Download O5: Lenses and the refractor telescope

Physics 1140
O5: Lenses and the refractor telescope
Introduction
In this experiment, you will study converging lenses and the lens equation. You
will make several measurements of the focal length of lenses and you will construct a
simple astronomical telescope. The components you will use can be purchased as a kit
from Learning Technologies, Inc. (see their website at http://www.starlab.com). At the
end of the experiment, the telescope is yours to keep.
Theory
When a bundle of parallel light rays enters a converging lens, the rays are focused
to a point in space a distance f, the focal length, from the lens. A converging lens is
convex in shape, that is, thick in the middle and thin at the edges. A diverging lens is
concave in shape, i.e. thin in the middle and thicker near the edges.
f
f
Converging lens, convex
Diverging lens, concave
A converging lens can be used to form an image on a screen of an object. The lens
equation relates the focal length f of a lens, the object distance do and the image distance
di ,
1
1
1
=
+
.
(1)
f
do
di
(This equation can used for both converging and diverging lens; the only difference is
that the focal length f is positive for converging lenses, negative for diverging lenses.)
do
ho
f
object
f
image
hi
optic axis
di
Copyright © 2005 University of Colorado, Department of Physics
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In the diagram above, the points labeled f are the focal points of the lens. The lateral
hi
magnification, m, of the image is defined as m =
. From the diagram above, one
ho
di
.
do
The diagram also shows you how a camera works: You have an object on one side of the
lens and you have film or (for digital cameras) a CCD or CMOS imaging array on the
other side. You focus the camera by adjusting the spacing between the lens and the film.
Most cameras have multiple lenses to allow them to produce excellent images with short
space between the lens and the film, but the principle of operation is as above.
can show that m can also be written as: m =
Measuring focal lengths
As you can tell, focal length is one of the most important things you might need to
know about a lens. In this lab, you will use three different techniques to measure focal
lengths.
Method I: Use the lens to form an image of an object, measure the distances di and do,
and then use eq'n (1) to compute f.
Method II: Place a point source of light at the focal point of a converging lens to produce
a collimated (parallel) beam of light, as shown below. When the location of the point
source is adjusted so that the beam coming from the lens is perfectly parallel, then the
distance from lens to source is the focal length.
do = f
point
source
Equation (1) can be applied to this situation. The point source is the object and the object
distance do is the focal length f. With do = f, equation (1) predicts that d i = ∞ . The rays
from the lens converge "at infinity".
Method III: Once a point source and a lens has been set up to produce a collimated
beam, as shown above, then the focal length of another lens can be easily measured.
Place the new lens (lens B) in the collimated beam and see where the rays are brought to
focus. The distance from lens B to the focal point is fB, the focal length of lens B.
screen
fA
f
B
point
source
lens A
lens B
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Refractor telescope
In the last part of this lab, you will construct a simple astronomical telescope, the
refractor telescope. The name comes from the fact that we will use lenses, which
produce images by refracting light. Astronomers distinguish between refractors, which
use lenses to form the image, and reflectors (invented by Isaac Newton), which use a
curved mirror to create the image. Most large telescopes are reflectors due to the fact that
it’s easier to make the single reflecting surface than to make the pair of surfaces needed
for a lens. Galileo’s telescope was a type of refractor. The telescope you will build
consists of two lenses: an objective lens with a long focal length fo, and an eyepiece lens
with a short focal length fe. The objective lens forms an image of a distant object (when
viewing astronomical objects, we recognize that the object is very far away, so we think
of the object as "at infinity"). By the lens equation, if the object distance is d o = ∞ , then
the image distance is d i = fo . This image, which appears a distance fo behind the
objective lens, is called an intermediate image, because it is intermediate between the
objective and eyepiece lens. The observer views this image through the eyepiece lens,
which acts as a magnifying glass. Note that a magnifying glass produces the largest
magnification when the magnified object (i.e., the intermediate image in this case) is held
a distance fe from the magnifying glass.
θ'
The angular magnification M of the telescope is defined as the ratio M = ,
θ
'
where (as shown in the diagram below) θ is the angular size of the image as viewed by
the observer through the telescope and θ is the angular size of the distant object as
viewed without the telescope.
objective
fo
distant
object
fe
intermediate
image
θ
θ'
eyepiece
Notice that the distance between the two lenses is A = f o + f e . From the diagram above,
f
one can show that the angular magnification can also be written as M = o , the ratio of
fe
the focal lengths of the objective and eyepiece lenses.
Light collection and diffraction
One other major issue requires more discussion: In essentially all telescopes, you
find that the objective lens is big and the eyepiece is small. This fact does not seem to be
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required by the diagrams above (it isn’t), but it’s what you always see… Why? Just to be
clear: Once you’ve been introduced to the telescope and you’ve had the fun of viewing a
few things, most of us try looking through the telescope the wrong way. It’s usually not
very satisfying. You see a tiny image of the other side of the room. That’s not what we
mean. Rather, you could imagine that the long focal length lens was of small diameter
and the eyepiece was large. Such a telescope WOULD produce a magnified image, but
you NEVER see such a design in practice. There are two reasons why we always try to
use the largest objective lens possible: 1) A large objective lens allows us to collect more
light. More light means a brighter image and brighter images make it easier to see details
even in a small image. 2) Light diffracts and a large objective lens causes smaller
diffraction angles and less obvious diffraction effects in the image. Neither of these issues
follow from the geometrical optics above, but both are important in designing a good
telescope. You will study the brightness issues in this lab. Diffraction effects can be
studied further in Lab O4 ‘Diffraction’.
Other lens systems
The diagram above shows that the eyepiece allows you to see an enlarged image
of the intermediate image, in other words, that the lenses work together to give you a
view of the distant object, a view that extends over a larger angular region of your visual
field of view. The final image appears larger (it fills an angular region, θ ′ ) than your
original view of the distant object without the lenses (it filled an angular region, θ ) and
you can therefore see more detail.
At some level, the entire process simply follows from the lens equation, but at
another level, it can be fairly mysterious why this combination of lenses works. Various
questions might occur to you. For example, might there be other combinations of lenses
that produce an image (there are). Why do we use this one? Answer: It’s simple and
produces a good image with simple lenses. The fact that the image is inverted is a minor
irritation. Could we find a combination of lenses that produces an upright image?
Answer: We can produce upright images with the addition of a prism (expensive, but
found in many telescopes designed for ‘terrestrial viewing’ e.g. for bird watching) or by
using a different combination of lenses. For example, Galileo’s telescope produced
upright images with a combination of converging and diverging lenses, but the image
quality can be poor). The issue of designing lens systems is a modern topic and somewhat
of an art. Certainly, we can’t do full justice to the subject here, but if you’re interested,
modern optics is a fascinating area to study further.
Experimental Procedure
In this lab, you will use an optics bench, which is simply a rail, on which lenses are
placed, with a ruler on the side for measuring distances. The other equipment includes a
bright light source, which acts nearly like a point source, and a refractor telescope kit
containing two converging lenses and associated cardboard tubes, spacers, and foam
pieces. You will also need some rubber bands and wax paper for viewing images.
Finally, there is a metal plate with an aperture (a hole) in the shape of an arrow. The hole
is covered with a frosted, translucent material (scotch tape). When this aperture is placed
in front of the light source, it forms a convenient object for image-forming experiments.
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Assembly of the kit components
Your Learning Technologies, Inc. telescope kit comes with the following items:
1) A pair of cardboard tubes, 30 cm in length, with one tube that fits neatly into the
other. Together, these tubes hold the lenses and allow you to adjust the focus of the
telescope.
2) A bag that contains the objective lens, the eyepiece lens, a red plastic cap to hold the
larger lens onto the larger cardboard tube, a foam donut that holds the smaller lens in
the smaller cardboard tube, a cardboard sleeve the fits inside the foam (more below),
and a white cardboard aperture.
Before proceeding with the experimental measurements, you need to assemble the
lenses into their holders. The most important thing to remember is that dirty lenses will
degrade the performance of your telescope. Try your best not to touch the lenses
anywhere but around their edges. If you do accidentally touch the lenses, immediately
clean them with dishwashing soap and water, and dry them with tissue (lens tissue is
best). Finger oils are somewhat acidic and will eventually eat into the lens surfaces. If
you clean off a finger print quickly, your lens will be fine. If you wait a day or so, the
finger print will be permanent.
Start by assembling the large lens onto the end of the larger cardboard tube. Stand
the tube up on end, set the large lens on the end of the tube, and slip the red plastic cap
onto the end of the tube, thus holding the lens in place. The red cap is a tight fit on the
cardboard, so you will need to work at it a bit. Rolling the edge of the cardboard tube on
a table top to taper it slightly may help.
Next, you should assemble the small lens into the foam holder: Stand the smallest
cardboard tube (the one that came in the small bag with the lenses) on end, place the
small lens curved surface down on the end of the smallest cardboard tube, and press the
foam piece down over the tube/lens stack so that both the smallest cardboard tube and the
small lens end up inserted into the foam piece. It may be helpful to use a Q-tip to firmly
hold the lens down while sliding the foam piece on, to prevent the lens from tipping.
Now both lenses are attached to something that you can handle easily.
Part I. Images and Magnification.
In this part, you will use Method I to measure the focal length of large and small lenses.
(1)
Place the light source at the end of the optics bench and attach it with the
thumbscrew in the slot. Place the arrow aperture on the front of the light source;
there is a magnet to hold it in place. It will save a little trouble in your
calculations if you position the source so that the object (the frosted arrow) is
exactly beside an integer mark (e.g. 2.0 cm) on the scale of the bench. Gently
tighten the thumbscrew to secure the source, and record the position of the object.
Connect the light source to the power supply and momentarily depress the "start"
switch to turn on the light.
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Place the frosted screen at the far end of the bench and note its position, as
indicated by the ring inscribed on the housing. Again, it will save some trouble if
you locate it a convenient integer mark, like 90.0 cm or 92.0 cm.
Now put small lens in the foam holder on the bench close to the light source and
move it slowly away from the source until you see a clear image on the screen.
The image is most easily seen looking through the screen towards the light source,
but it can also be seen from the other side. Adjust the position of the lens to give
the sharpest image and record the position of the lens (you will need to measure
how far inside the foam housing your lens is positioned, etc.). Calculate do, di,
and from equation (1), the focal length, fB, of the small lens.
(2)
If the image is not centered on the screen, adjust the position of the object plate on
the front of the light source until the image is centered. Now measure ho and hi,
the heights of the object and image. Compute the lateral magnification
d
hi
m =
and compare with the expected value i .
do
ho
(3)
Next, move the foam lens holder to the screen end of the bench and then slide it
away from the screen until you get a sharp image on the screen. Repeat the
measurements above for do, di, ho, and hi. Recompute fB and m. Compare your
measurements in parts (2) and (3).
(4)
Measure the focal length fA of large lens using Method I (don't bother with the
magnification m). Place the large lens in its cardboard holder on the optical rail.
Instead of the illuminated arrow we used for the small lens, use an object that is
several meters from the lens. We have several arrows mounted on the wall, or you
can use any other object you like. It may be useful to illuminate the object. Slide
the image screen from the far end of the rail toward the lens until you see a
focused image. Measure the object and image distances and determine the focal
length of the large lens.
When you perform this step of measuring the focal length of the large lens, you are
seeing the first step of making a telescope, namely, the formation of the intermediate
image. Place your eye roughly even with the image and look across the room at your
selected object. Look at the detail you can see with your naked eye. Now, look at the
image. Does it look brighter? Are you able to see more detail? Perhaps use the lens in the
foam holder as a magnifying glass and look closely at the image on the screen. Can you
see more details?
Part II. Use of a collimated beam
(1)
Here you will use Method II to measure the focal length of the large lens.
Remove the frosted arrow plate from the light source and put the light source back
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on the optical rail. The source itself is very small and can be considered to be a
point source. Readjust its position so that the source is at a convenient integer
mark on your bench. Now place the large lens close to the source and slide it
away until it produces a parallel, collimated beam. A good way to check that it is
parallel is this: Point the beam at a nearby wall where it will produce a disc of
light. Adjust the position of lens until the diameter of the disc is exactly that of
the lens opening. Now measure the distance from the point source to the lens; this
is the focal length, fA. Compare your measurement of fA with your previous
value. Compute a final best value.
Now you will use Method III to measure the focal length of the small lens.
(2)
Without moving the large lens (it is now producing a collimated beam), place the
small lens in its foam holder just beyond the large lens, at a convenient integer
mark, and place the frosted screen beyond the foam holder. Now move the screen
until you jet a sharp image of the point source on the screen. The distance from
the small lens to the screen is fB.
Part III. The refractor telescope.
Choose the lens with the longest focal length. This will be the objective lens with
focal length fo. Pretty clearly, this should be the larger lens mounted on the cardboard
tube. Then choose the lens with the shortest focal length, namely the small lens in the
foam holder. This will be the eyepiece with focal length fe. Place the short focal length
eyepiece foam holder into one end of the smaller of the two nesting cardboard tubes.
Then insert the smaller cardboard tube into the end of the larger, to complete the
assembly of your telescope. The large objective lens should be at one end and the
eyepiece lens at the other end of the final assembly. Place your telescope on the optical
rail and slide the two cardboard tubes together until the objective lens is a distance
A = f o + f e from the eyepiece.
Aim the telescope on the rail towards the far end of the room, where there is an
arrow and a graduated scale mounted on the wall, and adjust the telescope position until
you can see the arrow through the telescope. It may be difficult to find the image since
your telescope has a narrow field of view. Also, you may need to adjust the cardboard
tubes to get a sharp image. Is the final image you are looking at upright or inverted?
Explain your answers with words and diagrams.
Once you have found a clear image looking through the telescope, open your
other eye so that you can look simultaneously at the enlarged image with one eye and the
unenlarged image with the other eye. If you position the telescope just right, you can see
the two scales side by side and hence estimate the angular magnification M of your
telescope. Using several objects in the room, measure the size of the object, distance to
the object, the angle subtended by the object and the apparent angular size of the object in
the image. Thus, determine the magnification of your telescope. Compare your answer
with the theoretical value of M.
The telescope is yours to keep. It should produce great views of the Moon and Jupiter.
PLEASE DON’T EVER LOOK AT THE SUN THROUGH YOUR TELESCOPE!!
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Pre-Lab Questions.
1. What is the definition of the focal length of a converging lens? Illustrate your answer
with a diagram.
2. Consider a lens with a focal length of f =20.0 cm which is used to image an object of
height ho = 4.0 cm, a distance do = 40.0 cm away. On graph paper, draw a diagram
showing the size (hi) and position (di) of the image formed by this lens. Check that
the value of di obtained from your graph agrees with a calculation of di made by
equation (1).
d
hi
3. In the diagram on page 1, show that i =
.
do
ho
4. What is a collimated beam? Draw a sketch showing how you could produce one.
Could you make a collimated beam using only a point source and a diverging lens?
5. A refractor telescope has an objective lens with a focal length of 600mm and an
eyepiece lens with a focal length of 30mm. What is the angular magnification of this
telescope?
6. The Earth’s moon as seen from the surface of the Earth has an angular size of roughly
0.5 degrees. We say that it ‘subtends’ an angle of 0.5 degrees. What is this angle in
radians? The radian angular measurement is convenient because (by definition) the
angle in radians is the ratio of the length of arc subtended to the radius of the circle.
7. If the Moon is roughly 385,000 km from the Earth, what must the Moon’s diameter
be to subtend this angle? How does that diameter compare to the Earth’s diameter?
How does the angle subtended by the Moon compare with the angle subtended by
your thumbnail when you hold your arm straight out?
8. During a full solar eclipse, the Moon almost exactly covers the disk of the Sun. In
other words, they subtend the same angle. If the distance from the Earth to the Sun is
roughly 150 million kilometers, what is the diameter of the Sun?
9. (Counts for two problems) The Alpha Centauri star system is the closest stellar
neighbor to our own solar system. You can easily see this system with the naked eye
from anywhere on Earth south of roughly 25 degrees south latitude in the
constellation Centarus. It is an extremely interesting system as it is both very close (in
galactic terms) and has great potential for having possibly earth-like planets. The
Alpha Centauri system actually has three stars: Alpha Centauri and Beta Centauri
form a binary system roughly 4.35 light years (9.5x1012 km in 1 light year) from
Earth. They orbit each other at a distance of 11 astronomical units (Earth-Sun
distances, Saturn orbits the Sun at about 9.5 astronomical units) or roughly
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1.6x109 km every 80 years. The third star, Proxima Centauri, is a small red dwarf not
discovered until 1915 and, at 4.2 light years from Earth is the closest star to us. You
can see that Proxima is 0.15 light years from alpha and beta. We don’t know whether
Proxima is just passing through, or might be orbiting alpha and beta. Both Alpha and
Beta Centauri are similar enough to our own Sun (Alpha is a nearly perfect match)
that we are confident that they could support life on an appropriately placed planet.
Also, for a binary star system like Alpha and Beta Centauri, orbital calculations
indicate that planets closer than roughly 2 AU to either star would have stable orbits.
Venus is 0.7 AU, Earth is 1.0AU, and Mars is 1.5 AU from the Sun, so a planet like
Earth could exist in a stable orbit around Alpha Centauri, Beta Centauri, or both.
Consider these two questions: a) What magnification would your telescope need to
make an image of Alpha and Beta Centauri’s separation that subtends an angle equal
to the naked eye size of the Moon? b) Make an argument as to whether you would
expect it to be easy or hard to see an Earth-like planet in a telescope with this
magnification.
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