Geometric Optics – Part 4 – Thin Lenses


Good afternoon ladies and gentlemen, we are
now going to start the next portion of geometric optics, which is about thin lenses. Now, lenses
are different than mirrors, because mirrors were based upon the idea of reflection about
the normal line, but lenses doesn’t deal with reflection, it’s entirely about refraction.
About light, actually traveling through the lens itself, traveling through, and bending
when it goes through, so it’s not about reflection. And because it’s not about reflection
we’re going to see that the material of the lens will actually have an effect on the
image that is being formed. The shape isn’t just the only that’s important, it’s also
about the type of material, or in other words: the index of refraction. Well let’s go back to this picture for a
moment. There are two main types of lenses: converging lenses and diverging lenses. The
first set, these are converging lenses. The main thing that distinguishes what a converging
lens is, is that it is thicker in the middle, and thinner around the edges, so as you can
see throughout all of those examples. The most obvious example of a converging lens
is, of course, a magnifying glass (and we’re going to talk about the type of image that
is being formed by a magnifying glass later on). But let’s just write that down, an
example is a magnifying glass. The other type of lens that we are going to be talking about
is what we call a diverging lens. So if I just raise this picture up a little bit so
we can see, you can see a diverging lens is the exact opposite of a converging lens. So
basically, a diverging lens is thinner in the middle, and thicker around the edges.
And, the most obvious example would actually be the glasses that I am wearing right over
here. These are actually diverging lenses. If you look at the side of lens itself, you’ll
see how the lens has a thicker portion of the edge where the frame is, and in the middle
you actually have a thinner portion. So it’s actually similar to this last example over
here, where it’s thinner in the middle and thicker on the edge, okay, that’s a diverging
lens. And of course, the words converging and diverging
automatically they make you think about what’s happening to light. Cause with mirrors, when
you had a concave mirror, it was a converging mirror, and a convex mirror was a diverging
mirror. So with a lens, okay, a converging lens basically has the parallel rays of light,
going through, and they all bend so that way they all converge together. As we can see,
they converge through, what we call again our main focal point, okay. But you have to
realize this is different than a mirror, in that in a mirror, when the parallel rays struck
the mirror, they reflected back, and the focal point was on the same side as where the rays
came from. Now the rays of light are actually going through. Okay. So, with a diverging
lens, we can see that when parallel rays of light strikes the diverging lens they do diverge
and separate from each other as they’re going through the lens; and they’re not
going to come together at all on the real side. But rather, they’re going to come
together on the virtual side. So if we bring those lines back, we can see that there is
a virtual focal point for a diverging lens. Okay, and the virtual focal point is of course
not really there, but it’s what we’re going to use for our calculations. Now, a converging lens, I want you guys to
realize it doesn’t even have to be glass or be plastic. It can even be something as
simple as a water bottle. If you watch this demonstration of what happens when you use
the sun, and you use your bottle to focus the rays of the sun, to light a fire. Okay, so basically, the round shape of the
water bottle is a converging lens. It’s thinner in the middle, it’s thicker on the
edge and so ultimately it’s able to focus the rays of the sun at the focal point. So when we actually do this mathematically,
we have to be careful how we draw this. For the most part, it’s actually very similar
to how we draw mirrors, but there are a few key differences. First of all, when you’re
starting your drawing I want you guys to always draw a straight down the middle of your lens
(well as straight as you can at least). And, understand, that when light does travel through
it is going to be focusing through some sort of, well, focal point. But, rather than writing
“C” like we did for mirrors, we’re going to write simply 2F, so it’s still equidistant.
If this is F, 2F, it would be equidistant from F to the lens and from F to 2F. Now the
reason we write 2F and not C is because it’s not necessarily going to form a circular shape
anymore, it’s about the material of the lens itself that chooses where the focal point
is going to be. For example, if I use something with a higher
index of refraction, so I’ll write, if “n” is increased. That means, before when the
ray of light came across, it would converge through the focal point. But if I send green
light, which has a higher focal point, I’m sorry, higher index of refraction than red,
it’s actually going to bend more. So in this case it’s actually going to look like
this, and F for green is over here, and 2F would be, approximately over here. So that’s
part of the reason why I don’t actually write down F and C, I only use F and 2F because
it changes entirely based on what the index of refraction is. So over here, okay, so again: line straight
down the middle, F, 2F. Now, the thing is, the lens is curved on both sides, so on the
other side I’m just going to write F´ and 2F´. Now, they’re basically going to be
our virtual, or sorry, our secondary focal points. We’re not really going to be using
it, but if light were to travel from the other direction it would travel through to F. And
it’s going to be important for us when we actually start doing ray tracing.
Okay. Now, we go to the diverging lens. The diverging lens is the same thing where we
have a line going straight down the middle. But now remember, when light travels through
a diverging lens, they all diverge outward. So our main focal point is not going to be
on the right side, instead, our main focal point is going to be on the left side. Since
the main focal point is going to be on the left side, I’m going to label F over there,
and 2F will of course be equidistant; while, on the other side, we’ll have F´, 2F´.
Okay, so just keep that in mind for once we start doing ray tracing. Okay. Just like before we’re going to follow
certain rules for how we draw the lines, and the rules for lenses are almost the same as
the rules for how we draw mirrors. The rules for ray tracing lines, are, we first go parallel
along the axis. And after it strikes the lens, it’s going to go through F. Notice I said
F, not F´. It goes through the main, primary focal point. The second line is through F´,
not F, but though F´, then parallel along the axis. So, technically that’s always
going to be our second line, but in reality it’s a little hard to draw, and I know that
you guys tend to get a little messy, so thought we’re going to skip the second line, we’re
always going to go to our third line, our check line. The third line is the easiest
one. All it is, draw a line straight through the center of the lens. Okay, because when
you hit the center of the lens, and if you take a look at a third drawing here, when
hits hit the center of lens, technically is striking right on the normal line. And whenever
you strike exactly on the normal line, it’s like with refraction, it goes straight through.
So in this case it would actually go straight through for our image formation, okay.
So let’s try this out a little bit, and take a look at a few different examples. The
first example we’re going to deal with is the magnifying glass. Okay, so the magnifying
glass is our converging lens, but you have to be careful about where the object is versus
where the image is. Okay, if you take a look at the magnifying glass itself, you’ll notice
that where the object, okay so right now let’s use me as the object, Okay you guys are looking
at me, so when it’s up close to me, when I’m very up close, you guys could just take
a look at my eye, I’m sure my eye looks enlarged. But when the magnifying glass is
further away from the object, and if guys are looking at my face now, you should see
that face actually turns upside down in the lens. And, this is a real image that’s being
generated by the lens itself, and this is going to you, to your eye over there.
So let’s just do a little ray tracing and see what happens. Okay. Let’s first actually
do where we hold the object further away from the lens itself. Okay, so that’s this example
over here. Now, a few things to note, and we’ll need this for mathematical purposes,
okay, which side is real, which is virtual? Just like before, real means inverted, virtual
means upright, but you have to keep in mind where is the eye. So right now, if this is
the arrow on the left over here, I want you guys to take note. That means you guys are
looking through the magnifying glass in the middle at the object. So just like when you
guys are looking at me, right, you see that I’m upside down, and I probably appear smaller
than my actually size. So that’s what your drawing should look like as well, that’s
what it should indicate. So if I label this as F and 2F, I know that the right side, wherever
the eye this that’s always you’re real side and the other side is going to be my
virtual side so I have F´ and 2F´. I follow my basic ray rules, so parallel and through
F, parallel, then through F. My second line, I could do through F´ then parallel, but
it’s easier to always just go straight through the center. So just line up your ruler with
center of the lens, and just draw it going straight through. And wherever the lines converge,
that’s where the image is. So over here, I have a real smaller image. And notice, this
is smaller, this distance of the image from the lens is smaller than the distance from
the object. If you remember, the magnification is directly proportional to how far away it
is from the lens also. So we actually have a smaller real image, so that’s what you
guys see when you’re looking at me. If you take a look at the next example, you’ll
notice this object is on 2F´. Now I want you guys to be drawing in eye as well because
its good practice and it’ll help you with the math. If you remember with lenses, I’m
sorry, with mirrors; when the object was on C where was the image? Well we’ll see here,
parallel, then through F, through the center, the image is exactly on 2F. So, because this
distance of the object and distance of the image are exactly the same, we know that it’s
going to be
the same size. Okay. Now one more example, okay, now this one’s
interesting. Because, you’re actually looking at this example right now, you just don’t
realize it. We have an object that’s now closer to F´ than before. And as you remember,
with concave mirrors, the closer the object was to the focal point, the larger the image
appeared. And the same is exactly true for lenses, the closer it to F the larger the
object sorry the larger the image appears. And this will work out the same way. So let
me just draw my eye, positive, negative, okay, parallel and through F, straight through the
center, and you can see my image, all the way over here. That’s real and larger. Okay
and you guys might be thinking what does it mean that it’s real and it’s larger? Well,
the image is actually there. When you, in fact, when you look at the Smart Board projector
or any projector system. If I were to just sketch this out for you guys normally there’s
some sort of projector, with a lens, okay and then the screen over here and on the screen
you have the image being projected onto the screen, like let’s say a tree or something.
What’s inside your LCD projector is actually a little mini LCD screen. Meaning that inside,
there’s a picture screen, with a tree on it so if course, since every real image becomes
inverted, they actually start this image upside down. So there’s a little tree that’s
actually upside down, that’s then projected through the lens, onto the screen. And the
screen is exactly the distance of the image. And the distance of the object would simply
be of the little miniature LCD screen to the lens. And this would be, if you were to actually
draw this out as a real lens, the object, would be between F´ and 2F´. In fact, let’s show you guys a quick video
on this. Let me erase this, okay. Now in this one, in this video, video the person is actually
making his own homemade projector using and iPhone and a magnifying glass. He’s putting
the magnifying glass that way, he cuts it out. Okay let’s drown out the music. He
puts his phone inside, he turns up the brightness all the way as much as possible, he makes
sure that his phone is in between the focal point, F´, and 2F´, and when he uses it
he will actually project an image onto the wall behind the projector. So when we take
a quick look at the end of this video it’s not going to be very bright, because honestly
an iPhone isn’t very bright, but I’ll just pause it right here you guys can see
that there’s an actual image being projected over here, okay. But that image, if you were
to take a look at his phone, is upside down the way his phone is actually doing right
now, because it’s a real inverted image. I know it’s a little hard, but hopefully
you guys see that, basically, it’s a real image. This last example over here, well these last
two examples over here, are again similar to concave mirrors. F, 2F, what happens again
when an object is placed on 2F´? Well just like with concave mirrors when an object is
placed on F, hold on let me draw me eye cause it’s good habit, parallel then through F,
straight through the center, are those two lines ever going to meet? No, they’re completely
parallel to each other. So in this case, it’s going to be no image, just like with the concave
mirror. Okay, and the last example is about holding
a magnifying glass up close to something so that the object actually is between the lens
and the focal point itself just like my eye is right now. So, just like with a concave
mirror, which is like a makeup mirror, when it’s held up close to something you’re
going to create a magnified image and as you can see it was virtual and it was larger than
before. So, let me draw the eye, positive, negative, okay. Parallel then through F, oh
that’s a terrible line let me just redo that one, and then through the center. Okay,
now they’re diverging apart meaning that they’re never going to meet on the real
side, so, let’s back track them a little bit. And there’s the image, a virtual image.
So this isn’t something that’s being projected through the lens as in the case of the projector,
this is something that you can only see within the lens itself because it’s an illusion
that the mind makes up within itself to create this virtual image. So this is going to be
virtual, and larger. Okay. Now, let’s talk about diverging lenses,
okay. A diverging lens is similar to a convex mirror, or a security mirror. So if you take
a look at my glasses, at my face, right, hopefully you guys can see, what kind of image is being
produced. And hopefully you guys can see that I’m upright, meaning it’s virtual, and
that the image is smaller than before. Again, just like a convex mirror or a security mirror.
So in this case over here when you have the eye in the right hand side, looking at the
object, you see a virtual smaller image. So let’s actually trace this out, let’s trace
it first and then we’ll talk about the example. Line down the middle, now remember, light
diverges when it hits a diverging lens, meaning that it’s going to bend a way. So that means
this is my main focal point, on the left. Even though my eye is still over here on the
right hand side, and this is positive, and this is negative, my main focal point is on
the virtual side, meaning that it’s a negative value. F´ will be on the other side, of course
there’s also F´ and 2F´, but they’re not that important in this case. That’s
because I want you guys to see we have our first line, parallel, divergent, then back
through F. And then our second line straight through the center. And where those lines
meet, that’s where we have our image, which is right over there, okay. So we have a virtual,
smaller image, and that is the only type of image that is being produced, just like with
a convex mirror or a diverging mirror. So, let’s see, let’s take a quick look
at some animations just to double check our work, and to show you guys what’s going
on. Okay, over here I have a converging lens. You can see when the object is on 2F, or really
2F´, that it’s the same exact size. When you move the object closer and closer to F´,
the image gets larger and larger. If you go onto F´, there’s no image being formed
because the rays of light are parallel to each other. And if you go in between, just
like with a magnifying glass, keep in mind the entire time the eye is over there, okay,
so because the eye is over there so a larger, more magnified image. And the closer the object
is to F´, the larger and larger the magnification is. In fact it’s so large, you wouldn’t
see the entire object. Just like when you’re looking at my eye you can’t really see my
entire face, because it only magnifies a certain part to be much larger. When have it much
further back, okay, we have a real image being produced on the right hand side. The farther
away it is from F´, the smaller image, just like when you’re looking at me through this
lens over here. Okay, so a smaller image that’s upside down, which means it’s real and inverted. If you take a look instead at, let’s see,
sorry, give me a moment, a diverging lens; I want you guys to realize, the eye is over
here, so this is positive, this is negative, and no matter where the object is, it’s
still always going to be a virtual, smaller image. Okay, but parallel through F, straight
through the center so this one’s easy to draw, it’s not as complicated as a converging
lens, because there’s only one image produced. Cause mathematically, you have to remember
that this is a negative main focal point. Now, what else can I talk about in terms of
an easy example to understand? And the best one, of course, the human eye itself. So if
we talk about the human eye, okay, I want you guys to realize, the lens in your eye
is actually a converging lens. Well let’s just play this video quickly, which is in
German, but we’re not going to worry about that too much. So normally when light strikes the eye, it
converges, I’ll pause it for a moment, onto the optic nerve, which is at the back of the
eye. And so, when you see a picture of a tree, it’ll actually show a little real image
of tree that’s upside down behind the eye. Except that your eye, your brain, has been
trained to take that image, and flip it right-side up, which is why we see everything upright,
even though technically, it’s being focused in an inverted way on the back of our eye,
it’s going to be focused right-side up. Now, the thing is, if you happen to be near-sighted,
okay, that means that the lens in your eye is either a little bit fatter, or that your
eyeball is slightly longer. But because it’s slightly longer, the image focuses at the
wrong location, and, everything looks a little bit blurry because the image has to be project
onto the optic nerve perfect for that to work. So to fix that issue, we can, I wear glasses.
I myself am near-sighted, and let’s see hopefully, there we go. If we put a diverging
lens in front of the eye, the light will actually diverge outward first, before converging exactly
onto the optic nerve, and that’s how glasses work. That’s how my glasses work, for someone
who has myopia, or in other words, I am near-sighted. For someone who happens to be far-sighted,
the image is actually projected behind the eye. So to fix that, they’re actually going
to put a converging lens in front of the eyeball, which brings the light together, so that way
it’ll focus it directly onto the optic nerve.

12 thoughts on “Geometric Optics – Part 4 – Thin Lenses

  1.  Amazing.The handwriting is being captured onto the screen.That's a pretty cool teaching tool. What is the software and what are the tools that were used.

  2. I have my laptop connected to the SmartBoard while using screen capture software. The software I use is called Screencast-o-matic (http://www.screencast-o-matic.com/). That way my laptop camera records me writing on my SmartBoard, which automatically translates to the screen.

  3. I am having some difficulty with a project that requires viewing a cell phone and characters from an approx. distance of 4" from the eye.  I can easily place a lens with a diopter of 6 directly in front of the eye and read the characters but the project requires the lens/ lenses to be placed close to the face of the cell phone.  Would you recommend a combination using converging and diverging lenses or would it be best to go with a single large magnifier?

  4. Nice demonstration. Please correct: If the object is at the focal point, there is a virtual image. It is at infinity, similar to the sun and other distant objects whose light comes to us in parallel rays.

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