### Video Transcript

In this video, we will learn how to
identify the optical properties of convex lenses. Before getting into convex lenses
though, let’s think about what is a lens. Every lens shares two things in
common. First, lenses are transparent. That means we can see through them,
that light passes through lenses. So, for example, a lens might be a
bit of glass like this, something we can see through. The second property of lenses is
that they bend or refract light. Here, we have a ray of light that
was traveling like this until it reached the lens. Then, ever so slightly, the lens
bent the light, sending it off in a different direction. This bending or refraction actually
happens twice: when the ray enters the lens and when it leaves.

Lenses can be any shape, but often
they’re in the shapes we see in eyeglasses. Lenses like these are often made of
glass or plastic. When we look at them straight on,
lenses and glasses can look like circles. What we’re going to do though as we
study convex lenses is look at lenses from the side. Looked at that way, this is how a
convex lens appears. Notice that the lens is thicker in
the middle and thinner as we get to the top and bottom. Actually, the shape of a convex
lens is a very precise thing. The way it works, this part, this
side of a lens, is one section of a big circle. So that’s what makes this part of a
lens curve like it does. And then the same thing is true for
the other side of a convex lens.

So we have these two big circles,
and our convex lens shows where the circles overlap. One other thing about these circles
is that each of them has a center. The center of the one is here and
the center of the other right here. The name of this point is the
center of curvature; that makes sense. Each of these points is the center
of the curving line that makes up the circle. So this point here is the center of
curvature of this circle, and this point here is the center of curvature for this
one.

For a convex lens, a center of
curvature is always the same distance away from any point on the far surface of the
lens. That means, for example, that this
distance here is the same as this distance here or, say, this distance. The name for this distance is the
radius of curvature. It’s just the radius of the
imaginary circle that our convex lens is a part of. So if someone said, “what’s the
radius of this circle?” the answer would be the radius of curvature.

Now, let’s look again at the two
centers of curvature. If we draw a line that passes
between these two points, we see that this line also passes through the exact center
of our convex lens. This line is called the optical or
principal axis. It’s useful because it helps us
understand how rays of light will interact with our lens. Remember that one thing all lenses
do is bend or refract light. That means if we have a ray of
light that comes and lands on the lens, instead of just going straight through, the
ray will be refracted — bent — from its original path.

A convex lens bends light in a
particular way. Say we have another ray of light
that approaches the lens and that, like the first ray, this second ray is also
parallel to the optical axis. Once again, the convex lens bends
this ray of light, but notice that it doesn’t bend it quite so much as it did this
ray. And also, these two refracted rays
cross at this point. Keeping that point in mind, if we
let even more parallel rays reach this lens, when they pass through and were
refracted, we would see that all of these rays cross at the same point. Since this is where all the rays
come to a focus, it’s called the focal point.

And notice what our convex lens had
to do so that this focal point exists. The lens had to bend all of these
incoming rays of light so that they got closer and closer together. When rays of light do that, they’re
said to be converging. Another way to describe a convex
lens is as a converging lens. By the way, after the rays converge
and cross at the focal point, then they spread out from one another. This is called diverging. But anyway, a convex lens converges
incoming parallel rays of light so that they cross at a point. That’s the focal point. And if we were to look at the
distance between the center of the lens and this focal point, we would be looking at
what’s called the lens focal length. This length is used to describe a
convex lens.

For example, if we had a set of
convex lenses, they might be organized by focal length. Now, in this case, we’ve drawn rays
of light as approaching our lens from the left. But we could just as well have
picked them to come in from the other direction. That is, the parallel incoming rays
could’ve been traveling like this. The convex lens would still refract
or bend these rays. And like before, the place where
they cross is called the focal point. So there’s actually a focal point
on either side of our convex lens. Sometimes one of these points is
called a focus. And if we’re talking about more
than one focus, the word for that is foci. So this lens has two foci or two
focal points. Notice that both of these points,
just like the centers of curvature, lie along the optical or principal axis. Knowing all this about convex
lenses, let’s look now at a few examples.

Which of the following is a convex
lens?

We see here these four lenses. And let’s remember that, in
general, a lens is some transparent object, that is, light passes through it, that
also bends light as the light moves through the lens. All four of our possible answer
options meet those conditions. So we need to figure out which one
of these lenses is convex. The shape of a convex lens is made
from the overlap of two circles. So say we had a circle like this
and then another circle of the same size that overlapped with the first one a little
bit. This shape right here is the shape
of a convex lens. We see the lens is wider in the
middle, and it’s narrower at the top and bottom. Looking over our answer options,
the only one that matches this shape is option (C). This is the only lens that could be
formed by the overlap of two similar circles. And we notice it’s also thicker in
the middle and thinner towards the ends. Answer option (C) is a convex
lens.

Let’s look now at another
example.

The diagram shows a convex
lens. Light rays pass through the lens in
the vertical direction. Which line shows the optical axis
of the lens?

So here we see a convex lens. And all these lines, line one, line
two, line three, line four, and line five, represent rays of light. We’re told that these rays pass
through the lens in the vertical direction. That just means that these lines
don’t go into the screen or out of the screen at all. They only move in the plane of our
screen. We want to figure out which of
these five lines shows the optical axis of the lens. Another name for the optical axis
is the principal axis. We can remember that the optical
axis of a lens passes right through the exact center of that lens. But here that doesn’t narrow down
our options. We see that all five lines pass
right through the very center of this convex lens. The other thing that’s true though
about the optical or principal axis is that it passes through what are called the
centers of curvature of a convex lens.

To see what this means, let’s
consider the two surfaces of our lens. This is one surface here, and then
this is the second surface. Each one of these surfaces is part
of a larger circle. This, for example, is a larger part
of the pink circle, while this is a larger part of the orange circle. Each one of these two circles will
have a center. That center is called the center of
curvature. The optical axis joins those two
points together. Now, we don’t see here exactly
where these two centers of curvature will be. The circles we would need to draw
are just a bit too big. However, we can see that those two
points will be somewhere along this blue line. That’s because this line passes
through the middle of each circle.

If we made a rough guess for where
the two centers of curvature of these circles would be, the pink circle center might
be located here. The orange circle center might be
here. Like we mentioned, the line that
joins these two points is the optical or principal axis. We can see that this line is
identified as line two, and so that’s our answer. The optical or principal axis of a
lens passes through the center of a lens and also the centers of curvature of the
two circles that help make up the surface of the lens. In this diagram, the line for that
axis is line two.

Let’s look now at another
example.

The diagram shows a convex lens as
the intersection of two circles. Which points on the diagram mark
the centers of curvature of the lens? Select all that apply.

In this diagram, we see the convex
lens shaded in blue. Indeed, it is the overlap or
intersection of two circles. We want to identify the point or
points on the diagram corresponding to the centers of curvature of the lens. The important thing to remember
about a center of curvature is that it’s the center of a circle that helps make up
the convex lens shape. That is, one center of curvature
will be at the center of this circle. And another center of curvature of
the lens will be at the center of this one. We see that point one is at the
center of the first circle and point five at the center of the second. These are indeed the centers of
curvature of the lens.

Notice then that the center of
curvature of a lens is not at the center of the lens. In other words, point three is not
a center of curvature. Also, the top and bottom of the
lens, we could call them, are not centers of curvature either. Instead, we have to look back at
the two circles that helped make up the surface of the convex lens. It’s the centers of those circles,
in this case corresponding to points one and five, that are the centers of curvature
of the lens.

Let’s look now at one last
example.

The diagram shows a convex
lens. Which line shows the radius of
curvature of the lens?

Here, in our diagram, we have line
one, line two, and line three. We want to figure out which line
shows us the radius of curvature of our convex lens. Let’s notice, first of all, that
this surface of our lens is part of a larger circle, the one that’s dashed in an
orange line. This circle has a center. It’s right at this point. The name given to this point is
center of curvature. We call it this because it’s the
center of the entire curving line that makes up the circle.

Like any circle, this one has a
radius. The radius of curvature is the
distance between the center of curvature and the circle. Looking at our three lines, we see
that line three covers the distance from the center of curvature to the inside edge
of our lens. Line one, on the other hand, goes
from the center of curvature to the very center of the lens. However, it’s only line two that
goes from the center of curvature, the center of this orange circle, to a point on
the circle’s circumference along the orange dashed line. We know then that it’s line two
that shows the radius of curvature of the lens. This is our final answer.

Let’s now finish this lesson by
reviewing a few key points. In this video, we learn that a
convex lens is a piece of transparent material with this shape. The shape of a convex lens comes
from the fact that it’s the overlap between two circles. The center of each circle is a
center of curvature of the lens. The line passing through these two
points is called the optical or principal axis.

We also learned that a convex lens
converges or brings together incoming parallel rays of light. When the refracted rays cross, they
do so at a point called the focal point. The straight-line distance between
the focal point and the center of the lens is called the focal length. And lastly, we learned that the
distance between the center of curvature of a lens and that lens’s surface is called
the radius of curvature. This is a summary of convex
lenses.