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Physics E-1b
Expt 4: Geometric Optics
Spring 2007
Introduction
Preparation: Before coming to lab, read the lab handout and Chapter 23 in Giancoli.
Then complete the pinhole experiments in Part I and answer the bold numbered questions
that appear throughout this handout. Answers to these pre-lab questions are to be handed
in at the beginning of your lab. See the Lab Companion, page 3, for further information.
Be sure to bring to lab, writing paper, a ruler, a calculator and your copy of the Lab
Companion. Graph paper will be supplied.
Post Lab Questions: At the start of your lab section, you will be given an additional
handout with a series of questions to be answered during the lab and handed in at the end
of the experiment. Try to answer these questions with one or two concise sentences.
Experimental Setup Part I of the experiment is to be done on your own. However
aluminum foil and needles are available in the help room for you to make a pinhole.
Parts II, III, and IV are performed in the lab. A water filled aquarium and a laser light
source are used in the first experiment. A thin double convex lens is used in Part II while
a thick spherical lens made from a water filled glass flask is used in Part III. In Part IV
you combine these two lenses to build a microscope. Candles and samples of text (fine
print) are available to be used as objects for all of the experiments.
Objective: In this lab you study the magnification properties of a pinhole. You observe
the bending or refracting of light in water and determine the index of refraction of the
water. The behavior of a thin double convex glass and thick spherical water lens are
studied and compared to calculated results obtained with the thin-lens formula. The lens
makers formula is used to find the index of refraction of water in your thick lens. Finally
you utilize your data from these experiments to construct a microscope with the thin and
thick lenses.
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Part I: Pinhole Imaging
Pinhole Magnifier
First check and note how close you can bring this printed page to your eye before you can
no longer focus, and the print becomes fuzzy. Next, take a piece of aluminum foil and
make in it a pinhole of about 1/2-mm diameter or less. Hold it close in front of your eye.
Now look at the well-illuminated printed page through the pinhole. (If you wear glasses,
take them off. You don’t need them and they won’t do any good.) Bring the page up
closer and closer to your eye. Notice that the word you are looking at stays “in focus”
and is magnified as it is brought closer! (It finally gets fuzzy because your pinhole is not
small enough.) You have made a pinhole camera using your eye
(a) The magnification is easily calculated if you make some assumptions about the
distance between the pinhole and the retina of your eye. 1→ Draw a ray diagram that
explains how your pinhole magnifier works and derive a formula for the lateral
magnification.
Do you really see things upside down?
(b) Here is a way you can convince yourself that the image on your retina is inverted.
Look through your pinhole at a broad light source (like the bright sky or ceiling
fluorescent lamps). Hold a pencil point in front of the pinhole and look at its shadow on
your retina. Everything behaves as expected. Now reverse the order and put the pencil
point between the pinhole and your eye. Move the pencil and notice the direction of
motion of the shadow! 2→ Make a sketch and explain what’s happening.
Exercising the pupils...
(This part is just to do and notice - nothing to measure.) When you look at a broad light
source through your pinhole, you see a bright circle. That circle is the projection of your
pupil on your retina, not the outline of the pinhole. You can study the dilation and
contraction of your pupil by covering and uncovering your other eye, the eye that is not
looking through the pinhole. When you uncover the other eye so that light enters it, its
pupil contracts. So does the pupil of the eye looking through the pinhole! You can easily
see these “sympathetic” pupil contractions. Notice that it takes a time of the order of 1/2
sec for the pupil to contract or dilate when the light intensity is suddenly changed.1
1Eye-pupil size and mental activity. If someone shows you a picture of a good-looking individual of the
opposite sex, your eye-pupil diameter may increase by as much as 30%, according to Eckhard H. Hess,
Scientific American p. 46 (April 1965). This large a change is very easy to detect in your own pupil by
using this pinhole technique. You may wish to try this: Perhaps by just thinking, you can vary your pupil
size, depending on what you think about. Have someone read to you. (Concentrate on listening, not on the
pupil size.)
2
Part II: Index of Refraction of Water
In this experiment you will measure the index of refraction of water by measuring heights
and applying the small angle approximation. Refer to the sketch below.
Laser Light Source
Medium 1
(Air)
d
!i
h'
!r
Medium 2
(Water or Air)
h
C
Begin with the aquarium tank that is full of water. Mount the laser pointer in the clamp
and adjust it so that the ray strikes the bottom edge of the tank as shown in the diagram.
Use masking tape to locate the position of the tank on the lab bench. Measure the
distances d and h. Arrange the geometry so that d/h is about 3/4 — this value is not
critical. Replace the filled tank with the empty one, carefully positioning it in the
location determined by the masking tape. Without moving the laser, locate the point on
the face of the tank where the ray strikes (a piece of masking tape will help) and measure
h’. Place the full tank in its original position. Adjust the laser so that d/h is now about
1/5. Repeat the measurements. Calculate the values of h/h’ for the two sets of
measurements. Be sure to include the uncertainties. The index of refraction, n, of the
water is approximately equal to the ratio of the height h to the height h’.
3→ Show that the ratio h/h’ approaches the value of the index of refraction as d
becomes smaller. To prove this, relate h/h’ to the tangents of the angles of incidence and
refraction, and then apply the small angle approximation. That is, for a small angles
measured in radians, θ ≅ sinθ ≅ tanθ. To better understand when it is reasonable to use the
small angle approximation, complete the following plot before coming to the laboratory.
4→ Calculate values for θ, sinθ, and tanθ for θ from 0 to 0.9 radians in 0.1 radian
intervals and then plot all of these values (three plots) versus θ (in radians) on a
single piece of graph paper [Remember that the small angle approximation requires
that the angle be measured in radians.]
3
Part III: Lens Equations
Thin double-convex lens.
Simple thin lenses are designed by using the so-called lens-maker’s formula:
1 "$ n 2 %' "$ 1 1 %'
=
!1
+
f # n1 & # r 1 r2 &
Where f is the focal length, n2 is the index of refraction of the lens material, n1 is the
index of the surrounding material, and r1 and r2 are the radii of the respective surfaces.
For a double convex glass lens, n2 is about 1.5 and n1 is 1, r1 and r2 are positive, so f is
positive.
(a) Measure the focal length of the glass lens directly by obtaining the image of a distant
object on a piece of paper and measuring the image distance.
(b) Now determine if the lens behaves according to the thin-lens formula:
1 1 1
= +
f i o
where f is the focal length; i and o are the image and object distances from the center of
the lens. Light a candle and place it near the lens such that its flame is approximately at
the same level as the optical axis of the lens. 2 Hold a piece of paper on the opposite side
of the lens and focus the image of the candle on it by moving the paper toward or away
from the lens. You may have to raise the lens a bit (by placing it on a book, for example)
if your candle is too tall. Measure the object (candle) distance and the image (paper)
distance from the center of the lens. Repeat this for several configurations; choose a
range in which the candle is close to as well as far from the lens. Record the object and
image distances in a table and calculate their inverses. The object and image distances
will be your independent and dependent variables, respectively (refer to page 13 of the
Lab Companion). A plot of o versus i will not yield a straight line, and so we will use the
“trick” explained on page 15 of the Lab Companion — we cast the thin-lens formula into
the canonical straight-line form (y = mx + b) so that it reads 1/o = -1/i + 1/f. From this
we see that a plot of 1/o versus 1/i should yield a straight line of slope –1, and 1/f will be
the y-intercept. Determine the focal length of the lens from such a plot.
(c) Next separate the candle and paper by a distance greater than four times the focal
length of the lens and note that there are two possible positions for the lens, which yield
an image at the screen. In homework assignment #9 (Giancoli problem 23-82), you will
prove mathematically that there are two values of o and i that satisfy the thin-lens
formula only if the distance between the object and image (screen) is greater than 4f (i.e.,
d = o+i > 4f). As a follow-up, see what happens if the candle and paper are separated by
a distance < 4f. What happens when o+i = 4f?
2
It's best if the room lights are turned off at this point...hopefully all the other groups are at the same point
of the lab so you won't interfere with each other.
4
Thick spherical lenses.
A round-bottom (Florence) flask filled with water makes a good short focal-length lens. It
is obviously not a thin lens and thus one wouldn’t expect the thin-lens formula to be
totally correct.
(a) Fill the Florence flask with water and place it in the cork support ring on the lab
bench. Place the lit candle near the flask. Again, measure the object distance and the
image distance from the center of the lens. Repeat the procedure you used for the glass
lens above and again plot 1/o vs. 1/i and determine the focal length of the spherical lens
from your plot.
(b) If the candle is at the focal point of the lens, then the image is at infinity and the light
rays on the image side of the lens should be parallel. Try it. Are the candle light rays
grazing along the bench top parallel? The candle needs to be at the center of the lens for
this to work.
(c) You can also quickly and accurately determine the focal length of the lens by imaging
a distant (at least across the room) object onto a piece of paper.
(d) What kind of image (if any) do you get if the candle is placed closer than the focal
length of the lens? Instead of imaging onto a piece of paper, place your head where the
paper is. Have your lab partner slowly move the candle from far away toward the lens.
(e) A derivation of the lens maker’s formula applicable to our spherical lens is given at
the end of this lab write-up. The focal length of a spherical lens is
f =
r ( 2 ! n 2 n1 )
2 ( n 2 n1 ! 1)
Where r is the radius of the sphere, n2 is the index of refraction of the lens material, and
n1 is index of refraction of the surrounding medium.3 Here f is measured from the surface
of the lens to the focal point. This equation can be solved for n2:
! 1+ f r $
n2 = 2n1 #
" 1 + 2 f r &%
Measure the diameter of the flask and determine the radius, r then using your values for
the focal length4 and r, determine the index of refraction of the lens.
3
Note the difference between this result and that which would be obtained from the thin-lens lens maker's
r
formula, letting n1=1, and r1=r2=r: f = 2 (n-1) . Furthermore, f is measured from the center of the lens in
the thin-lens case.
4 In this derivation it is reckoned from the flask surface.
5
Part IV: Compound Microscope
Now combine your two lenses to make a microscope. The short focal-length objective
lens will be the round flask, and the longer focal-length ocular (eyepiece) lens will be the
glass lens. The object to be examined should be placed just slightly outside the focal
plane of the objective. The image is formed a long distance, io, from the objective. An
object of height h1 located at distance oo from the objective gives a real image of height h2
= (io/oo)h1 at the objective image point. This image is a distance oe from the eyepiece and
subtends an angle h2/oe there. If the object were examined with the naked eye at a
distance 25 cm (the standard near point), it would subtend angle h1/25 cm. The
magnifying power of a microscope can be defined as the ratio of the angular size of the
object when viewed with the microscope to the angular size of the object when viewed at
25 cm from the naked eye. Thus the magnification M = (h2/oe)/(h1/25) or, since h2 =
(io/oo)h1 then M = 25(io/oooe). In principle, any magnification may be obtained with any
two converging lenses; in practice, the maximum magnification is severely limited by lens
aberrations.
io
oo
object h1
fo
fo
oe
fe
spherical flask
(objective)
objective image h 2
fe
Thin glass lens
(eyepiece or ocular)
Combine the lenses according to the recipe (see sketch above) to make yourself a
microscope. Use some small printed material as your object and position it just beyond the
focal point of the objective. Knowing the object distance, you can determine the
intermediate image distance from your previous data (graph). Set up the ocular so that the
intermediate image is within its focal length. Now look through your microscope at the
printed material. Start with your eye close to the ocular; you should see the printed
material in the central portion of the objective and greatly distorted images in the rest of
the objective. Move your eye away slowly; keeping the central portion of the objective
lined up with the center of the ocular. At one position the field of view of the central area
seems to just fill the eyepiece lens. This is the optimum position for the eye, called the
“exit pupil.”
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Appendix
Focal length of a sphere (fat lens-maker’s formula)...
h
h'
F
index = n1
index = n2
Example of a "thick" lens. The focal
point F is located a distance f' beyond
the last surface.
f'
2r
Let us trace parallel rays through the sphere. The ray through the center of the sphere is
not deviated. The ray at transverse distance h from the center line makes an angle of
incidence θi given by θi = h/r, for h/r << 1. The deviation δ of this ray at the first surface
is equal to the angle of incidence θi minus the angle of refraction θr. For small angles,
Snell’s law, n1sinθ1 = n2sinθ2 , becomes n1θ1 = n2θ2. Then the deviation toward the
normal at one surface is given by
$ " '
$
n '
! = "1 # " 2 = "1 & 1# 2 ) = "1 & 1# 1 )
% "1 (
% n2 (
This equation is general (for small angles) and is useful for tracing rays through
complicated systems. In the present example, we find for the deviation at the first surface
!=
h #%
n &
1" 1 (
r $ n2 '
Now follow the ray to the rear surface. It gets closer to the axis by an amount 2r times
the deviation δ. Thus it reaches the rear surface at distance h’ that is
#
# n
n &
&
h' = h ! 2r" = h ! 2h% 1! 1 ( = h % 2 1 ! 1(
$ n2 '
$ n2 '
At the rear surface, the ray is again deviated toward the axis. By the symmetry of a circle
about a chord, the deviation upon emerging is the same as that upon entering. Thus, the
ray emerges at an angle 2δ to the axis and at lateral distance h’. It will therefore hit the
axis at distance f’ beyond the last surface, where
7
2! =
h'
f'
The previous three equations then give
# n
#%
&
n &
h% 2 1 " 1(
2 " 2(
$ n2 ' r $
n1 '
h'
f' =
=
=
2!
h#
n & 2 #% n 2 &(
2 % 1" 1 (
"1
r$
n2 '
$ n1 '
Notice that if n2 = 1.33 and n1 = 1, then the water filled flask, f’ is positive. However, if
n2 = 1 and n1 = 1.33 (the case of a flask filled with air and submerged under water), f’ is
negative and we have created a diverging lens!
Scotchlite retrodirective reflector.
If n = 2, then according to this last equation a paraxial ray entering at transverse distance
h strikes the rear surface of the sphere at distance h’ = 0 (see figure). Thus, a parallel
beam is focused exactly at the rear surface. The beam is partly reflected and partly
transmitted there. The reflected part is eventually directed back at 180 degrees to the
original direction. The transmitted light at the rear surface can be largely reflected back
into the glass by covering the rear surface with a silvery reflector.
A reflecting material called Scotchlite, which uses this principle, can be obtained in most
hardware stores. It is used for bright road signs, the paint used for the white lines on
highways, license plates, safety reflectors, and decorating, among other things. Examine
it with a magnifying glass. You will see that it consists of many tiny glass spheres
embedded on a sticky silvery surface and then painted with clear red shellac (for red
Scotchlite) or something else for other effects. It turns out that the largest index that one
can easily get with glass is about n = 1.9. This is close enough to 2 so that it works fairly
well.
For your amusement, a large piece of Scotchlite is taped onto the blackboard in the lab.
You can easily measure the retrodirective properties of it. Shine a flashlight beam on it,
with the flashlight held right in front of your nose so you can see the light that is
retrodirected through 180 degrees Now gradually move the flashlight out to the side
while still shining the beam on the Scotchlite. Note how the intensity falls off
dramatically. This way you can estimate the angular width of the retrodirected beam.
Why would one expect some angular width; i.e., why doesn’t it give perfect
retrodirection?
Dew heiligenschein...
A drop of water makes a good converging lens with its focal point just a short distance
beyond the drop. If a dewdrop is held at this distance in front of a blade of grass by the
fine hairs on the grass, sunlight is focused on the blade of grass. The blade of grass
reflects this light back toward the sun, reversing the path of the incident light—the
dewdrop on the grass forms a retroreflector. With the sun behind you, the only
8
retroreflected light you will see will be from the grass straight in front of you, around the
shadow of your head. Hence, there will be more light coming from the region around
your head’s shadow than from any other region of the grass, a phenomenon call
heiligenschein.5 The sixteenth-century artist Benvenuto Cellini concluded that this was
evidence of his genius. Actually, a nearby friend would see the glow only around her
own head’s shadow, and conclude that she is the genius.
Spherical glass-bottle lenses were used to concentrate the light of a candle on fine work,
such as lace making. Such a “lace-makers’ condenser” was described by John White in
1651 under the heading “How to make a glorious light with a candle like sunshine.
Round bottles of water, left in an open window, have been known to start fires when the
sun moved into the right position.
5
German, light of the holy ones.
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