Video Transcript
In this video, we will learn how to
draw diagrams of light rays interacting with convex mirrors. We’ll start by seeing how a convex
mirror works. Here is a convex mirror, a
three-dimensional object. If we shine a ray of light onto the
mirror, it reflects off of the surface that looks a bit like a bowl. Because this mirror is convex, we
will only consider the light approaching the mirror from the right, as that’s the
light that hits the convex part of the mirror. Any light approaching from the left
will be interacting with a concave face, so we won’t consider that here.
In three dimensions, a convex
mirror looks like a bowl, and if we were to look at the mirror from the side, then a
section of the convex mirror would appear this way. This mirror is part of a larger
circle. The center of that circle is
here. This is called the mirror’s center
of curvature. The distance between the center of
curvature and any point on the surface of the mirror is the same. That is, these pink lines all have
the same length. If we draw a horizontal line from
the center of curvature to the mirror, then we reach the center point of the
mirror. These two points define an
imaginary line called the optical axis. This axis serves as a reference for
rays of light that are incident on the mirror.
For example, this ray of light is
parallel to the optical axis. When it reaches the mirror, it
reflects off. One way to figure out the direction
of the reflected ray is to pretend that at the point where the original ray ran into
the mirror, the mirror at that location was a flat surface like this. So if we had a second ray like this
one, also parallel to the optical axis, we could use a similar approach of
approximating the mirror to be flat at the point where the ray meets it. This way, we’re able to draw the
direction of the reflected ray more accurately.
Now, notice something about these
two reflected rays. They don’t intersect. And what’s more, as they travel
farther away from the mirror, they move farther apart. In other words, these reflected
rays will never intersect. But, and here’s the secret to how
convex mirrors work, if we trace the reflected rays backwards as though they had
come from a point inside the mirror, we see that these lines do cross. This is called the focal point of
the mirror. Any time a ray of light comes in
parallel to the optical axis, we can trace the reflection of that ray backward. And the trace will cross through
the focal point. None of these dashed lines are
actual rays of light.
To be clear, no light makes it on
the left side of the mirror as we’ve drawn it. But if we imagine a gigantic
eyeball over here that was able to see all of these reflected rays, that eye would
naturally trace these rays backward to the point where they would cross. Even though these rays don’t
actually come from the focal point, the eye would perceive that they do. So that’s the focal point for a
convex mirror. No light actually reaches this
point. But when parallel rays of light are
incident on the mirror and then reflected off, these reflected rays seem to be
coming from the focal point.
As we’ve seen, another important
point is the center of curvature. Say that a ray of light is incident
on our convex mirror so that if it could continue going in a straight line, it would
pass through the center of curvature. When a ray travels like this, it
reflects off of the mirror along the same path that it came. Indeed, this will happen for any
ray of light lined up so that it would pass through the center of curvature. When we’re able to draw in rays of
light that would pass through the center of curvature or would come from the focal
point, we become better able to draw ray diagrams of convex mirrors.
The purpose of a ray diagram is to
show how the image of some object is formed. An object can be anything in front
of a mirror. It might be a book or a stick or an
animal or literally anything. Here we have some object that
extends above and below the optical axis. Rays of light will travel from the
object, and they’ll be incident on our mirror surface. Then, depending on how the mirror
reflects those incident rays, an image of the object will be formed. This idea of object and image is
actually very familiar. For example, when we stand in front
of a mirror brushing our teeth, we ourselves are the object, and we look at our
image to see how to brush. In the case of this object, we’re
letting it be anything and color-coding its top and its bottom so we can keep those
straight.
As we mentioned, rays of light come
from the object toward the mirror, and they actually come from every point on the
object. To avoid cluttering our diagram
now, we’ll start by considering only the rays of light that move from the tip of our
object. And in fact, we’ll consider just
two of the many rays that come off of our object’s tip. One of these rays of light travels
parallel to the optical axis. The second one we’ll consider moves
straight towards the center of curvature of our mirror. We know that this second ray will
be reflected straight backward along the path it came. The first light ray, though, will
be reflected so that if we trace the reflected ray backward, we cross through the
focal point.
Now the only way an image of our
object can form is if two of the reflected rays or of the dashed lines we traced
backwards intersect. We see that the reflected rays off
of the surface of the mirror will not cross. However, when we trace them
backward, we find these lines do intersect. The point where they do is where
the image of this part of our object forms. To see what the entire image of our
object looks like, let’s trace rays of light coming from the lowest point on our
object. Once again, we’ll use rays that are
parallel to the optical axis and a ray that’s traveling straight toward the center
of curvature of the mirror. This ray, we know, will reflect
straight backward, while the incoming ray parallel to the optical axis will be
reflected so that we trace it back and see that it passes through the focal
point.
Once again, we look for a point of
intersection between these reflected rays. The actual reflected rays get
farther and farther apart from one another. But tracing these rays backward, we
see that those lines do intersect. That green dot is the bottommost
part of our image. Overall then, the image of our
object looks like this.
Let’s notice a few things about
this image. First, the blue arrow in our image
points up just like the blue arrow in our object. Likewise, the green arrow in our
image points down just like our object’s green arrow does. This means that our image is right
side up, also called upright. The alternative to an upright image
is an inverted one. In the case of our object, this
would be an image where the green arrow pointed up and the blue arrow pointed
down. Here, though, our image is
upright. A second thing to notice is that
our image height is smaller than our object height. We say that such an image is
reduced in size. That simply means that it’s smaller
than the object that made it.
Lastly, remember, we said that no
rays of light are able to pass on to this side of the mirror. The mirror reflects all the
incoming rays off and keeps them on this side. This means that all of these dashed
lines we’ve drawn here are not real rays of light. When it’s these dashed lines that
show us where the image forms, that means the image is called a virtual image. This is in contrast to a real image
because these aren’t actual rays of light intersecting to form this image. There’s no way we could take a
screen, say, a piece of white paper, and project this image onto that screen. That’s what it means for the image
to be virtual. It means we can’t create it by
shining rays of light on a screen or flat surface. So the image produced by this
object is upright, reduced in size, and virtual.
And here’s something
interesting. Every image produced by a convex
mirror can be described this way. In other words, convex mirrors only
produce upright, virtual, reduced-in-size images. To see this more clearly, let’s try
taking this object and moving it closer to the mirror. What we’ve done here is we left our
original image drawn in. That was the image due to this
object. But we’ve moved our actual object
to a position here.
As always, to figure out how the
image of this object forms, we’ll draw in two rays of light coming from both its top
and its bottom. One of the top rays is parallel to
the optical axis, and the other is headed straight towards the center of
curvature. These rays reflect, and if we trace
back the reflected rays, we see those dashed lines cross right here. We’re now going to do something
similar with the bottom of our object: two rays, one parallel to the optical axis
and one headed straight towards the center of curvature. The rays reflect, and if we trace
the reflected rays backward, they intersect right here. So, here is our updated image. This is the image of the object
when it’s at this location closer to the mirror.
Note that even though this image is
bigger than the one before, it’s still smaller than the object overall. Therefore, it’s still reduced in
size. And just like before, it is an
upright image and it’s virtual. It appears on the opposite side of
the mirror as the object. It really is the case that all
images formed by convex mirrors meet these three descriptions. The opposite of an upright image is
an inverted one. The opposite of an image reduced in
size is a magnified image. And the opposite of a virtual image
is a real image. With this kind of mirror though, we
never encounter an inverted, magnified, or real image. Knowing all this about convex
mirrors, let’s look now at an example.
Can the image produced by a convex
mirror be larger than the imaged object?
Seen from the side, a convex mirror
can look like this, where the rays of light approach the mirror from the right. All points on the surface of the
mirror are the same distance away from a point called the center of curvature. Between the mirror and the center
of curvature is another point called the focal point. The horizontal line joining these
two points is called the optical axis. Now we bring all this up because
these two points and the optical axis can help us answer this question. When this convex mirror produces an
image, we want to know if it’s ever possible for that image to be larger than the
object it’s an image of.
Say that this is our object, and
this could literally be any physical object. We can find out what the image of
this object looks like by tracing rays of light from the tip of the object. First, let’s consider a ray of
light that runs parallel to the optical axis. That ray will be reflected like
this, and we can trace it back and see that that trace passes through the focal
point. The other ray from our object is
one headed straight toward the center of curvature of our mirror. This ray will reflect backward
along the path it came. The top of our image will form
where these two traced lines intersect at this point here. Overall then, our image will look
like this. And we see that indeed it is
smaller than our object.
Our question, though, is asking if
it can ever happen that the image produced be larger than the object. Maybe, we might think, for some
differently sized object or a differently positioned object, the image would indeed
be larger. But let’s consider this. No matter where our object is, so
long as it’s on the right side of the mirror, and no matter how large or small our
object is, we know that the image of that object must appear somewhere along this
line that is traveling from the tip of the object to the center of curvature of the
mirror.
In the case of our object being
located here and having this size, we’ve seen that that point is right here. But then notice something about
this line. The line always slopes
downward. That means that by the time we pass
behind the mirror, whatever the height of our image will be, it must be less than
the height of the object. In fact, the same sort of thinking
applies to an object that looked, say, like this. A ray of light from the tip of this
object that’s traveling toward the center of curvature would look this way.
Once again, the tip of the image of
this object would lie somewhere along this line behind the mirror. Since this line is sloping toward
the optical axis, we can see the image will once again be smaller than the
object. We see then that in general, the
image produced by a convex mirror must be smaller than the object. Such an image is said to be reduced
in size. So for our answer, we’ll say that
no, the image produced by a convex mirror cannot be larger than the imaged
object.
Let’s finish our lesson now by
reviewing a few key points. In this video, we learn that a
convex mirror has a center of curvature, a point that is the same distance from all
points on the surface of the mirror, as well as a focal point. These points lie along what’s
called the optical axis of the mirror. If a ray of light parallel to the
optical axis is incident on the mirror, that ray is reflected and the trace backward
of the reflected ray crosses through the focal point. Similarly, if a ray of light is
incident on the mirror that would have passed through the center of curvature had it
not been reflected, then the reflected ray will travel back along the path of the
incident ray.
For a convex mirror, no real rays
of light reach the focal point or the center of curvature. These are only the seeming origin
points of certain reflected rays. We learned further that to figure
out the image created by an object, we allow two rays to leave the tip of that
object. One ray travels parallel to the
optical axis. The other heads directly towards
the center of curvature of the mirror. The image is formed where these two
dashed lines, imaginary rays of light, cross. Lastly, we learned that images made
by convex mirrors are always upright, not inverted, virtual, not real, and reduced
in size rather than magnified. This is a summary of drawing ray
diagrams for convex mirrors.
