### Video Transcript

In this video, we will learn how to
draw diagrams of light rays interacting with convex lenses.

Let’s begin by recalling what we
mean by a convex lens. A lens is a piece of transparent
material, that is, material that lets light pass through it. Lenses have particular shapes that
can be used to change the direction of light rays. A convex lens is a type of lens
with a shape like this. It’s thinner at the edges and
thicker in the middle. It is important to point out that
this shape is how the lens looks when we view it from the side.

Typically, if we were to draw a
diagram of light rays passing through a convex lens, then we would have our lens
like this. And we would have our light rays
coming in from the side like this. The dashed horizontal line that
we’ve drawn, which passes through the center of the lens, is known as the optical
axis. Again, it’s worth being clear that
this is all a view from the side. If instead we were to look along
the direction that the light rays are traveling, then the convex lens would look
like a circle. When we’re drawing ray diagrams,
we’ll stick to this side-on view because it provides a clear way of seeing the
direction of the light rays.

Speaking of the direction of light
rays, we know that when light rays are just traveling through air, they move in
straight lines. However, we have also said that
lenses can change the direction of these light rays. Now, it’s not all that much use
knowing that a lens can change the direction of a light ray unless we also know how
that direction is going to change. So the question that we want to
answer is, just how does a convex lens affect the light rays passing through it?

It turns out that there are a few
simple rules for how this works. So let’s clear some space and look
at what these rules tell us. The first rule is that any light
ray that passes through the center of a convex lens does not change direction. When we talk about the center of a
convex lens, we mean this point here, which is on the optical axis and also an equal
distance away from the front surface and the back surface of the lens. If instead we were to look along
the direction of the optical axis so that the lens looks like a circle, then the
center of the lens would be this point in the center of the circle, which is on the
optical axis.

What this first rule is saying is
that whatever direction a light ray is initially traveling in, if it goes through
this center of the lens, then the light ray keeps traveling in that same
direction. So a light ray traveling
horizontally along the optical axis goes through this center of the lens and
continues traveling horizontally. A ray through the center initially
traveling in this direction will keep traveling in this same direction.

At this point, it’s worth
mentioning that in reality it’s a little bit more complicated than this. A light ray traveling through a
convex lens experiences refraction when it enters the lens through the front surface
and again when it leaves the lens through the back surface. In this diagram here, we have drawn
a particularly wide convex lens, which we can use to help us see the effects of the
refraction more clearly.

Let’s consider the path of this
light ray here. In the absence of any refraction,
this ray would pass through the center of the lens. But in reality, refraction at this
front surface means that the ray’s direction bends by a small amount when it enters
the lens. Likewise, refraction at the back
surface of the lens causes the direction of the ray to get bent back again so that
it’s pretty much the same as the ray’s direction before it entered the lens. If instead we had ignored these
refraction effects, then the light ray would have followed the path shown by this
green line here.

We should make it clear that in
this diagram we have used a deliberately wide lens to exaggerate the effect. For very thick lenses, this effect
is pretty strong. However, for very thin lenses, the
effect is negligible. This means that for very thin
lenses, the effect is so small that we can ignore it and continue to use rule
one.

Now, let’s look at the second
rule. This rule applies to light rays
which are parallel to the optical axis. That means any light rays which
before they reach the lens are traveling parallel or side by side with this dashed
line here. Now, if we have a ray traveling
exactly along this optical axis, then this ray passes through the center of the
lens. And so we know from the first rule
that this ray keeps traveling in the same direction. So this second rule is going to be
useful for rays which are parallel to this line but not exactly along the line
itself, for example, rays like these two shown here.

The second rule says that any light
ray which is initially traveling parallel to the optical axis has its direction
changed by the lens so that it ends up going through a position known as the lens’s
focal point. The focal point of a lens is a
point on the optical axis which is a horizontal distance of one focal length away
from the center of the lens. The focal length distance is a
property of the lens itself. Different lenses will in general
have different focal lengths. These light rays change direction
as a result of refraction at both the front surface and the back surface of the
lens.

The second rule tells us that the
combined effect of these two direction changes is that both the rays will end up
going through this focal point. So the direction of the rays will
change a bit as they enter the lens. And then it will change a bit more
as they leave the lens so that the rays end up passing through the focal point. Every light ray which is initially
parallel to these ones before it enters the lens will also end up going through this
focal point. We can say that the rays are
focused or converged to this point by the convex lens.

The third and final rule is kind of
like the reverse of the second rule. To understand this rule, it’s
important to recall that the lens has a focal point on both sides of it. In each case, the distance from the
lens to the focal point is one focal length.

The third rule says that any ray
which passes through the focal point before going through the lens has its direction
changed so that it ends up parallel to the optical axis. Let’s consider this light ray
here. We can see that this ray passes
through the focal point before it gets to the lens. So we know from the third rule that
the lens is going to change the direction of this ray so that it ends up parallel to
the optical axis. This change in direction happens in
two stages. The ray’s direction changes once at
the front surface of the lens and a second time at the back surface of the lens. The overall effect of these two
direction changes is that the ray comes out from the lens parallel to the optical
axis, like this.

The third rule applies to any light
ray which goes through this focal point as long as that ray then also goes through
the lens. So for this light ray here, after
two direction changes, it will come out from the lens looking like this. We could also imagine any number
more light rays all passing through this initial focal point.

These three rules for light rays
passing through a convex lens can also be used to help us figure out what happens to
light rays coming from an object. Let’s clear some space so we can
see how this works. We’ve kept the diagrams which
illustrate each of the three rules on the screen, because we’ll find that these
diagrams are going to be useful.

Let’s suppose that we have a convex
lens and that some distance away we have an object. We have drawn this object in a
position such that the bottom of the object is on the optical axis and the top is
some distance above this axis. The object that we have drawn looks
like a straight vertical line. But it’s important to remember that
we’re looking at this situation from the side. Perhaps this object is actually a
piece of paper with a drawing on it. And we just can’t see that drawing
because we’re looking at the paper along its edge.

We want to know what effect this
convex lens has on the light rays coming from this object. Let’s suppose that the lens has
focal points here and here. We will begin by considering some
light rays coming from the top of the object. From the first rule, we know that
the light ray which passes through the center of the lens keeps traveling in the
same direction. The second rule tells us that the
light ray which is initially parallel to the optical axis gets its direction changed
by the lens so that it ends up going through the lens’s focal point. The third rule is about the light
ray which passes through the focal point on the near side of the lens. In this case, when we draw that ray
in, we see that it doesn’t actually pass through the convex lens. So this light ray isn’t going to be
any help to us.

But it turns out that that’s
actually okay and that we only need these two light rays in order to figure out
what’s going on. The two light rays come from the
same point on the top of the object. When these two light rays meet each
other again on the far side of the lens, they form an image point which corresponds
to this point on the object. So let’s mark this position on our
diagram. We can do exactly the same thing
for any other point on the object.

Let’s now consider the bottom of
the object. The first rule tells us that the
light ray which goes through the center of the lens keeps traveling in the same
direction. The second rule is about the ray
which is initially parallel to the optical axis. And the third rule is about the ray
which goes through the focal point on the near side of the lens. In this case, these are both the
same as the ray that we have already drawn.

Let’s look at one more point on the
object halfway between the bottom and the top. Since the diagram is getting a
little busy, we’ll draw the light rays from this point in a different color. Using the first rule, we know that
the light ray through the center of the lens keeps going in the same direction. Using the second rule, we know that
the ray which starts out parallel to the optical axis gets its direction changed so
it ends up going through the focal point. The position at which these two
rays cross each other on the far side of the lens is the image point corresponding
to the point on the object where these two rays came from.

We could go on and do this for
every single possible point on the object. And we would find that they all
have a corresponding image point. Of course, in practice, we don’t
actually need to do that. It turns out that if we trace the
rays from the top of the object, we can find the position of the top of the
image. And this is enough to draw in what
the image looks like. In this way, the convex lens has
produced an actual image of the object. This image formed here can be
viewed with the human eye or projected onto a screen. We call it a real image.

We can see that this image is
inverted. That is, it’s upside down compared
to the object. We can also see that the image and
the object are not the same size. The image is bigger than the
object. If we’re looking along the
direction of the optical axis so that we can see this drawing on the object, then in
the image everything in this drawing will be bigger and the whole thing will be
upside down.

Let’s illustrate that with a much
simpler drawing consisting of an arrow pointing directly upward. The image of the paper is bigger
than the object. And so the drawing of the arrow in
this image is also bigger. Since the image is inverted
relative to the object, the arrow in the image now points down instead of up.

Let’s now quickly consider what
happens when we move the object further away from the convex lens. We know that the light ray through
the center of the lens continues on in the same direction. And we know that the ray parallel
to the optical axis gets its direction changed so that it goes through the focal
point. These two light rays came from a
point on the top of the object. So the position on the far side of
the lens where they cross corresponds to the same point on the image.

We can use this information to draw
in the image. We can see that in this case the
image is still inverted compared to the object. But now the image is smaller than
it was before. And in fact it’s actually smaller
than the object. So the size of the image depends on
how far away the object is from the convex lens. By moving the object further away
from the convex lens, we make the image smaller. And it turns out that the image can
be either bigger or smaller than the object depending on this distance.

Let’s now finish up by summarizing
what we have learnt in this video. First, we saw that a convex lens is
a lens that has a shape like this when we look at it from the side. Then we learnt that there are three
rules for what happens to light rays when they pass through a convex lens.

The first rule is that any ray
which passes through the center of the lens continues traveling in the same
direction. The second rule tells us that any
light ray which is initially parallel to the optical axis, so that’s this blue
dashed line here, has its direction changed by the lens so that it goes through a
position on the optical axis known as the lens’s focal point. In our diagram, that’s this green
dot here. The third rule says that any ray
that goes through the focal point before passing through the lens has its direction
changed so that it comes out from the lens parallel to the optical axis.

Finally, we learnt that a convex
lens can be used to form an image of an object. We saw how we could use our rules
for light rays passing through a convex lens to work out where the image would be
formed and what it will look like. We discovered that the size of the
image formed depends on the distance between the object and the convex lens.