Properties of Lenses and Telescopes
Overview:
SAFETY WARNING: NEVER LOOK AT THE SUN THROUGH A TELESCOPE.
In this laboratory exercise we will investigate the behavior of lenses and mirrors that are used in
optical telescopes using a basic telescope kit. When you finish this exercise you will learn how a
real image is formed by a lens and measure the focal length of the lens. We will also learn how a
virtual image is formed. We will relate the size and distance of the image from the lens to the
size and distance of the object from the lens. We will learn how to construct ray diagrams and
determine the magnification that results when two lenses of different focal lengths are used to
form a telescope. Finally, the increase in light gathered with the kit telescope will be computed.
Learner objectives:
Make observations with an astronomical telescope.
Understand the operation of the parts of the telescope.
Make quantitative measurements of image formation by the lenses of the telescope and
compute its magnifying power from these measurements.
Measure the magnification and compare to the computed power.
Materials Needed:
You will need:
The telescope kit
“Star and Planet Locater”
Ruler with cm scale.
Masking and transparent tape (and also possibly paper glue)
Long sturdy rubber bands
Three boxes of aluminum foil, small 25 square feet size the same thickness as the larger tube.
Image Formation by a Converging Lens
In Figure 3.1 a light ray is shown entering a prism-shaped piece of glass. The dotted lines are at a
90 degree angle to the surface at the point where the ray enters and leaves the glass. Note how
the ray of light is bent TOWARD this line as it enters the glass. If you stick a drinking straw or
pencil into a cup of water, the pencil will appear bent by the same process which is called
refraction. This bending happens each time a light ray travels from a medium of one density into
another of different density (in this case air and glass).
Figure 1. Bending of a light ray by a prism
Now note how the ray is bent AWAY from the 90 degree line as the light leaves the glass on the
other side of the prism. Finally, you can seehow the beam had a net change in direction
downward after it left the prism. Similarly, if the prism were turned upside down, the beam
would be bent upward. This net bending depends on the angle between the two faces of the
prism. The larger it is the more the beam is bent. If the two faces are parallel, no net change in
direction occurs (like light passing through a window pane).
Now a convex lens is much like two such prisms placed base to base. We sketch a convex lens in
Figure 2 with three parallel rays entering the lens at the top, middle and bottom. Parallel rays like
this are typical for extremely distant astronomical sources.
Figure 2 Focal point of a convex lens
The top and bottom rays are bent down and upward as mentioned earlier. The middle ray hits
the portion of the lens with parallel faces that does not bend the ray at all. If the lens is made
correctly, all the parallel beams cross one another at a single point, F, called the ``Focus''
(We will use these two facts later). For initially parallel rays, the distance from point F to the
center of the lens is called the ``focal length'' (f). A white card (screen) held at point F will show
a point of light. This point of light is called a ``real image'' because light rays cross at that
location. Lenses that are fatter in the middle which bring light rays to a focus as shown are often
called “converging” lenses.
As shown in Figure 3 below, there are two lenses with the telescope kit, you purchased for this
course. Both are converging but one bulges out more in the middle than the other.
Figure 3. Telescope kit with smaller “fat” and larger “flatter” lenses plus tubes. “Fat” is
eyepiece lens. “Flatter” is objective.
Figure 4 Set-up studying focal length and image. Several thick books, a ruler, and a small Postit note pad were used. Photographed outside using the Sun. Use an indoor set-up with an
illuminated window on the other side of the room to measure image size of and whether the
image is up-right etc. You may not need books on the left if the source is horizontal.
Take the flatter, wider of the pair of convex lenses and examine it. If a lens is used to form an
image of something infinitely far away, the distance from the lens to the image is defined to be
the focal length, f. For now, infinitely far will mean the view through a window>>f.
Use the lens to form an image of the view through a window. To do this, hold the lens in front of
a blank page on a small pad until a sharp image is formed on it as shown in Figure 2 and 4. Read
off the focal length using the scale on the ruler as shown with the ruler pointing at the source. Put
zero cm at the pad location and make sure the pad is 90o to the ruler to read off the focal length
from the position of the lens. Also do this for the smaller “fatter” lens.
Focal Length Section. See if you can answer these questions and measure this data.
First “fat” lens data:
Focal length in cm= fest,1 = ____________ Do the best you can for the “fat” lens.
Is the image right side up?__________
Measure the image size.__________
Is the image real ? _______
Is the image right-side up or up-side down?____________________
Second larger “flat” lens data:
Focal length= fest,2 = ____________
Is the image right side up?_________
_
Measure the image size.__________
Is the image real ? _______
Is the image right-side up or up-side down?____________________
1. Which lens in the Telescope kit had the shorter focal length, the “fat” lens or the
“flatter” less bulgy lens?
a) fat or more bulgy
b) flatter or less bulgy
c) Both the same focal length.
2.
a)
b)
c)
Which lens in the Telescope kit had the larger image of the distant object?
the long focal length lens.
the shorter focal length lens.
There was no difference because both had the same focal length.
3. Relative to the original distant object or light source, what was the image
orientation when you formed them with the Telescope kit lenses?
a) right-side up (same orientation as original)
b) up-side-down (inverted relative to original)
c) sideways (turned through 90 degrees relative to original).
Non-celestial, everyday point light sources such as a candle flame or light bulb are not at an
infinite distance from the lens. In Figure 5, we sketch three rays from a point source that is not at
a large distance from the lens but is farther than the focal length, f, from the lens.
Figure 5. Image and Object Distances
The middle beam traveling along the symmetry axis of the lens is unbent and the other two are
bent to bring the point light source to a focus as before. However, with the parallel rays in Figure
4, the image was at the focal length, f, from the lens. In this case the incoming rays are diverging
and not parallel so the lens forms an image a distance larger than f.
The increase in the distance of the focal point if the object is closer has an effect on our vision.
The human eye can adjust its lens shape to accommodate an object being closer but there is a
limit. For most people, the closest distance to comfortably view an object is about 25 cm (unless
one is very near sighted).
Image formation processes
Now there can be, of course, more than one source of diverging light rays and they do not have to
be on the central line of symmetry of the lens. In Figure 6, a lens is shown imaging two sources
of light which also could correspond to points on the ends of an arrow with the head on top at O
and the tail below at O’. Following the dashed beams from O, the middle beam passing through
the center parallel faces of the lens is not bent. However, the other two are bent so that all three
come to a focus I below the axis of the lens. By tracing the three solid rays from the tail, you can
see that those from the tail O’ below are brought to a focus I’, above the axis i.e., the point images
are upside down or inverted. By imagining light coming from points all along the arrow, Figure 6
shows how an inverted image of an extended object like an arrow is formed.
Figure 6. More than one point source or sources at different locations on an extended object..
Solid rays come from lower end of objet; dashed from upper end. They converge on opposite
(reversed) ends of the image as shown.
An observer near the axis of the lens and on the same side of the lens but beyond the focus of the
rays will see what appears to be an arrow extending from above the lens axis to a point below. A
white card at the focus will display an arrow to an observer off to one side. This type image
where rays cross is called a “real” image.
Drawing and Using Ray Diagrams
The method used in Figure 6, a ray diagram, is a useful way of checking some of the results you
observe with lenses in this exercise. The question is, when light from an object passes through a
converging lens, how and where does the image form? Figure 7 shows how you use knowledge of
the focal length, f, of the lens to make this judgment. Points F and F' are at a distance equal to f
from the lens. We have also drawn an arrow with the tip of the arrow marked O. The position of
the image can be located using two of three rays from each point on the object as illustrated in
Figure 7 for point O.
Figure 7. Ray diagram.
Carefully referring to Figure 7, to find the image position of any other point on the arrow you
need to draw two rays with the third serving as a check, where the three rays are defined as
follows:
(i) Draw a ray from the point on the object to the lens parallel to the axis of symmetry of the lens.
After passing through the lens this ray must pass through F. In Figure 7 this is line OL.
(ii) Draw a ray from the point “O” on the object through the point F' toward the lens. After
passing through the lens this ray must emerge on a path parallel to the symmetry axis of the lens.
The intersection of the two rays locates the image position of a point. In Figure 7 point O on the
arrow is images at point I behind the lens.
(iii) A third ray which will also locate the image position of a point can be drawn from the point
on the object to the center of the lens. It will emerge from the lens with no change in direction.
Note this in Figure 7.
(d) The intersection of this central ray with either of the two previous rays will locate the image
of the point on the object. As you can see from Figure 7 all three rays from point O intersect at
point I, but only two of the three need to be drawn to locate point I.
When the rays cross, a real image is formed. A screen at crossing would show the image.
Figure 8 Ray diagram for the upper point of the object O and lower part of image I. Space is left
to complete the diagram for lower part of object O” and upper part of image I”. See questions
below.
Try answering the following questions. Refering to the figure above, pick a, b, c, d and e
answers which match in Figure 8 and questions 4 and 5.
4. Partially construct the upper end of the image to the right of a lens due to an upwadly
pointing vertical arrow outside the focal length to the left of a convex converging lens. Draw
a ray from the low tail of the arrow to the lens parallel to the axis of symmetry of the lens.
This ray must pass through a point _______
a) one focal length on the side of the lens toward the arrow.
b) in the middle of the lens.
c) one focal length from the lens on the side away from the arrow.
d) in the upper half of the lens but not as high as the arrow point.
e) in the lower half of the lens below below the middle a distance equal to the arrow tail.
5. Assume a vertical arrow (pointing upward) to the left of a convex converging lens
outside the focal length. One step in using a ray diagram to construct the upper end of the
image to the right of the lens is to draw a ray from the lower tail toward the lens parallel to
the axis of symmetry of the lens. After passing through the lens, this ray must pass
through a point
a) one focal length on the side of the lens toward the arrow
b) in the middle of the lens
c) one focal length from the lens on the side away from the arrow
6. Assume a vertical arrow (pointing upward) to the left of a convex converging lens outside
the focal length. If you draw a ray from the end of the arrow point through the focal point
one focal length from the lens toward the arrow, the ray will encounter the lower half of the
lens. After passing through the lens, this ray must emerge on a path _______
a) parallel to the symmetry axis of the lens
b) 90 degrees to the axis of symmetry of the lens
c) downward through the focal point on the opposite side of the lens.
7. A ray which will help locate the image position of a point can be drawn from the lowest
point on the object to the center of the lens. It will emerge from the lens
_________________
a) parallel to the symmetry axis of the lens
b) 90 degrees to the axis of symmetry of the lens
c) proceeding in the same direction that it entered the lens.
8.For an object to the left of a convex lens farther than the focal length, the image is found
___________ where the light rays cross.
a) to the right of the lens
b) to the left of the lens
c) at the center of the lens
The Telescope Eyepiece (Magnifier) and a Virtual Image
In astronomical telescopes the front lens or mirror forms an image of a celestial object as we have
observed. This front objective lens is called the “objective” lens or mirror. In many cases a CCD
(like in a camera) is placed where the real image is formed. In most every-day telescopes/ the
real image is viewed by a person using an eyepiece lens which magnifies the image revealing
small details. This is simply the familiar magnifying glass. Larger aperture objective shows
finer detail than the smaller aperture eye alone.
Typically, one moves the eyepiece moved inward and outward until the eye comfortably views
the image through the magnifier i.e. the object is seen sharply. This arrangement is shown in
Figure 9 of an objective lens and magnifying eyepiece in a telescope.
Typically the observer’s view is sharpest and most comfortable when the eye piece is somewhat
closer to the intermediate real image than the eye piece’s focal length. In Figures 6, 7 and 8, the
object is farther than the focal length from the lens. If the object is closer than the focal length it
is not possible to form a real image that can be seen on a screen. Figure 10 shows the situation in
a telescope where the real image “source” is closer than the focal length from the eyepiece lens.
Figure 9. Arrangement for a visual telescope. From left: object, objective lens which forms a
real image.. , real image formed by it and magnifying eyepiece, then finally the observer. The
virtual image created by the eyepiece is seen. The f’s of objective and eyepiece are shown.
Image
or objectthe
ateye
comfortable
25 focus
cm viewing
It down to about 25
Figure
10a. Typically,
can comfortably
on objectsdistance.
very far away
looks
tiny.
cm but not closer. The small angular size is indicated by lines to the point and tail.
Convex lens permits bringing the image or object closer.
Ray diagram shows how giant virtual image at 25 cm
results.
Figure 10b Magnified virtual Image ray diagram. Convex lens permits bringing the image or
object closer than 25 cm. Ray diagram shows how a giant virtual image at 25 cm results. The
object is closer to the lens than the focal length. F.
In this case, the lens is unable to make all the rays passing through the lens converge after
passage through it, and no image is formed that can be projected on a screen on the right side of
the lens. Figure 10 shows a ray diagram of the formation of a virtual image of an object closer
than the focal length F. In particular two red rays are shown from the tail of the arrow, one from
the bottom through the center and another from the tail parallel to the axis which is bent slightly
upward to pass through F on the observer side. The back extrapolation of these two red lines
diverge from a point about 25 cm from the observer. So as the red and black lines show, an
observer can see an enlarged or MAGNIFIED image with the ends of the image arrow being
farther from one another than the ends of the REAL arrow. In other words, the eye sees an
apparently larger image because it perceives the final direction that the light rays come from, and
has no awareness of any bending that might have happened to the rays on their way from the
object to the eye. Because the light rays do not actually cross in this type image and an image
cannot be formed on a screen this type of image is called a VIRTUAL image.
Try this yourself. Place the Telescope kit short focal length convex lens on, say, a printed
page or other object and move it outward with your eye at a constant distance from (fairly close
to) the lens Note when the magnified image appears and its qualitative properties when the lens
is less than one focal length away from the page. Continue to move the lens away and note its
distance from the paper when the image ceases to be magnified..
Eyepiece View Section.
Close-up view:
“Fat” eyepiece lens data:
Focal length= fest,fat = ____________
Is the image real or virtual? _______
Is the image right-side up or up-side down?____________________
Use these results above to try answering the following question. Refer to the figure above
and pick a, b, c, and d answers.
9. Assume you have a magnifier, such as the telescope short focal length convex eye piece
lens which you want to use to get a magnified view of a printed page. The eypepiece lens
placed, close to, a printed page, should show a (an) _______ magnified image of the page if
the lens is within ________ of the printed page.
a)upright; 3 times the focal length
b) upright, 2 times the focal length
c) upright: somewhat less than the focal length.
e)inverted; less than the focal length
.
Telescope
Optical Assembly and Arrangement of Telescope:
Now construct your telescope following the directions in the kit. Refer to Figure 3 earlier and
Figure 11 below. You can follow the printed assembly instructions exactly if you wish using glue
but note that “thin” and “thick” refer to the width of the lenses. Here is a a simplified assembly
procedure not requiring glue that you may wish to use. Slide the smaller eye piece tube into the
larger objective tube. Slide the yellow mount tubes onto the black tube ends almost all the way
so they stick out about ¼ inch. Holding the larger black tube vertically with the yellow tube at the
top, drop the wider, flatter objective lens into the opening. Push in the two black rings to keep the
objective lens from falling out. Slide the yellow tube in until the rings are flush with the outer
end of the yellow tube. As shown in Figure 11, secure the rings with small tabs of tape that don’t
touch the lens along with a tabs to keep the yellow tube from sliding. Do the same for the
eyepiece end using the fatter eyepiece lens and blue rings. Again, secure the rings and yellow
tube by small tape tabs. When you look into the eye piece, your eye should be back from the eye
piece a small amount to maximize clarity.
Now that you have constructed your telescope, point the long focal length objective lens end of
the tube away from you toward the object to be observed.
Turn short focal length eyepiece lens end of tube toward you.
Set up as shown
<) ()______________________()______________
Eye Eyepiece Magnifier
Objective Lens
Object
The eyepiece magnifies the real image formed by the objective.
Starting with a distance between the eyepiece and objective a distance apart slightly less than
fe + fo , slide the objective toward the eyepiece while looking through the eyepiece at the end.
Stop when the distant object comes comfortably into focus. The eyepiece acts as a magnifier of
the objective image. Examine the image carefully.
Figure 11. Assembled telescope plus simple mount using three small aluminum foil boxes and
two large rubber bands. Objective is in red fitting slipped over cardboard tube. Eyepiece is at
other end of smaller tube inserted into objective tube.
See if you can answer the following questions. Refer to Figure 9 and your observations
through the telescope.
10. The telescope objective lens in the “Telescope” module produces a (an)________ image.
a)upright; real
b) upright, virtual
c) inverted; real
e)inverted; virtual
.
11. Assume you have a magnifier, such as your kit telescope eye piece lens which you use
with an objective lens like the one in your kit to observe an object. Using this telescope,
through the eye piece, you see a (an)_______image of the object.
a)upright; real
b) upright, virtual
c) inverted; real
e)inverted; virtual
Magnification of Telescope:
Use a page with equally spaced marks fastened to a wall with masking tape. A brick wall or fence
with boards or bricks all the same width or height would also work. A pattern to use for the
magnification exercise is on the last page of this module.
Pointing your telescope at the wall, look through it with one eye and keep the other eye open.
Alternative blinking is another possibility. Count the number of divisions seen without the
telescope inside one or two or three etc. as seen through the telescope
Example: The horizontal bars are divisions of the scale.
Telescope Eye alone (not looking through the telescope)
--------------------------------------------------------------------------------------In our graphic example above we see that two magnified divisions corresponds to four
unmagnified divisions. Therefore, 4/2 = 2x angular size magnification.
Record your results below.
Magnification = (# without telescope)/(# with telescope) =
(_____________) / (____________) = _________
The theoretical magnification of a telescope is approximately equal to the focal length of
the objective divided by the focal length of the eyepiece.
Record the objective lens focal length fo = ___________________(units?) from your earlier
measurement.
Record the eyepiece focal length fe = ___________________(units?) from your earlier
measurement.
Compare your result above to the theoretical telescope angular magnification mag =
fo/fe =
_____________ / _____________ = _________________
See if you can answer the following question now.
12. If the objective of a lens telescope has a focal length of 60 cm and the eyepiece a focal
length of 10 cm, what is the magnification of the telescope
a) 10 cm/ 60cm = 1/6
b) 10cm x 60 cm = 600
c) 60 cm/ 10 cm = 6
d) (60 cm/10 cm)2= 36
Change in Objective Size:
You know the telescope objective forms a real image. Now mentally see if you can predict what
will happen if you cover up half the objective.
See if you can fill in the answers below about “Light Gathering Power.”
Now, looking through the telescope eyepiece with it focused on a distant object, cover up
first ¼ then ½ the objective.
Describe how the image changed as you covered up more of the objective.
__________________________
Considering how the image changed, explain why this happens.
______________________
The amount of light gathered by a telescope objective depends on its area.
Measure the diameter of your telescope’s objective lens in mm or cm, diameter =_________
Compute the radius, r = (diameter/2) =____________________
Compute area of objective =
πro² = _______________
The radius of the opened iris of the dark adapted eye is about 3 mm or 0.3 cm.
Compute the light gathering power of your telescope compared to the eye =
πro²/(πre²) = (ro/re) ² =
_____________________
From these results explain why astronomers build telescopes with 10 m diameter objectives
to explore deep into space.
Now see if you can answer the following question.
13. If the objective of a lens telescope has a diameter of 6 cm and the dark adapted eye 3
mm What is the light gathering power of the telescope compared to the eye ? The
telescope’s light gathering power is _______ the eye. Note that 3 mm=0.3 cm.
a)1/20
b)20 x
c)400
Summary and Additional Remarks.
We have defined focal length of a convex converging lens and measured it for the lenses of our
telescope kit.. Ray diagrams show how a convex lens forms a real inverted image if the object
distance is longer than the focal length. The image is larger the longer the focal length. This is
the situation for a telescope objective lens examining an object at great distance. For
astronomical observations, usually a CCD or other device is placed at the focus to record the real
image. The real image formed by the larger objective contains more detail than the smaller eye
alone can reveal. For more every-day observations, a magnifying eyepiece is placed just beyond
the real image a distance less than its shorter focal length. We show with ray diagrams how the
eyepiece forms an enlarged virtual image. The magnification of the telescope can be computed
from the focal lengths of the objective and eyepiece. The light gathering power of our telescope
depends on the area of the objective thus revealing fainter stars than the eye alone can see..
Again, a pattern to use for the magnification exercise is on the last page of this module.
Later we will learn how to make observations with our telescope or your own binoculars or
telescope. You can go ahead and try it on your own.
The skymaps.com web site has lists of binocular objects that your kit telescope will also reveal
under dark sky conditions.
Many of these e.g. M42 Orion Nebula, M31 globular star cluster, M31 Andromeda Galaxy are
marked on your Star and Planet locator.