Download Reflection and Refraction of Light

LPC Physics
Reflection and Refraction of Light
Reflection and Refraction of Light
Purpose:
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To become acquainted with the reflective properties of convex and concave mirrors.
To become familiar with the properties of refraction.
To become familiar with the focal properties of reflection and refraction.
To measure the focal length of a lens.
To set up a refracting telescope and compare 2 different eyepieces
Equipment:
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Pasco optical bench and accessories, holders
Rive Ray Box
Ray Box Accessories
Red reading lights
Lasers
Calipers
Theory:
Reflection
The reflection of light from a plane surface is described by the law of reflection, which
states that the angle of incidence, θi, is equal to the angle of reflection, θr, as can be
derived by Fermat’s Principle. By convention, these angles are measured with respect to
a line perpendicular to the plane surface. Reflection from a plane mirror or a flat
transparent plane surface of a piece of glass or plastic are the most easily demonstrated
examples of the law of reflection.
In Figure 1(a) several rays are shown incident on a plane surface, and in each case
the reflected ray is also shown. For each ray, the angle of incidence θi is seen to be equal
to the angle of reflection θr.
Reflection at Plane Surface
Refraction at Plane Surface
Reflecting Surface
n1 sinθ1 = n2 sinθ2
θ2
n2
Ray 1
Ray 1
n1
Ray 2
Ray 2
Ray 3
Ray 3
θ1
Normal
to Surface
Normal
to Surface
(a)
(b)
Figure 1 Illustration of reflection and refraction of light rays at a plane surface.
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Refraction
A light ray incident on a plane surface that forms the boundary between two transparent
media can be reflected at the surface or transmitted into the second medium. In general,
light rays incident on a plane interface will be partially reflected and partially transmitted.
The ray that is transmitted does not continue in the same direction. Instead it undergoes a
change in direction. The ray is said to be refracted. This is illustrated in Figure 1 (b).
The angle of incidence is θ1, and the angle of refraction is θ2. The change in direction of
the light ray is caused by the fact that the speed of light is different within different
media.
The speed of light in a vacuum is c, and it represents the maximum possible speed
of light. For any material, the speed of light is v, where v < c. A quantity called the
“index of refraction,” or n, for any medium is defined by n = c/v. Note that since v < c
then the only allowed values of n are n > 1.
The relationship between the angle of incidence θ1 and the refracted angle θ2 is an
expression involving the indices of refraction of the two media, n1 and n2. The
relationship is known as Snell’s law, and it is
n1 sin θ 1 = n2 sin θ 2
Eq. 1
When n1 > n2, Eq. 1 implies that θ1 < θ2. This states that a ray going from a
medium of a given index of refraction to one of a smaller index of refraction is bent away
from the normal. If n1 < n2 then it must be true that θ1 > θ2, and thus a ray going into a
medium of larger index of refraction is bent toward the normal.
Focal Properties of Reflection and Refraction
Descriptions of the focal properties of reflection from spherical mirrors are shown in
Figure 2. When parallel rays are incident on a concave spherical surface, reflected rays
are converging and come to an approximate point focus. If R is the radius of curvature of
the spherical surface, the focal point is a distance f from the vertex of the spherical
mirror, where f = R/2. The distance f is called the “focal length” of the mirror and is
positive by convention for a concave converging mirror. The reflection of parallel rays
incident on a convex spherical mirror are diverging, but they appear to have come from a
single point. The distance from the vertex of the mirror to that point is called the “focal
length,” and its magnitude is given by f = R/2. By convention, the focal length is
negative for a convex diverging mirror.
Convex Diverging
Spherical Mirror
Concave Converging
Spherical Mirror
R
R
f
f
Figure 2 The focal properties of spherical mirrors for incident parallel rays.
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Light incident on a converging lens parallel to the axis of the lens intersects the axis at a
point called the focal point, F. The distance from the lens to the focal point is called the
focal length, f, --see Figure 3 below. In general, the object distance, s, the image distance
s', and the focal length are related by the "lens maker's formula,"
1 1 1
+ =
Eq. 2
s s'
f
where s, and s' are as indicated in Figure 3.
If an image is formed on the opposite side of the lens from the object (right side in
Figure 3(a) ) then the image is said to be real and the value of s' is positive.
If, as in Figure 3(b), the image is formed on the same side as the object, then the image is
said to be virtual, and the value of s' is negative.
A diverging lens also has a focal length, but this is found by tracing diverging rays back
until they intersect the axis of the lens (see Figure 3(c) below). The focal length for a
diverging lens is always negative.
The magnification of a lens, or of a system of lens is defined to be the ratio of the image
size to the object size, as is often more convenient, the ratio of the angular diameter of
the image to the angular diameter of the object. For a telescope composed of two
lenses of focal length f1 and f2, the magnification, m, should be equal to the ratio
m=−
f1
f2
Eq. 3
where f1 is the objective and f2 is the eyepiece.
image
object
f
f
(3a) An object outside of the focal length of a converging lens produces a real
image. The image is inverted and magnified.
f
image object
f
(3b) An object inside the focal length of a converging lens produces a virtual
image. The image in upright and magnified.
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object
Reflection and Refraction of Light
f
image
f
(3c) An object outside the focal length of a diverging lens produces a virtual
image. The image is upright and diminished.
f
object
image
f
(3d) An object inside the focal length of a diverging lens produces a virtual
image. The image is upright and diminished.
Figure 3 Using principal rays to determine the image location.
Exploring Telescopes
There are a few factors to consider when shopping for a telescope, or simply comparing
different types. The primary consideration is the size of the Objective (lens or Mirror).
In general, the larger the objective, the greater the amount of the light the telescope can
gather (and hence the shorter the exposure time), the greater the resolving power (and
hence the more detail that can be seen), and the greater the “effective”maximum
magnifying ability of the telescope.
Magnification
Magnification depends on both the focal length of the telescope and the focal length of
the eyepiece according to the relation:
f
m = − tel
f eye
Thus, telescopes with the same focal lengths will have different magnifications with
different eyepieces. For example, a telescope with a focal length of 100 centimeters and
an eyepiece with focal length of ten millimeters will have a magnification of
100
m=
= 100
1.0
Note that eyepiece focal lengths are typically given in millimeters. Also notice that both
focal lengths must be in the same units (i.e. centimeters). As you can see, the “smaller”
the eyepiece, the greater the magnification it can provide. The focal length of a telescope
depends on the curvature its lens or mirror.
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The F Ratio
It is standard in the telescope-making world to list both the diameter of a telescope and its
“f ratio.” While this sounds confusing, it is really quite simple. The “f ratio” is simply
the ratio of the focal length of the telescope to the diameter of its objective. If the
telescope described above (with a focal length of 100 cm) had a mirror 10 cm in
diameter, then its f ratio would be:
100
F ratio =
= 10
10
F ratios are commonly written is a special format. For this particular telescope, its f ratio
would be written as f/10. Commonly, the diameter and the f ratio are listed together-- for
example “a 10 cm objective, and f/10”. These two bits of information allow you to easily
calculate the focal length (10 x the diameter = 100 cm). Telescopes with small f ratios
are considered to be “fast” while telescopes with large f ratios are not so fast, less fast, or
downright slow…
The designation of fast vs. slow has to do with imaging and exposure times. The shorter
the necessary exposure time, the faster the telescope. A 10-cm, f/10 would provide high
magnification but would cut down on the light available “per pixel”. A 10 cm f/5 would
provide less magnification, but would provide twice as much light per pixel and would
thus be twice as fast, or require one half of the exposure time.
Resolving Power and Magnification
Returning to the issue of magnification, recall that the magnification for a given telescope
depends on the focal length of the eyepiece being used. For the telescope described
above, changing from a 10-mm eyepiece, to a 5 mm eyepiece would change the
magnification from 100 to 200. With this in mind, why do we usually use longer focal
length eyepieces (i.e. around 20 to 50 mm), as opposed to much shorter eyepieces that
would provide unlimited magnification? The answer has to do with the amount of detail
that can be seen. Basically, if you magnify a slightly fuzzy image, you will get a larger,
very fuzzy image. Technically, this is described as the resolving power of the telescope:
λ
α = 1.22
D
where α is the smallest angular separation between two objects that can be resolved, λ is
the wavelength of the light being viewed, and D is the diameter of the Telescope’s
objective. If we assume green light, then a handy formula is:
14
α=
D
where D is in centimeters, and α is in arc seconds. Thus a 10-centimeter objective would
provide resolving power of 1.4 arc seconds. If the telescope has a resolving power of 1.4
arc seconds, it would make no sense to magnify an image so large that angular
separations smaller than 1.4 arc sections are visible. Such small details would be
“blurred out” in the magnified image.
Effective Maximum Magnification
Problems arise when the atmosphere comes into play, as it always does on Earth.
Typically, light from a distant source is refracted by the moving atmosphere and creates a
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“seeing disk” on the retina of the viewer, or on the detector. This seeing disk is typically
larger than one arc second, meaning that a 14 cm telescope would achieve the same
theoretical resolution as a 14-meter telescope because of atmospheric limitations. That’s
not the whole story, however, since the effective magnification of the telescope is also
related to the size of the objective (in general, the more light the clearer the image) by the
relation:
mmax = 20D where D is in centimeters.
Thus the telescope described above would have a maximum effective magnification of
200
Experiment:
Part A: Reflection
1. Adjust the screen in the front of the Ray Box so that one ray is emitted.
2. Place a 60.0o prism on a piece of white paper with one of its triangle-faces down.
Draw a straight line along the face of the prism for one of these surfaces where it
touches the paper as shown in the figure. Place a small dot in the center of the line as
shown.
Line drawn along
prism face
Reflected Ray
Incident ray
Figure 4
3. Place the ray box about 0.15 m away from the prism and adjust the single ray to strike
the plane surface at the position of the dot at an angle of incidence estimated to be
about 60o. With a straight edge, draw a line in the direction of the incident ray and
one in the direction of the reflected ray. Repeat this process two more times, once for
an incident ray of about 45o, and again for an incident ray of about 30o.
4. At the point of the dot, construct a perpendicular to the face of the prism. Extend all
six of the lines showing the ray directions until they intersect at the point of the dot.
Using a protractor, measure the incident angles θ1i, θ2i, and θ3i and the reflected
angles θ1r, θ2r, and θ3r for each of the rays. Record all these angles (to the nearest
0.1o) in the Data Table.
Part B: Refraction
1. Place the prism on the paper so that the face of the prism on the paper is an equilateral
triangle. Draw straight lines on the paper along two adjacent faces of the prism. The
prism may have one side that is frosted. Make sure the beam is not set to travel
through this side.
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2. Place the ray box about 0.15 m away from the prism. Adjust the direction of the ray
box so that the incident ray strikes one face of the prism at an angle of about 50o
relative to the normal to the prism face. Using a straight edge, draw a line in the
direction of the incident ray and one in the direction of the refracted ray.
(a)
(b)
Incident ray
Reflected ray
Lines drawn at
faces of the prism
Figure 5
3. Using a separate sheet of paper for each ray, repeat steps 1 and 2 for two other rays,
one incident at an angle of about 60o with respect to the normal and the other incident
at about 70o with respect to the normal.
4. Construct the lines tracing the path of each of the incident rays through the prism, and
draw in a perpendicular at the point of intersection of the rays to the prism face. This
will produce a figure from which angles θ1, θ2, θ3 and θ4 can be determined with a
protractor. Measure these angles for each of the three cases and record the values of
all four angles (to the nearest 0.1o) in the Data Table.
Part C: Focal Properties of Reflection and Refraction
1. Turn the screen so that five rays are produced. You may need to adjust the distance
of the bulb from the screen to produce a set of parallel rays. This is done by simply
sliding the green metal box within the gray plastic base.
2. Place the circularly shaped metal reflector on a piece of white paper and trace its
outline on the paper. Place the ray box about 0.15 m away on the concave side of the
reflector. Align the five parallel rays with the center of the reflector to produce a
pattern like the one in Figure 2(a). Make a tracing of this pattern, and from it,
measure the focal length of the concave reflector. Record it in the Data Table as fcon.
3. Turn the reflector around and repeat Step 2 on another piece of paper with the
reflector now acting as a convex mirror. Trace the pattern, which should look like
that of Figure 2(b). Extend the reflected rays back to the point from which they
appear to come. Measure the focal length and record it in the Data Table as fdiv.
4. Using a compass, construct a circular arc that is the same radius of curvature as the
reflector. Record in the Data Table the radius of that constructed circle as R, the
radius of curvature of the reflector.
5. Place the plastic converging and diverging lenses on separate pieces of paper and
trace the ray patterns produced by the parallel beam of rays from the ray box.
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Measure the focal length of the converging lens and record it in the Data Table as fcon.
Measure the diverging lens focal length and record it in the Data Table as fdiv.
Part D: Measuring focal length of convex (positive) lens
1. Set up the optical bench with the light source on the far left. Place the crossed-arrow
target (9121) and the diffuser (9120) on the same standard component carrier close to
the light source. The diffuser should be on the light-source side.
2. Place the 18-mm lens (9132) to the right of the target-object. Vary its distance from
the object from more than 2f (2x18-mm) to less than f and follow the position of its
real image on a screen or an index card (no need to measure anything yet).
3. Locate the lens in a convenient position and measure the locations of the object, lens
and the real image. The object distance is determined by the position of the screen
when the crossed arrow target comes into focus.
4. From the object and image distance, calculate the focal length.
5. Repeat twice with different object distances. Calculate favg for the three trials, as well
as δf.
6. Repeat for one setting with the 48-mm lens (9133). For the convenience of Part E,
step 1, find the proper object distance so that the image distance is about 200 mm.
7. Without moving the source, lens or screen, calculate the focal length.
Part E: Measuring the focal length of a concave (negative) lens:
1. Using the image formed by the 48-mm lens in Part D, Step 6 as a virtual object,
determine the focal length of the 22-mm lens (9131). Insert the negative lens
between the 48-mm lens and its image. That image then becomes the virtual object.
2. Adjust the location of the negative lens and the screen to find the real image formed
by the negative lens.
3. Calculate the focal length of the negative lens from the object and image distance.
Try other settings without making calculations.
Part F (Optional): Beam Expanders
1. Build a beam expander (as shown in Problem #25, Chapter 35 in Halliday, Resnik
and Walker), each using a different pair of converging lenses.
2. Using a laser, verify the following equation:
f
w2 = ( 2 )w1
f1
Eq. 4
where w1 and w2 are the widths of the incoming and outgoing beams respectively.
You can measure w1 and w2 using calipers and a screen.
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3. Repeat Step 1 using a diverging and a converging lens. Be sure to sketch your set-up
for Steps 1 and 2 in your lab book and write-up.
Part G (Optional): Exploring Telescopes
There will be various telescopes located around the room. Answer the following
questions, and use the tables to fill in your answers.
When you are finished, go outside with one of the telescopes and look at a star or planet,
and a streetlight. Estimate the magnification by compared what you see through the
telescope, with what you see with the naked eye. Your estimate will be very rough, but
you can at least compare it with theory…give it your best shot!
1. What types of telescopes are available? Are they reflecting or refracting? What are
the brands?
2. What is the diameter of the objective lens or mirror for the telescopes (you may have
to measure it)
3. What are the focal lengths of each telescope (if it isn’t printed on the telescope, then
you can assume that the focal length is the distance from the eyepiece to the mirror,
counting reflections—i.e. the total light path length.
4. What are the f-ratios for each telescope?
5. What are the theoretical resolving power (s) of each telescope?
6. What is the maximum possible magnification for each telescope?
7. What size eyepiece would provide the maximum possible magnification for each
telescope (i.e. the smallest useful eyepiece)?
8. Do objects really appear as much larger or smaller as the formulas indicate?
Analysis
Part A: Reflection
Calculate the difference (|θi – θr|) between the measured values of the incident
angle and the reflected angle for each of the three rays and record them in the
Calculations Table.
Part B: Refraction
1. According to Snell’s Law, for refraction at the first surface the relationship (1)sinθ1 =
nsinθ2 should be true. The value of n = 1 has been used for air, and n stands for the
index of refraction of the prism. At the second surface the equation is nsinθ3 = (1)
sinθ4. Solving these two equations for n gives
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n=
sin θ 1
sin θ 2
and n =
sin θ 4
sin θ 3
Eq. 5
2. Using these two equations, calculate two values of n for each of the incident rays. It
turns out that these are not independent measurements because the errors made in
drawing the rays to determine the angle tend to produce two values of n with errors in
the opposite direction. Therefore, take the average of the two values calculated by
Equations 5 as a single measurement with that incident ray. Record in the
Calculations Table these averaged values of n as the measurements for n for each ray.
3. Calculate the mean n and standard error αn for these three measurements of n.
Record these values in the Calculations Table.
4. Here is a list of some common optical materials, and their indices of refraction:
o Pyrex:
1.47
o BK7:
1.517
o Crown K5:
1.522
o Flint Glass:
1.57 – 1.75
Based on your calculations, which material do you think most likely composes the
prism? If none of these match, do some research on your own to decide.
Part C, D and E: Focal Properties
1. Calculate the percentage difference between the value of fcon and the value of fdiv for
the reflector.
2. According to theory, the value of the focal length for the reflector should be equal to
R/2. Calculate the percentage difference between the measured value of R/2 and the
focal lengths fcon and fdiv for the reflector.
Results:
Write at least one paragraph describing the following:
• what you expected to learn about the lab (i.e. what was the reason for conducting
the experiment?)
• your results, and what you learned from them
• Think of at least one other experiment might you perform to verify these results
• Think of at least one new question or problem that could be answered with the
physics you have learned in this laboratory, or be extrapolated from the ideas in
this laboratory.
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Clean-Up:
Before you can leave the classroom, you must clean up your equipment, and have your
instructor sign below. How you divide clean-up duties between lab members is up to you.
Clean-up involves:
• Completely dismantling the experimental setup
• Removing tape from anything you put tape on
• Drying-off any wet equipment
• Putting away equipment in proper boxes (if applicable)
• Returning equipment to proper cabinets, or to the cart at the front of the room
• Throwing away pieces of string, paper, and other detritus (i.e. your water bottles)
• Shutting down the computer
• Anything else that needs to be done to return the room to its pristine, pre lab form.
I certify that the equipment used by ________________________ has been cleaned up.
(student’s name)
______________________________ , _______________.
(instructor’s name)
(date)
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Data Table
Reflection
Ray
1
2
3
θi
Refraction
Ray
θ1
1
2
3
Calculations Table
θr
θ2
Difference in Angle
θ3
n
θ4
αn
n
Mirrors
f (cm)
% diff fs
R (cm)
% diff R/2
Concave Mirror fcon
Convex Mirror fdiv
Lenses
f (cm)
Converging Lens
Diverging Lens
Part G: Exploring Telescopes
Table 1
Type of Telescope
(brand &classification)
Diameter of
objective
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Focal
Length
f-ratio
(f-number)
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Reflection and Refraction of Light
Table 2
Type of Telescope
Type of
telescope
focal
length of
eyepiece
Theoretical
resolving power
Object
viewed
Maximum possible
magnification
Table 3
Theoretical
magnification
Estimated
magnification
Smallest useful
eyepiece.
Percent
difference
♣
♣
Parts of this lab were adapted from Physics Laboratory Manual, David H. Loyd, Saunders College
Publishing, 1990.
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