Physics 103

Lens Combinations: Telescopes

Introduction

The rules of ray-tracing have a simple consequence for lens combinations: if
two lenses are mounted one after the other, then the image formed
by the first lens becomes the object for the second lens. Thus one can
apply the thin lens equation twice to find the image formed by the system of two
thin lenses. With each application you have to measure *p* and *q*
from the appropriate lens and use the *f* for that lens. You also have to
remember the sign conventions for object position *p* and image position *q*
(opposite!) If necessary,
REVIEW the thin lens equation. Check your understanding of these
ideas by deriving the following useful fact: two lenses
which are at the same place -- essentially on top of each other -- have an effective
focal length *f* which obeys

[Note this is NOT TRUE if the lenses are at different places!]

One of the simplest and most useful lens combinations is the
astronomical telescope (below).
The lens at the left is called the *objective*, and the lens at the
right is called the *eyepiece* (the one you would put your eye up to). The
object is at infinity, and the image is also at infinity! What good is that? you may
wonder. Look at the *angles* in the simulation below. In particular,
select the "source" at the far left by clicking it, and then drag the rays to change
their angle. You will see that
the telescope magnifies angles -- and if you think about how we see, you will
realize that this is what we mean intuitively by magnification.

Another way to think about this is to
add an "eye" at the far right
to look through the telescope. The eye is a *third* lens and a "retina".
The object for this third lens is the image formed by the second lens, so if that
image is at infinity, it will be focussed nicely onto the retina (the relaxed eye
can easily form images of things far away). Again select the source and
drag the rays around to see
how angle translates into position on the retina. The eye is seeing the image
of distant stars as points, and their angular separation is magnified.
If the eye were looking at a fixed constellation of
stars, would the constellation look inverted through the telescope or not?
Explain carefully!

How far apart should the two lenses be to make a telescope? Find *q _{o}*
and

You might be interested in what Galileo said
about his methods for making quantitative measurements with the telescope. He had
no clear theory of geometrical optics, but he knew that the telescope magnified
angles. Here
is his description in its entirety, from his 1610 book **The Starry Messenger**:

He never wrote about the theory of the telescope again.