Refraction form a Glass Prism
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Index of
Refraction
Table 4 Furnish a Heading
Surface
Angle of
Incidence θi
Sinθi
Angle of
Refraction θr’
Sinθr’
Index of
Refraction
(1)
(2)
C. Lateral displacement of the beam.
Thickness of the rhomboid_______________
Angle of incident on the first face __________
Displacement of the beam____________
IV. Results
A. Mirror.
Angle of Incidence and Reflection.
Carefully compare the angles θi and θr for each trial. Find the general rule for reflection of a
light ray from the surface of a smooth plane mirror. Considering the precision of your
measuring instruments, estimate the error in your measurement of θi and θr. Express your error
estimates as ± some number of degrees or tenths of degrees. (Example: 16.3E ± 3.5E).
Position of the image of a source in front of a plane mirror.
Calculate the percent difference between your two values for the object and image distances in
the trials.
B. Index of refraction from D-shaped lens.
Index of refraction using angles for incident and refracted ray and from the angle for total
internal reflection. Percent error from the given value in each case.
Percent difference in index of refraction by the two methods______________
Prism
Side(1)___________ Side (2)__________
Average_________
% Diff__________
C. Lateral displacement of the beam
Lateral displacement of the beam from measurement and Eq. 5 and percent difference.
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V. Conclusion and Discussion
A. Discuss the verification of the law of reflection for plane mirrors. Considering the accuracy
of your measuring instruments, estimate the error in your measurement of θi and θr. Express
your error estimates as ± some number of degrees or tenths of degrees. (Example: 16.3E ±
3.5E). Has the experiment verified the law of reflection for plane mirrors?
Discuss the position and orientation of the image of an object in front of the mirror. What
is your percentage error? Is it sufficiently small so as to verify Eq. 2?
B. Do the indices of refraction for the D-shaped lens obtained by the two procedure compare
favorably? Given their respective percent errors and the different procedures used to obtain
the index of refraction, is one of the method’s preferable?
Index of refraction for the prism and error.
C. Does the measured value of the lateral displacement of the beam verify the distance given
by Eq. 5? Discuss any uncertainties in the measurement.
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