### Video Transcript

How many foci does a concave lens
have?

In this question, we are asked to
look at a concave lens and figure out how many foci it has. First, let’s remember that a
concave lens is a lens that is thicker along the edges and thinner in the
middle. We can think of this shape as a
cylinder on its side with its circular edges along the horizontal axis like this,
which is being overlapped by two spheres like shown here. If we were to remove the overlapped
material from the cylinder, it would then have the same shape as a concave lens.

Notice that concave lenses are
symmetrical, both horizontally and vertically. If we were to split this lens along
the horizontal or vertical axis, we would have two identical halves. This symmetry will be important for
our solution, as we will soon see. For now though, let’s consider
these circles outside the lens.

These circles represent the
curvature of our lens. Notice how their edges line up
perfectly with the curve of our lens. Let’s mark the centers of these
circles with points. These points are called the centers
of curvature, and both are an equal distance from the lens. If we were to connect these two
points with a line, it would pass directly through the center of our lens. This line is called the optical
axis. If a ray of light is traveling
along the optical axis and goes through the lens, its direction will not change. In fact, any ray of light that
travels through the center point of the lens, which is also the halfway point
between our centers of curvature along the optical axis, will pass through without
having its direction change.

But now, let’s consider light rays
that are traveling parallel to the optical axis. A concave lens is a diverging lens,
which means that as light rays pass through it, they are directed away from each
other.

Recall that foci are points at
which light rays converge or meet. How can we have a focus, a
convergence, with diverging rays? Let’s look closer at these light
rays that are traveling through the lens. If we trace these diverging rays
back along their newest path direction, they both appear to have started from the
same point on the other side of the lens. This is the focal point of a
concave lens. And its distance to the center of
the lens is called the focal length.

Now, remember that concave lenses
are symmetric about the vertical axis. The centers of curvature are the
same distance from the center of the lens on both sides. This means if we have a focal point
on one side of our lens, we can expect to have one on the other side as well due to
symmetry.

We can prove this by tracing back
the final direction of rays when they come from the other side of the lens. They also converge on a single
point, which is the second focal point of this concave lens. This means that the answer to “How
many foci a concave lens has?” is two, one on either side of the lens. This gives one focus for each of
the two centers of curvature.