"Unexpected" Image Formation with the help of our eyes

[DaTario]

Hi All,

It is quite familiar the result bellow concerning the image of an object when it is on the focal point of a concave mirror (ref: http://www.physicsclassroom.com/Class/refln/u13l3e.cfm):
"When the object is located at the focal point, no image is formed. (...) light rays from the same point on the object will reflect off the mirror and neither converge nor diverge. After reflecting, the light rays are traveling parallel to each other and do not result in the formation of an image."

Ok, but if our eyes are the "measuring aparatus", their convergent lens will make some sort of image to appear to us. Is it the case that one has to teach this subject appealing to a concept of image which excludes us as observers?

Best wishes,

DaTario

 

[CrazyNinja]

I could make perfect sense of your case till I read :

Is it the case that one has to teach this subject appealing to a concept of image which excludes us as observers?

Could you simplify the question? Then I believe I could help you better.

 

[DaTario]

Yes, I'll try.

My question has to do with teaching.

How are we to teach the no image formation when an object is placed at the focal point of a concave mirror if in this condition our eyes are capable of mounting an image?
Do we have to explicitly deny our eyes as an admissible apparatus?
Or do we have to explain that the eyes form an extra optical setup that, joining the already existent (object + mirror) system, redefines the list of situations where no images are formed?

Best Regards

DaTario

 

[jtbell]

I would call that an "image at infinity", not "no image". It can be used as an "object at infinity" for the next lens in a multiple-lens system. The lens of the eye is only one example of such a "next lens".

 

[DaTario]

I would call that an "image at infinity", not "no image". It can be used as an "object at infinity" for the next lens in a multiple-lens system. The lens of the eye is only one example of such a "next lens".

In this case, may this image at infinity be interpreted, by our eyes, as an infinitely distant object?

 

[Merlin3189]

I would not attach too much importance to this statement. This is someone attempting to explain and simplify to beginners.
As presented in the link you give, I think it is a reasonable statement.
On the other hand I agree with jtbell and would use such a description in other contexts.

Generally, I would not think in those terms. I don't think it is helpful once you understand what is going on.
If you think of a Gallilean telscope, the front converging lens attempts to form a real image, but the rear diverging lens gets in the way and it is never formed. The diverging lens may create a virtual image, but to see it, your eye still has to change it to a real image on your retina. Similarly with a Cassegrain, though the final eyepiece lens (or usually combination) may take the real image from the diverging mirror and convert it to a "not-image" at infinity, to make it easier for a normal eye to get the real image on its retina.
In many complex lens systems, it is only the final lens (maybe the eye) which actually gets to create a real image. The earlier lenses may create virtual images, but some will simply be foiled in their attempts to create real images.

What I think this lesson is trying to do is simply to give newcomers a general feel for what happens with a single mirror (or lens.)
If an object is at infinity, a real image is produced at the focus. As the object gets closer, the real image moves further away towards +infinity. When an object is very close to the mirror, it forms a virtual image behind the mirror. As the object moves further away from the mirror towards the focus, the virtual image moves further behind the mirror towards -infinity. Around the focus the image suddenly flips from virtual near -infinity to real near +infinity. What can we say about the image when the object is at the focus? Perhaps it's best to say nothing! But saying an image is not formed may be no worse than saying a virtual image is formed at infinity.

How are we to teach the no image formation when an object is placed at the focal point of a concave mirror if in this condition our eyes are capable of mounting an image?

I would say "the mirror does not form a real image", not that "no image is formed". And really I would say that the objective in the Gallilean telescope does form a real image, even though the eyepiece gets in the way and converts it to a virtual one.
If I were teaching, I would explain clearly what a mirror or lens does, where it IS clear, and talk about the interesting singularity of the behaviour at the focus. I found that the idea of a virtual image was the main difficulty. If people accept that, then a virtual image at infinity is no more difficult.

Edit:

may this image at infinity be interpreted, by our eyes, as an infinitely distant object?

I don't think our eyes interpret anything. The lens just forms a real image with the rays entering the eye (if possible). Any interpretation our brain puts on what it sees, depends on a lot more than the rays of light.

 

[jtbell]

As the object moves further away from the mirror towards the focus, the virtual image moves further behind the mirror towards -infinity. Around the focus the image suddenly flips from virtual near -infinity to real near +infinity. What can we say about the image when the object is at the focus? Perhaps it's best to say nothing!

It may be useful to point out that the "sudden flip" from virtual to real is an artifact of this particular description of the system. If we focus (heh ) on the convergence or divergence of the collection of light rays from a point object, after reflection off a concave mirror, there is no sudden transition.

Let the object start at a point close to the mirror. The reflected rays diverge. Move the object slowly away from the mirror towards the focal point. The reflected rays become less and less diverging and more and more nearly parallel. At the instant that the object is exactly at the focal point, the reflected rays are exactly parallel. As the object continues to move away from the mirror and from the focal point, the reflected rays start to converge. This transition is a continuous process, when you consider it in terms of the angles that the light rays make with the optical axis.

 

[DaTario]

Ok, Merlin3189, I agree with with you, sorry for the loose way of speaking, I was referring to what comes out after neurological processing of the physical optical input ou stimulus. It seems that usually one can attach main stream interpretations to some of these optical inputs on someone's eyes.

Let me just go further and ask you if you agree with the following modification in your statement :

your original statement:
"I would say "the mirror does not form a real image", not that "no image is formed."

your statement with the modification:
"I would say "the mirror does form neither a real image nor a virtual one", not that "no image is formed."

 

[Merlin3189]

Yes it's only the $v=\frac{fu}{f-u}$ which goes wrong at u=f
I didn't really mean to imply the lens did anything odd.

Edit: re the next post which slipped in while I was writing
I think I might agree with the rewording, but I do like the idea of a virtual image at infinity, so I'm not sure.
Actually, that was a bit quick, off the cuff. If you say it does not form a real image nor a virtual image, presumably it doesn't form an image at all. So maybe you say "the lens does not form an image", rather than "an image is not formed". But were getting into hair splitting. Can't we just say what happens to the light, as jtbell says?

See what others think.

 

[DaTario]

It may be useful to point out that the "sudden flip" from virtual to real is an artifact of this particular description of the system. If we focus (heh ) on the convergence or divergence of the collection of light rays from a point object, after reflection off a concave mirror, there is no sudden transition.

Let the object start at a point close to the mirror. The reflected rays diverge. Move the object slowly away from the mirror towards the focal point. The reflected rays become less and less diverging and more and more nearly parallel. At the instant that the object is exactly at the focal point, the reflected rays are exactly parallel. As the object continues to move away from the mirror and from the focal point, the reflected rays start to converge. This transition is a continuous process, when you consider it in terms of the angles that the light rays make with the optical axis.

I agree that in terms of some angle parameter there is no apparent sudden transition, but I guess the question has to do with the ontology of this thing we call the image. Images are produced by real or virtual interceptions of light rays, so, when the light rays get parallel to each other we may say that our image is gone to the infinity (and, hence, has disappeared), much in the same sense that in changing the angle of incidence of a flashlight beam on a wall, the transition from the ellipse to the hyperbola has the singular midway state of a parabola. Thus one may say that when the flashlight orientation smoothly crosses $\pi / 2$ there happened a sudden flip from ellipse to parabola and then to hyperbola.

 

[Merlin3189]

That sounds quite a nice analogy. The change is similarly gradual and sudden at the same time! And it happens when the ellipse is infinitely long. (I'm struggling a bit to work out just how the change from parabola to hyperbola looks, apart from the extra conic intersection.)

 

[DaTario]

It seems that parabola requires very restrictive conditions regarding its proportions, while ellipses and hyperbola seems to be lesse restrictive in the same sense.
Just in the same sense that improper image is related to its real and virtual counterpart.

 

[Drakkith]

I usually prefer to think about optics problems as "what is the light actually doing" instead of "what is this image doing". I'm sure thinking about it in terms of images can help simplify certain situations, but most of the optics related threads I see here on PF are about some confusion the OP is having with "images".

 

[DaTario]

I usually prefer to think about optics problems as "what is the light actually doing" instead of "what is this image doing". I'm sure thinking about it in terms of images can help simplify certain situations, but most of the optics related threads I see here on PF are about some confusion the OP is having with "images".

I like this way of thinking as well. In fact, in a rather deeper level, I prefer to think in terms of "what are the charges actually doing". But returning to your point, I guess I understand that thinking about the rays make things simpler than trying to keep track of images.

But images can help to "compress" the knowledge, acting as an unifying principle: if you know where the image is, it gets very simple to predict what direction will be taken by any given ray.

 

[Drakkith]

But images can help to "compress" the knowledge, acting as an unifying principle: if you know where the image is, it gets very simple to predict what direction will be taken by any given ray.

Sure. Like I said in my post, I'm sure there are situations where thinking about images is simpler than thinking about rays.