After our in-class exercise with ray-tracking, you already know how to do it. However, I’d like to add some extra comments, explaining in detail the mea- ning of the arrows we draw for the “object” and the “image” – what is exactly their role in the ray-tracing diagrams.
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After our in-class exercise with ray-tracking, you
already know how to do it. However, I’d like to add
some extra comments, explaining in detail the mea-
ning of the arrows we draw for the “object” and the
“image” – what is exactly their role in the
ray-tracing diagrams.
Below is a ray-tracing diagram for a converging lens – something you
already know very well. But let’s take a closer look at the object and
at the image, using a magnifying glass:
There is a point source of light, and
the image is also a point.
The ray-tracing method enables one
to find the point image of a point object
formed by the lens. The left arrow is not
a part of the object, and the right one is
not a part of the image!
Then, what are these arrows for?! Is it really necessarry draw them?
Wouldn’t it be OK to make ray-tracing diagrams just like this one?
Well, such a diagram is “essentially correct”.
It looks “somewhat silly”, doesn’t it? And it may be confusing.
The arrows show where exactly the point object and the point image
are located. They add much clarity to the diagrams!
Therefore, we should always draw them -- however, keep in mind
that they are not themselves objects or images, just “helpful indicators”.
Point objects are interesting – but primarily for astronomers(stars
are good examples).
In most „real-life” situations, however, we deal with objects offinite size
– e.g., likethe “rod” pictured below. Can we use ray tracing for such
objects?
Sure! – why not?Simply think of the “rod” as of
a “chain” consisting of a large number of
point sources, and then do ray tracingfor each
“point source”, one by one!
Such a ray tracing procedure, though, would not be very convenient
if done on paper. The large number of rays drawn would make the
plot pretty messy – look:
However, it is not
necessary to do the
ray tracing for all
our “point sources”.
It’s enough to do the
tracing only for the
object endpoints –
and we will get the
image’s endpoints,
which is all we need.
Of course, the rod
needs not to at a
position symmetric
relative to the lens
axis – one may shift it
up or down, ray tracing
performed for the two
endpoints only always
give us the right posi-
tions of the image’s
endpoints.
And, of course, dividing the object into many “point sources” was
needed only to explain the underlying idea – having understood it,
we don’t need to plot individual “point sources” any more.
We can plot the rod “as it is”, and do the ray tracing only for its
ends – and then just plot the “image rod” by drawing a line
between the two endpoints we have obtained.
So much about the ray tracing procedures for large objects!
And now we switch to the next important topic – magnification.
First, let’s define the so-called “lateral magnification”:
B’
A
ho
O
hi
B
A’
xo
xi
The symbol “>>” means “much larger”, and “<<“ means “much smaller”.
However, for us a more interesting and more important
parameter is the so-called ANGULAR MAGNIFICATION
First. let’s define what we call the ANGULAR SIZE of an object – the
picture below explains what it is:
The angular size (AS) of an object depends on how far it is from the eye.
The closer is the object, the larger is its angular size.
The AS of a dime viewed from the distance of 1 yard is about 30 minutes
of arc. From 30 yards, it’s about a single minute of arc.
Human eye cannot resolve details smaller that a few minutes of arc.
Looking at a dime from 30 yards, you can probably recognize that it’s
a coin – but you rather won’t be able to tell whether it’s an American
coin, or a Canadian dime.
For “seeing things better”, we always want to bring them closer to
our eyes – i.e., we want to make their angular size bigger.
Angular magnification, not lateral magnification, is the
one that really matters when we talk about instruments
used for direct visual observations.
Last time, we did ray-tracing
for a simple two-lens microscope.
from the plot, it is clear, that the image
is indeed considerably magnified. But it
still does not show that the angular magnification
is big. In order to see that, we need to add the eye
of the observer to the picture – it’s on the next slide.
The angular size of
the object observed by
an unaided eye is the angle
between the lens’ axis and the
red line in the picture. The angu-
lar size of the image is the θ2 angle.
Angular magnification is particularly important
in the case of telescopes – it is, instruments used
for observing very distant objects.
Talking about lateral magnification in the case of
telescopes does not make much sense! Why?
The reason is simple: because we usually don’t know
how far the object is, and what are its dimensions.
It’s only the angular magnification that matters. If
you see a telescope in a store, with a label “ 60×”,
it means that the angular size of the of the object’s
image produced by this telescope is sixty times
the angular size of the same object viewed by an
unaided eye. For instance, the angular size of full
Moon is about 30 minutes of arc; watched by this
instrument it would be of the size of a vinyl LP record
held in extended hand.