### Video Transcript

In this video, we will learn how to
identify the optical properties of concave lenses. Before talking about this type of
lens, let’s understand what a lens is in the first place.

A lens can have any shape. The two important things about
lenses are, first, light can pass through them. So here we have a ray of light. And instead of being blocked or
reflected from the lens, we see it’s able to travel through it. That’s one property of lenses. The second thing all lenses have in
common is that they bend or refract light. We can see that, here, the ray of
light bends just a little bit when it crosses into the lens. And then, it also bends or refracts
when it leaves. So then, all lenses, first, are
transparent. That means we can see through
them. Lenses might be made of a material
like glass or clear plastic. And then second, all lenses bend or
refract rays of light.

Let’s now clear some space and
think about a particular kind of lens, a concave lens. Starting with a lens with this
shape, we can make a concave lens from it. What we do is make two big circles
of the same size. Then, we imagine cutting out
material from this block to follow the shapes of these circles. So, now our lens looks like
this. We now have a concave lens. Notice that this lens is thinner at
the middle, and it’s thicker at the top and bottom. Whenever we see that shape and the
sides of the lens follow a circular arc, then we know we have a concave lens.

Each of these two circles that
we’ve used to define our lens has a center point. These are the points on the circles
that are the same distance from any location on the circumference of the circle. That means then that each of these
center points is the same distance from any point on the surface of the lens closest
to that center. For this reason, each of these
points is called a center of curvature. There’s the lens’s center of
curvature on the left and the lens’s center of curvature on the right. The distance between each center of
curvature and the nearest surface of the lens is called the radius of curvature.

Since these two circles are the
same size, that is, they have the same radius, the radius of curvature of one is the
same as the radius of curvature of the other. Say we clear away everything but
the lens and its two centers of curvature. If we draw a line through these two
points, we have created what’s called the optical or principal axis. This is the line that travels
through the lens’s two centers of curvature, as well as passing through the exact
center of the lens. The optical axis is helpful for
showing us how rays of light will be refracted by this concave lens.

For example, say that we have an
incoming ray of light that’s moving parallel to the optical axis. When this ray reaches our lens,
rather than continuing on in a straight line, the lens will refract or bend the ray
of light. As we see, this happens twice, when
the ray enters and leaves the lens. So a concave lens does change the
path of a ray of light. And if we had another incoming ray
also parallel to the optical axis, we would see that once again the ray passes
through the lens, but it’s refracted. Adding in a few more of these
parallel rays, we see something interesting. The effect our concave lens has on
the light is to spread it out. These rays get farther and farther
away from one another after they pass through the lens.

When rays of light do this, we say
that they are diverging. This means the distance between
rays that are next to one another is increasing as the rays move forward in
space. But that then raises a
question. If our concave lens makes rays
diverge, how can they ever cross one another to form an image? If the rays of light never meet, if
they never intersect, then no image will be formed. To understand how this works, say
that we were to put our eye right over here on this side of our lens. That means all these diverging rays
of light would land on our eye. And what our eye would do is trace
these refracted rays backward, straight backward.

So, for example, for this top ray
of light, once it lands on our eye, our eye would perceive this ray of light to be
traveling on a path like this that we’ve shown with the dashed line. And then, likewise, for this ray of
light, our eye would trace it backward to perceive that it travels along this dashed
line. The same thing goes for the other
refracted rays. Our eye perceives them all to be
moving in a straight line. And so it sees a point of
intersection of those lines here. Since this is where all the rays
seem to come to a focus, it’s called the focal point. This is where our eye would think
all of these rays of light were originating, where they were coming from. This is where it sees an image.

The distance between the focal
point and the very center of the lens here, say we measure that straight-line
distance this way, is called the focal length. Now, here’s something very
important. Just as both sides of our concave
lens have a center of curvature, so both sides have a focal point and both sides
have a focal length. We would see this focal point and
this focal length get involved if we let rays of light approach the lens from this
side. And they can do that. Rays of light can reach a lens from
any side. Knowing all this about concave
lenses, let’s look now at a few examples.

Which of the following is a concave
lens?

We know that, in general, a lens is
an object that lets light through. That is, it’s transparent. And when light passes through a
lens, it is refracted or bent. All of our answer options (A), (B),
(C), and (D) show objects that do those two things. They let light through and they
refract it. All of these objects then are
lenses, and we need to figure out which one is a concave lens.

A concave lens is one that is
thinner in the middle portion and thicker at the top and bottom. The only one of our lenses shaped
that way is lens (D). We see it’s thinner at the middle
portion, wider or thicker at the top and bottom. And then, and this is another sign
of a concave lens, the edges of the lens look like they’re made up of part of a
circle. The lens in answer choice (D) does
show that shape. Our answer then is that option (D)
is a concave lens.

Let’s look now at another
example.

The diagram shows a concave
lens. Which line shows the optical axis
of the lens?

Okay, so we see these five lines —
line one, line two, line three, line four, and line five — passing through this
concave lens. We want to know which line shows
the lens’s optical axis. This axis passes directly through
the center of the lens. However, we can see that all five
lines do that. All of them pass right through that
center point. So let’s remember something else
about the optical axis of a concave lens. A concave lens, we could say, is
made up of two surfaces. Here, the dashed pink line shows
one surface, and the dashed orange line shows the other. Each of these surfaces is in the
shape of a part of a circle.

So, for example, the top surface is
part of a larger circle whose shape looks partly like this. The bottom surface of the lens is
part of a larger circle that looks a bit like this. Each of these two circles, and
we’re only seeing a part of each one, has a center point, some point which is the
same distance from all points on the circumference of the circle. We don’t know exactly where those
center points are. But if we had to make a rough
guess, for the pink circle, we might put its center point here. And then, for the orange circle,
its center might be located here. Wherever those center points are,
they’re the same distance from either side of the circle.

The optical axis of a lens is a
line that joins together the two centers of curvature of that lens. For our concave lens, this blue
line is about where its optical axis would be. And notice that this blue line
overlaps with one of our five lines. That then tells us the answer to
this question. It’s line one that shows us the
optical axis of the lens.

Let’s look now at another
exercise.

The diagram shows a concave
lens. Which points on the diagram mark
the centers of curvature of the lens? Select all that apply. (A) Point one, (B) point two, (C)
point three, (D) point four, (E) point five.

On our diagram, we see all of these
five points labeled. We want to figure out which point
or points mark the centers of curvature of this concave lens. Notice something about the
lens. Its side surfaces are part of a
larger circular shape. In other words, each side of our
concave lens has a circular arc to it. We see the two larger circles
sketched in in these orange dashed lines. Like any circle, these circles have
a center point. It’s a point that’s the same
distance from every point on the circumference of the circle.

When two circles define the shape
of a concave lens, like these two circles do here, those center points are called
the centers of curvature of the lens. We can see that the center point of
the circle on the left is right here. That’s labeled point two. Then, the center of the circle on
the right is located here. That’s at point five. These two points then, points two
and five, are the centers of curvature of the concave lens. For our answer, we choose option
(B), which is point two, and option (E), point five.

Let’s now look at one last
example.

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

We see here three lines indicated:
line one, line two, and line three. Before thinking about these lines,
though, notice that there are two dots indicated on this diagram. One dot is here. That’s the point at the center of
this dashed orange circle. The other dot is here, which is at
the center of this concave lens. Notice that if we drew a vertical
line through the center of the circle, then the beginnings of lines one, two, and
three are all along this line. In other words, all three line up
with the center of this orange circle.

But then, let’s look at where the
lines end. Line one ends here, which is on the
outer surface of this conclave lens. Line two though ends right here,
which is in line with this second dot. Line two goes from the center of
the circle to the center of the lens. Then, lastly, there’s line three,
which goes from the center of the circle to the inner surface of the lens. We want to pick which of these
three shows the lens’s radius of curvature.

There’s actually a name for this
point at the center of the orange circle. It’s called the center of curvature
of the lens. The distance from the center of
curvature to any point on the inner surface of the lens is the same. And then, notice what that distance
is. It’s from the center of the circle
to the circumference. It’s equal to the circle’s
radius. Because our circle helps to define
the shape of a lens, this distance is called the radius of curvature. It tells us how much or how little
the surface of the lens is curved.

Notice that what we’ve called the
radius of curvature does not lie on top of any of our three lines. That’s actually okay because the
radius of curvature is just a distance. Whichever of lines one, two, or
three indicates the same distance is the radius of curvature of the lens. Remember, we saw that line one goes
from a line through the center of curvature to the outer surface of the lens. Line one then is longer than the
radius of curvature. Line two goes from the center of
curvature to the center of the lens. This also is longer, just slightly
than the radius of curvature. Line three, though, begins at the
center of curvature and ends right here on the inner curved portion of the lens.

So then, the length of line three
is the same as the length of this line we’ve drawn here that we’ve called the radius
of curvature. And this gives us our answer. Line three is the line that shows
the radius of curvature of this concave lens.

Let’s now finish this lesson by
reviewing a few key points. In this video, we saw that a lens
is an object that is transparent — that is, light passes through it — and that
refracts or bends rays of light. A concave lens, a certain type of
lens, is shaped as follows. The sides of a concave lens are
shaped as part of a larger circle. Each of those circles has a center
point, and together they’re called the centers of curvature of the lens. A line passing through both centers
of curvature is called the optical axis or principal axis of the lens. When parallel rays of light reach
and then pass through the lens, they’re spread apart from one another. The rays are said to be
diverging.

However, if we trace the refracted
rays backward, we see that those traces do cross. They cross at a location known as
the focal point of the lens. Just as each side of the lens has a
center of curvature, so each side has a focal point. A single focal point is sometimes
called a focus. When we have more than one focus,
we call them foci. And finally, if we measure the
distance between one of the lens’s focal points and the center of the lens, that
straight line distance is equal to what is called the lens’s focal length. This is a summary of concave
lenses.