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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.

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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



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



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”:











Quick quiz ( not written, verbal):

  • Object far away from the lens (xo>> f ):
  • Is the magnification ML a large number ( >>1 ), or
  • a small number ( << 1 )?
  • Can you think of a device that is a good example of such situation?
  • 2. Object close to the lens ( xo only slightly larger than
  • the focal length f ):
  • Is the lateral magnification a large number, or a small one?
  • Can you think of a device that is a good example of such situation?
  • (Hint: one such device is here, in this very classroom!).

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.