The Telescope

The Telescope
Objective: In this lab, you will measure the properties of lenses and put them together to
make a telescope. Once you have built the telescope, it’s yours to keep! You will find
that the design is so simple that you may want to build a bigger one someday.
Background
There are many types of telescopes, which can be made with various combinations of
lenses and mirrors. The telescope that we will make is called a refractor, and is similar to
the kind use by Galileo. The design is shown in the figure below.
The refracting telescope is made by simply placing two lenses together so that their focal
points coincide. The first lens is called the objective (because it collects the light from
the object) and the second one the eyepiece; where the focal lengths are f O and f e ,
respectively. As such, the distance between the two lenses is given by f O + f e .
We can now use geometric optics and ray tracing to understand how the image is formed.
(This lab assumes that you are familiar with ray tracing, the lens-makers formula, and
simple geometry.)
The distance from the object to the objective, O, is usually very far away compared with
the size of the telescope. As such, light from the object will come to a focus near the
focal point of the objective. We can verify this fact by calculating the image distance, I,
from the lens-makers formula:
1 1 1
+ = . (1)
O i
f O

Solving for the image distance, we get:
2
f
O
f
O
i = ≈ f
O
+ , (2)
f
O O
1−
O
where we have made the approximation that the object distance is much larger then the
focal length of the objective. The figure shows the image of the objective, which we can
see is small and inverted. Clearly, when the object moves to infinity, the second term in
the above equation vanishes and we find that the image approaches f O . The distance
between the image and focal point is labeled x in the figure.
The image from the objective now acts as the object for the eyepiece. In a since, we can
think of the eyepiece as a magnifying lens that is used to look at the image produced by
the objective. Note that the image produced by the objective is a real image since light
actually focuses to a point. Now, we use the lens-makers formula to get the image
distance from the eyepiece, i’,
1 1 1
+
i'
f
= , (3)
2
e
f
e
f
e
−
O
2
fe
where we have used the fact that the image from the objective is at a distance fe
−
O
from the eyepiece (see the figure). Solving for i’, we get,
2
f f
i ' = e
e
≈ − ⋅O
1 f
, (4)
1−
2
O
f
O
f
O
1−
⋅
f
e
O
where we have again used the approximation that O>>f O . Note that from ray tracing, we
find that at the eyepiece, the rays appear to be originating from i’, but there is no real
light focused at that point. This is an example of a virtual image, therefore the negative
sign for i’.
The calculations above provide all of the information that we need to calculate the
magnification. The magnification is defined as the angle subtended of the object as
viewed through the telescope, θ e , divided by the angle subtended by the object as viewed
with the naked eye, θ O . Part b of the figure shows these two angles. For small angles,
tanθ =θ, so we have:
h
θ
O
= (5)
O
and
f
e
hO
hi
f
O
f
O
h0
θ
e
= = = ⋅ , (6)
2
i'
f
e
f
e
O
⋅O
2
f
O

where the second equal sign results from using Equation 4 in the denominator and the
numerator results from simple trigonometry * (see the figure). The magnification is then
given by the ratio of Equation 6 to Equation 5:
θ
e
f
O
m = = . (7)
θ f
Procedure
O
You will be given two lenses. Fix the lens in place and move a laser beam transverse to
the beam as shown in the figure below.
e
Watch the beam that is transmitted by the lens on a screen. If the screen as at the focal
point, the point on the screen will not move as the laser is translated. Measure the focal
length of both lenses in this way.
Using the focal lengths that your measured, mount the lenses confocally. When you
launch light into the pair of lenses and shown in the figure below, and move the laser
between two positions, the rays exiting should be parallel. This can be tested by
measuring the positions of the two spots for the two rays just after the confocal pair and
then on the other side of the room. If the lenses are confocal, the distance between the
beams should be the same. Make small adjustments to the lenses until the exit beams are
parallel and measure the distance between the lenses. How does this compare with the
sum of the focal lengths you measured?
* If h is the height of the image from the objective lens, then using similar triangles, we have h/h O = i/O and
h e /h =i’/(f e -x). Taking the product of these two equations, and assuming x small and i‘=f O (which is a good
approximation for small x), we get h e /h O = (f O /O)(i‘/f e ). Using Equation 4 to eliminate i‘, we get h e /h O =
f e /f O .

Next, use the tubes provided to mount the two lenses. Without looking through the
telescope, adjust the distance between the lenses so that they are confocal. Now look at a
distance object through the window at the end of the hall. Is it in focus? How much did
you need to adjust the telescope to bring it in focus?
Remove the objective; insert the aperture and objective together back into the telescope;
and look at a distant object. Do you see anything differently? You may want to remove
and add the aperture several times to see what differences you notice. (The simplest way
is to leave the aperture out of the telescope and hold it up to the objective by hand so that
you can remove it and add it quickly).
Finally, while viewing the distant object with one eye through the telescope, look at it
with the other eye without the telescope simultaneously. From this estimate the
magnification. How does this magnification compare with what you get from Equation
(7).
Take your telescope home and take a look at the moon and the stars…