Video Transcript
In this video, we will learn how to
describe the reflection of light rays from a concave mirror. Before we think about those
reflected light rays though, let’s consider what a concave mirror looks like. One way to picture a concave mirror
is to imagine that we’re standing in front of a shape like this. If we were to take a laser pointer
and shine it onto this reflective surface, this mirror, then the beam of light would
be reflected off of the mirror, something like this. If we were to look at all this from
a top-down perspective, then our concave mirror would look this way.
For a concave mirror, light always
approaches the mirror from this side, we could say that the light goes inside the
curve of the mirror. A concave mirror has a special
shape to it; it’s part of a circle. The center of that circle, the
point right here, is called the center of curvature of the mirror. The distance between the center of
curvature and any point on the surface of the mirror is the same. That distance is called the radius
of curvature. Notice this distance is just the
radius of the circle overall. We give it the special name, radius
of curvature, because it describes the radius of our curving mirror.
Now let’s say we clear away
everything except our concave mirror and our center of curvature. We know that mirrors are designed
to reflect light rays. So if a ray of light comes in and
lands on the mirror rather than passing through or being absorbed by the mirror,
this ray of light is reflected. The direction of the reflected ray
depends on which part of the mirror the original ray lands on. Here’s something interesting
though. Say that we have another ray of
light that’s parallel to the first ray that landed on our mirror. This second ray of light will also
be reflected. And when it is, it will cross over
the first reflected ray.
Things get even more interesting if
we add yet another ray that’s parallel to the first two. And we find that this reflected ray
passes through the same point where the first two crossed over. If we keep adding parallel incoming
rays and watching where they reflect, we find they all pass through this same point
right here. We can say that it’s at this point
that all of these reflected rays come to a focus. For that reason, we call this the
focal point of our mirror. Whenever parallel incoming rays of
light land on a concave mirror, the mirror reflects all those rays to and then
through its focal point.
Clearing away our rays of light, we
see we now have two points of interest for our mirror. And let’s remember that this
distance here is called the radius of curvature of our mirror. This distance is measured from the
center of curvature to the center of the surface of our mirror. It turns out that this distance
here, from the focal point to the center of the mirror’s surface, has its own
name. It’s called the focal length. There’s a relationship between the
focal length and the radius of curvature of a concave mirror. Writing it as a mathematical
equation, it looks like this. The focal length of the mirror is
equal to one-half times the radius of curvature.
Another way to say this is that the
radius of curvature is twice as large as the focal length. This relationship is always true
for a concave mirror. That’s very handy because it means
if we know either the focal length or the radius of curvature, we can solve for the
other distance. Knowing all this about concave
mirrors, let’s look now at a few examples.
Which of the mirrors in the diagram
below is a concave mirror? Incident light is shown by the
yellow lines.
Here we see two mirrors, Mirror 1
and Mirror 2. The difference between them is the
way that they’re curved with respect to the incoming light. Mirror 1, we see, is curved
outward, bulging towards the incoming light, we could say. Opposite to this is mirror 2, which
we see is curved inward. We want to know which of these two
mirrors is a concave mirror. One way we can remember the shape
of a concave mirror is to think about the word cave in this name. Just like a cave is something that
we travel into, a concave mirror, we could say, is something that light travels into
inside this curving arc. Because the light reaching Mirror 2
is entering into the curve of the mirror, we know that this mirror is concave.
Of these two mirrors then, Mirror 2
is the concave mirror.
Let’s look now at another
example.
For any spherical mirror, the
distance between the center of curvature and the center of the surface is just the
radius of that sphere. Which one of the following
sentences correctly describes the location of the center of curvature of a concave
mirror? (A) The center of curvature of a
concave mirror will always be on the side opposite to the observer. (B) The center of curvature of a
concave mirror will always be on the same side as the observer. (C) Depending on the path of the
light rays, the center of curvature of a concave mirror can sometimes be on the side
opposite to the observer and can sometimes be on the same side as the observer.
To start figuring out which of
these three answer options is correct, let’s sketch a curved mirror. And we’ll say that this is a
concave mirror. A concave mirror is one where light
always enters into the curved portion of the mirror, here on this side. This is the opposite of light being
incident on the mirror from this side. All of our possible answers talk
about which side of a concave mirror an observer will be on. In order to observe or see the way
a concave mirror reflects light, an observer always needs to be on the same side of
the mirror as the incoming rays of light. An observer of a concave mirror
then is always facing this curve of the mirror shape.
We can recall that this shape is
part of a larger circle. The center of that circle, located
here, is called the center of curvature of the mirror. For a concave mirror then, the
center of curvature of the mirror and an observer are always on the same side of the
mirror. What we’re finding is that answer
option (B) is the correct description of a concave mirror. The center of curvature of a
concave mirror will always be on the same side as the observer.
Let’s look now at another
example.
Below is a ray diagram for a
concave mirror. Which one of the five locations
along the optical axis represents the center of curvature of the spherical
mirror?
Here we see a concave spherical
mirror. There are two parallel rays of
light that are incident on the mirror. And along with this, we see this
line parallel to those rays and passing through the center of the surface of the
mirror. This line is called the optical
axis. For our purposes, the important
thing is these five locations marked out along the optical axis. We want to identify which one
represents the center of curvature of the spherical mirror. Looking at our diagram, we might
first think that it’s point 2. But we must be careful. For a concave mirror, the point
where parallel incoming rays of light are reflected to a focus is actually called
the focal point. This is different from the center
of curvature, so we won’t choose point 2 for our answer.
Knowing that point 2 is the focal
point though helps us find our answer. That’s because of this statement
here. The distance from the center of a
concave mirror’s surface to the focal point is one-half the distance from the center
of the mirror’s surface to the center of curvature. That’s a long statement. But here’s the idea. If we start here at point 3, the
center of our mirror’s surface, and we go from point 3 to point 2, our focal point,
then that total distance we’ve traveled, which by the way is called the focal length
of our mirror, is one-half the distance from the center of our mirror’s surface to
the center of curvature.
In other words, if we double the
length of this line here, we’ll go from point 3, the center of our mirror’s surface,
to the center of curvature of the mirror. Doubling the length of our line
gets us out to point 1. The curve of our concave mirror is
actually part of a larger circle. The name for the center of this
circle is the center of curvature. On our diagram, that point is
labeled as point 1. This then is our answer. It’s at point 1 along the optical
axis where the center of curvature of the spherical mirror is located.
Let’s look now at one last
example.
Which of the following sentences is
a correct description of what will happen to parallel rays incident on a concave
mirror? (A) They will continue
undisturbed. (B) They will be focused at the
center of curvature. (C) They will be focused at the
focal point. (D) They will not be focused at a
point at all.
To see which of these answer
options is correct, let’s sketch in a concave mirror. And because this mirror is concave,
we know that incoming rays of light will approach it from this side. Now our question specifically talks
about parallel rays of incoming light. So let’s say we have two such rays
like this. Answer option (A) says that when
these rays reach the mirror, they will continue on undisturbed. If that happened, the path of the
rays would look like these dashed lines. We know though that one property of
a mirror is to reflect light. The incoming rays then won’t follow
the original paths they were on. We can eliminate answer option (A)
from consideration.
Options (B) and (C) talk about
these incoming rays being focused. That means when the rays are
reflected, they cross at a point. Answer option (B) says that this
focusing happens at a point called the center of curvature. Let’s now recall that our mirror is
actually part of a larger spherical surface. The center of that sphere, which is
right here, is at the point called the center of curvature. For our parallel incoming rays of
light, we see that this is not the point where the rays are brought to a focus. We can cross out answer option
(B).
Option (C) says that the rays are
focused at what’s called the focal point. This indeed is a correct
description of what happens to these incoming parallel rays. The point where the reflected rays
are brought to a focus is called the focal point. We see then that option (C) is
correct. And in the process, we can tell
that option (D) is not correct. We’ve seen that these parallel
incoming rays really are brought to a focus after being reflected. For our answer, we choose option
(C). When parallel rays of light are
incident on a concave mirror, they will be focused at the focal point.
Let’s finish our lesson now by
reminding ourselves of a few key points. In this video, we studied concave
spherical mirrors. We saw that the surface of these
mirrors are part of a larger circle. The center of this circle is called
the center of curvature of the mirror. The distance between the center of
curvature and the center of the surface of the mirror is called the radius of
curvature. If parallel rays of light are
incident on the concave mirror, they reflect off the mirror through a common
point. The name for that point is the
focal point. And we learned that the distance
between the center of the surface of the mirror and the focal point is called the
focal length. Lastly, we saw that the focal
length is equal to one-half the radius of curvature of the mirror. This is a summary of concave
mirrors.
