🚀 go-pugleaf

I submit this to encourage the rest of you to submit some of your
favorite teaching scenarios and strategies, as resources for the rest of us.


A teaching scenario: Lenses, Images and Parallax.

Cliff Swartz has written a book of teaching strategies and
scenarios, which is soon to be published. I haven't seen it, but
I can imagine that it might include things such as this example of
mine below. I don't know what Cliff's book will cost, but this
comes to you at no charge, and is worth every penny. :-)

A demo/lecture scenario is most effective when it develops ideas
in a logical sequence, introducing new concepts building upon ones
already grasped. This requires a good deal of thought and planning.
It requires extreme care in choice of words and manner of
expression, not only for precision, but to focus the students'
minds on what you want them to notice, observe, and think about.

I find that students have a lot of trouble with virtual images and
virtual objects, and part of the problem is conceptual. They use
imprecise language in describing images and don't really grasp the
relation between the real situation and the lens equation. This
scenario attempts to give them more useful concepts. This isn't
polished, and anyone who wants to try something like this is well
advised to practice first, and adapt it to one's own teaching
style. The script is a bit lengthy because I don't have diagrams
which would save words. I find this can be completed in a 50 minute
class period, but could be split into two parts.

REAL IMAGES

Use a large diameter lens (4" or more, for bright image) of a focal
length about a foot or more, on a 1 meter optical bench with the
usual light box with + shaped mask. The + has an arrow pointing up,
and a dot on the right. Use a ground-glass screen (not an opaque
screen).

Place the screen at the far end of the bench. Cast on it a real
image of the object. Go through the usual moves to show that for
a given object-screen distance you can get a large or a small
image, depending on lens position

Ask students what sort of image is on the screen. They respond
"Real." If they don't also say "inverted" remind them of that.
Position the screen so that they are looking directly at the side
of the screen on which the light is incident. You are standing on
one side of the bench, say to the students left.  Ask whether the
image is inverted left-right. They may hesitate (which is good).
Swing the object box so it faces them, and note that the dot is to
their right, the arrow is up. Swing the box back, so the image is
seen on the screen, and note that the arrow is up and the dot is
to their right.

"Does this mean that the image is inverted up-down, but not right-
left?" you ask with straight face. "The whole apparatus is
symmetric about the optical axis, so why *should* up-down behave
differently from right-left?"

After someone suggests that the question wasn't fair (good point!)
because you shifted the reference frame for left-right when you
rotated the object-box, note that from the students' point of view
the dot on the object-box is on their left, but the dot is on their
right on the screen. So the image *has* been inverted when you keep
the reference frame for left-right constant. Also note that, as you
stand beside the bench, "The dot on the object box is on *my* side
of the bench, but the dot on the screen is on the *other* side of
the bench. The arrow is up on the object-box but down on the
screen. Clearly both are inverted."

Swing the entire apparatus around so the students face the object
box, and see the *back* side of the ground-glass screen. From this
reference frame, they see the dot on the object-box on their right,
but the dot is on their left on the screen.

You may wish to mention the question "Why does a mirror invert
right/left but not up/down." But if this has not *already* been
discussed in class, I think it would be distracting to discuss it
here in the middle of this scenario. Besides, the mirror question
has extra dimension to it. :-) (The mirror image is not flat.)

You might point out that a real image seen without a screen is
called an 'aerial' (in the air) image. But the image is there, and
of the same character, whatever we choose to name it!

When these points are made move to the next concept

VIRTUAL IMAGES

Position the bench perpendicular to the students' line of sight.
This might be a good time to move the screen a significant way
forward or back, so what they see on the screen is slightly `out
of focus' but still easily recognizable. Tell them you are moving
the screen. "Now where is the image?" If they say "On the screen"
they haven't yet gotten the point. Perhaps it is best not to
resolve that question, but proceed to the next step, then return
to this one.

Move the screen back to the position of best focus. "I've just
found, by focusing the screen, *approximately* where the image is.
The real image is very near here," you say, pointing to the screen.
(Note: don't say "on the screen" simply point to the screen.)
Remove the screen. "Now where is the image," you ask.

[We are trying to get across that what they see on the screen is
*not* the image, it is a pattern of light scattered from the screen
which looks very much like the image if the screen is close enough
to the image position.]

Some may say "Nowhere," or "There is no image."  Respond "Well, the
must be *somewhere*. Look at the lens equation 1/f = 1/p + 1/q; it
has a solution (q) for *any* value of p. Can this equation be lying
to us?" If you are a 'ham' pantomime looking around for the 'lost'
image. "Maybe it fell under the table."

Some may say "The image is still where it was before." Respond "How
do you know?" Someone says "Put the screen back and find it." Do
it, but say, "Good answer--sort of, but not complete. That's just
*one* way to find the image, an image which, by the way, is there
whether we look at it or not, and whether a screen is there or
not." Comment on the stupid semantic question "If a tree in the
forest falls and no one is there, does it make a sound?" if you are
philosophically inclined.

Many students think that the real image is the `picture' they see
cast on a screen. This exercise is intended to get them over that
naive' concrete' view, and to encourage more precise language.

Now I think it best to put the screen back in place and relocate
the lens so that it forms a *small* image on the screen, no larger
than a few cm. (You may need to change lenses, but stick with a
converging lens so you don't complicate the scenario.) Note that
the image is on the screen. Take the screen away. "The image is
still there, we now agree. Now let's find it another way."

Place a wooden pencil point-up on a lens-carriage just below the
image. Move it forward and back along the bench till it is
precisely at the image location. Move your head sideways until
there's no parallax (no relative motion) between pencil and image.
You can position the pencil point at a height so the image seems
'glued' to it. ==> Explain in detail what you are doing as you do
it. <==

Now have the students come up one by one (it's worth the time
spent) to look at the image on the pencil-point as you did. Insist
they move their heads left/right and up/down to experience the
parallax. I never cease to be amazed how many of them sincerely
think it is wonderful to see. Suggest that those waiting in line
hold their left and right hand index fingers at different distances
from their eye, close one eye, and experiment with parallax.
Suggest they try to touch the image, which seems so 'real'.

Now position yourself to observe the image again. Move your head
toward it. "I can observe this image from different distances, let
me try to get close. The image is getting a bit blurry when I am
closer than 25 cm to it." Explain the term 'near-point'. "Now, when
I'm too close, the image is *very* unclear, I can barely make it
out." Remove the pencil and move closer. "When the image is right
on my eyeball I can't make out its shape at all." Move closer to
the lens. "Can't see any image from here. Where is it? Is it still
there?" "Yes, behind your eye" students respond.

Now remove the screen. Move the lens to the optical bench end where
the screen was. Move the object box close to the lens, within the
lens' focal length. Now look through the lens. "I see an image. It
is much larger than before, but where is it? I move my head to
observe parallax. The image seems to be toward the other end of the
bench." Have a student place a finger near where you think the
image is, then move the finger forward and back at your direction.
"There, now you are pointing directly at it!" you announce.

In this process emphasize that you are looking at the image through
the lens, but looking at the student's finger directly (not through
the lens). Ask the student to place a long rod in a carriage at the
position you've found. The rod must stick up above the lens so you
can see it. To save time, make the final adjustment to eliminate
the parallax between image and rod by slightly adjusting the lens
position. "My arm isn't long enough to reach the rod, you explain."

Now have students file by to see it themselves. Explain that at the
location of this image, there's no light from the object box. So
obviously we can't put a screen there and find it." Actually put
a screen there to demonstrate that, if you think your students are
a bit slow to get this point. "Our eye thinks there's a source of
light at the image position because of the direction the rays enter
our eye. Our brain triangulates this to calculate where those rays
*seem to come from* That's what defines a *virtual* image. To see
it one must look through the lens toward the light source.

You may wish to use overheads of appropriate ray diagrams here, and
also at other points during this scenario.

This might be a good point to say that *all* images are defined by
the rays *emerging* from the lens. If they are convergent, the
image is real at the plane of convergence. If they are divergent,
the image is at the apparent plane they diverge from.

"Is there any other way we could find an image?" you ask.

Return to the real aerial image case you began with, but perhaps
use a longer bench, or a shorter focal length lens, so that there's
working room.

Put up the overhead of this case. "Note that the rays we looked at
when we could see the real image were *divergent* from the image
after passing through it. Our eye is a lens system (cornea and
lens) and a screen (retina). Let's substitute a lens and screen for
the eye."

Put a lens and screen so that the image from the lens on the bench
(Lens 1) becomes the object for this new lens (Lens 2) and forms
a real image on the screen.

When this is achieved, I like to take a 50 cm long rod and clamp
it to the lens holder of lens 2 (near the carriage) and likewise
to the screen holder, to hold both in fixed relative position. The
rod can be long enough so that a pointer can be placed at the
position of the image formed by lens 1 (which is the object
position for lens 2). "Now whenever there's anything giving off
light at the pointer position, it will be sharply imaged on this
screen. Even if it's an *image* at the pointer position, like it
is right now, we'll see another image of it on the screen. We have
made an 'image-finder' (lens 2, screen, rod and pointer)."

Demonstrate this for various positions of the real image, and
confirm each time that the image is at the pointer by placing a
ground glass screen there.

Now use this arrangement to locate some *virtual* images!

I've run out of steam tonight, but you see how this sort of demo-
scenario could help students realize that virtual images are just
as important as real images, and really not all that mysterious.

VIRTUAL OBJECTS

Now you may wish to repeat this scenario with a *diverging* lens.
Note that when the object is real, the image is always virtual.
"Can a diverging lens form a real image?" You ask.

"Let's look at the equation." Graph the equation. "It is an
hyperbola. Notice that it has two branches." [Don't outrage the
mathematicians by calling it `two hyperbolae' for it is
mathematically considered a single curve, with two branches.]
"Notice from the graph that we can get positive q (real image) only
when we have negative p. What does negative p mean? It means the
object is on the other side of the lens from the cases we've been
doing so far. How can we get it on the other side? Remember how we
defined image, by the convergence or divergence of rays coming from
the lens? We define object the same way, by the convergence or
divergence of rays *entering* the lens."

"In the previous cases the objects were real, emitting divergent
rays which entered the lens. To get a negative p, we have to have
rays convergent into the lens. How can we do that?" Students say,
"Use another lens to put a beam of convergent rays into the lens."
You ask "How will we know where the image is?" Students respond
"It's where they converge toward." [Don't you wish all students
were so astute?]

Emphasize that we are studying lens 1 and trying to test the lens
equation as applied to it. The other lens (lens 2) and the object-
box are merely tools for producing convergent rays. To take
measurements, we note the position of the screen, the position of
lens 1, and then remove lens 1 and use a screen to find where the
rays from lens 2 converge to. These three numbers are all we need
to confirm the lens equation for lens 1. We do *not* need to know
the position of lens 2, nor the position of the object box.

Now turn the students loose to apply this knowledge to the usual
`thin lens' laboratory. Insist that they study a converging, and
a diverging lens, and produce data for all possible images within
the span of the bench, real and virtual, and both real and virtual
objects.

In the spirit of our data analysis admonition: "Whey you can
transform the data to a linear relation, you should do so." Here
we easily can. Plot 1/q vs. 1/p. The intercepts of this straight
line will both be 1/f. This has transformed the hyperbola to a
straight line, re-emphasizing that it was *one* curve, after all.

I've been a little less detailed here near the end, for the 'moves'
are in the same spirit as before and get a bit repetitious to
relate.

As you all know, wrong answers from the class can often be
exploited for learning value. I once had an optics class so good
that I had to conspire in advance with one of the best students to
supply some wrong answers now and then. Haven't had to do *that*
in a long time.

        -- Donald

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Dr. Donald E. Simanek                            Office: 717-893-2079
Prof. of Physics                    Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745          CIS: 73147,2166
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++