Video Transcript
In this video, we will learn how to
draw diagrams of light rays interacting with concave mirrors. We’ll start by looking at just how
a concave mirror works.
We know that in general a mirror is
an object that reflects light. If a ray of light is incident on a
mirror then, it won’t go through but instead will bounce off. A concave mirror is a mirror with a
particular shape; it looks like a bowl. If a ray of light goes into the
bowl, it travels until it reaches the mirror’s surface and then is reflected. Imagine we’re looking at this
concave mirror from the side. From this angle, the mirror would
look like a two-dimensional shape. It would look a bit like this.
Now, here’s something important
about concave mirrors. The shape of the mirror that we see
here, this arc, is actually part of a circle. All concave mirrors are like
this. The curve of the mirror follows a
circular shape. Like any circle, this one has a
center point, right about here. Because this point is the center of
a circle that defines the curve of a mirror, it has this special name, the center of
curvature of the mirror.
Now let’s think about another point
on our sketch. The very center of our concave
mirror is located here. This is often called the mirror’s
vertex. If we draw a line passing through
the center of curvature and the mirror’s vertex, the name for this line is the
optical or principal axis. Knowing the optical axis is really
important for helping us draw ray diagrams. A ray diagram is just a sketch like
the one we have here that shows how rays of light are incident on the mirror and how
they’re reflected. If we have an incoming ray of light
that reaches this concave mirror, we can figure out how the mirror will reflect the
ray by looking at where on the mirror the ray lands.
For example, this ray lands right
at this spot on the mirror. We can pretend as though at that
spot, the mirror is a flat mirror like this. Thinking about the ray reflecting
from a flat mirror can give us a better idea of the direction the reflected ray will
travel in. We can use this strategy for any
ray coming into our mirror at any angle. Now say that we have an incoming
ray of light that’s traveling exactly parallel to the optical axis. The ray will arrive at the
mirror. And once again, we can pretend that
this small section of the mirror where the ray lands is a flat surface. This shows us that the ray will
reflect like this.
Now, remembering that this pink dot
represents the center of curvature, say that we have another different parallel ray
of light coming into our mirror. When it arrives, we treat the
mirror at that spot as though it is flat, which shows us that the ray reflects in
this direction. Notice that the two reflected rays
of light seemed to cross or intersect right at this point. Precisely speaking, these two
reflected rays don’t quite pass through the same exact point in space, but they are
so near to doing so we can effectively say that this blue dot is the focus of the
reflected rays. Another word for focus is focal
point. Any time there’s an incoming ray of
light that’s parallel to the mirror’s optical axis, the reflected ray will very
nearly pass through the focal point. We don’t need to draw a flat
surface on the mirror to figure out where the ray goes.
Now, here’s another interesting
thing about the focal point. Imagine we had a ray of light that
went right through the focal point as it reached the mirror. The reflected light ray will travel
exactly parallel to the optical axis. So whereas before we had a ray of
light reaching the mirror traveling this direction and we saw that that reflected
ray traveled out through the focal point, a ray traveling the opposite way will be
reflected parallel to the optical axis.
Along with our mirror’s focal
point, the blue dot, the center of curvature, the pink one, can help us draw ray
diagrams. Any time a ray of light is incident
on the mirror and it passes through the center of curvature, that ray of light will
be reflected back exactly along the path it traveled to reach the mirror. In other words, if it passes
through the center of curvature on the way in, it passes through the same point on
the way out. Knowing all this, we’re now ready
to start drawing ray diagrams with this concave mirror.
The point of a ray diagram is to
show how some object that’s in front of the mirror appears as an image. Those two terms, object and image,
are important to keep straight. Imagine that we’re standing in
front of a mirror brushing our teeth. In this case, we ourselves are the
object here, and the image is what we see in the mirror. So, an object is a real actual
thing in three-dimensional space, and the image is the picture of that object formed
by the mirror. Let’s now see how the image of some
object forms due to this concave mirror.
In general, an object can be
anything, and it can be anywhere. That is, as far as this mirror
goes, we could have an object over here or over here or over here or over here, or
anywhere in space. And also, as we said, the object
could be anything. It could be a stick or a book or an
animal. Anything our eyes can see can be an
object. Often, though, when we draw ray
diagrams, we position an object on or near the optical axis of a mirror. We do this mostly for simplicity’s
sake.
Imagine we have an object that
looks like this. Granted, this is a pretty odd
object. We’ve drawn our object this way to
make the image that it forms more clear. If it helps, though, remember that
this object could be anything. We’re just drawing it this
particular way for the sake of our ray diagram. Now that we have an object in front
of our mirror, we know that rays of light will leave the object, bounce off the
mirror, and form an image. In general, light will travel from
every single point on the surface of the object. If we drew all those rays in,
though, our diagram would quickly get too complicated.
So, for now, let’s just focus on
the rays of light that leave the tip of the object. Even if we look at only one point
on our object, though, we run into another issue. Technically, light from this point
travels in all directions. If we tried to follow each of these
rays, even just the ones that reach the mirror, still our diagram would quickly
become too complicated to follow. So, here’s what we’ll do; we’ll
only pay attention to two rays of light that come from the tip of our object. One ray travels parallel to our
optical axis, and the other is on a path to go straight through the center of
curvature of our mirror. If we follow the ray that’s
parallel to the optical axis, we know that when this ray reflects off the mirror, it
will reflect through the focal point. That’s the one reflected ray.
Now, considering the other incoming
ray, since it passes through the center of curvature, we know that when it reflects,
it will reflect back along the same path it came. Now look where our two reflected
rays cross, right here. This point is where the image of
the tip of our object will form. With this ray diagram, we figured
out what the image of one point on our object will look like. We still need to find the image of
the rest of our object though. Now, thankfully, we don’t have to
repeat this process for every single point on the surface of our object. We can actually fill in the rest of
our image by drawing a ray diagram of just one other point on our object, the
bottommost point. What we’ll do is leave this image
of the top of our object in place. And then from the bottom of our
object, follow two more rays: one that’s parallel to the optical axis and one that
passes through the center of curvature.
Following the ray parallel to the
optical axis, we know that this ray will reflect off of the mirror through the focal
point. Then for the other ray passing
through the center of curvature, when this ray reflects, it will pass back through
that same point. So, for these two reflected rays,
their point of intersection is here. We’ve colored this point of the
image green so that it matches the color of that point of our object. We can now fill in the rest of the
points in our image. It will look like this.
Notice a few things about this
image. Number one, the image overall is
smaller than our object overall. This can happen when light from an
object is reflected in a concave mirror. When an image is smaller than its
object, we say that the image is reduced in size. Something else we can say about the
image is that it’s upside down compared to the object. The top of our object is here while
that point is at the bottom of the image. And the same thing goes from the
bottom of the object and the top of the image. An image like this is called
inverted.
Notice one last thing about our
image. It’s on the same side of the mirror
as our object is. Believe it or not, this doesn’t
always happen. When it does happen though — when
the images on the same side of the mirror as the object — the image is said to be a
real image. Now we have to be a bit careful
with this word real. Our object is real in the sense of
being a three-dimensional shape in a three-dimensional space. We could touch it, feel it, weigh
it, and so on. Our image though is not like
that. It’s not something that we could,
say, hold in our hand. It is after all just an image. It’s where reflected rays of light
meet, but still these reflected light rays really do intersect.
If we were to put a screen in place
here, say a sheet of white paper, then the image of our object would be projected
onto that sheet. We could move the sheet around, and
the image would go in and out of focus. Whenever we can do that, the image
is said to be real. So, for this object in this
position relative to our concave mirror, the image produced is reduced in size,
inverted, and it’s real. Remember we said, though, that in
general an object can be at any position in front of a mirror. Let’s now see how the image of this
object forms when we move our object closer to the mirror so that it’s in between
the center of curvature and the focal point.
Here’s our object in its new
position. And just like before, we’ll trace
rays that come from the topmost and bottommost points of our object. Also as before, we’ll follow two
rays as they leave the tip of our object. One approaches the mirror traveling
parallel to the optical axis, and the other approaches the mirror on a line so that
if we traced this ray backwards, that trace would pass through the center of
curvature. We might wonder why this ray of
light doesn’t go from the tip of our object in this direction towards the center of
curvature. The reason for that is this ray
would never actually reflect off of the mirror. We would never get a reflected ray
to use to form an image. So we have these two rays from the
tip of our object.
As a side note, to find where the
image of this point in our object forms, technically we could draw any two rays
coming from this tip so long as they reflect off of the mirror. Wherever those reflected rays meet
is where the image of this point forms. The reason, though, that we always
choose these two particular rays, one that travels parallel to the optical axis and
one that’s on a path to go through the center of curvature, is simply because it’s
generally easier to figure out how these rays reflect off of the mirror. Knowing that then, let’s follow
these two rays as they do that.
The ray that leaves the object
traveling parallel to the optical axis will reflect back through the focal
point. The other ray traveling on a line
through the center of curvature will reflect back through that point. The point where these two reflected
rays intersect is where the image of the tip of our object forms. Knowing this, let’s clear away
these rays and follow a similar process for the lowermost point of our object. We have our two specific rays
leaving this point. The ray originally parallel to the
optical axis reflects through the focal point, and the ray on the line of the center
of curvature of the mirror reflects back through that point. The image of this point of our
object forms where these rays intersect.
Drawing in the rest of our image,
it looks like this. Now, our image is clearly much
bigger than our object. Any time this happens, the image is
said to be magnified. Like before though, our image is
upside down compared to our object. This means it is an inverted
image. And lastly, because the image is on
the same side of the mirror as the object, this image is also real. So far, we’ve considered what image
is formed when our object is farther from the mirror than the center of curvature
and also what kind of image forms when it’s between the center of curvature and the
focal point. Now let’s see what happens when our
object is between the focal point and the mirror.
Once again, we’ll follow rays that
leave the top and bottom of our object. Of these two rays, the one
traveling parallel to the optical axis reflects through the focal point, while the
other ray is reflected back through the center of curvature. Notice something here though. As these two reflected rays travel
farther and farther, they move farther and farther apart. They’re never going to cross, and
so they will never form an image of our object. So, does this mean that any time an
object is between the focal point of a mirror and the mirror, no image forms?
If we try this experiment ourself,
say, moving our hand closer and closer to a concave mirror, we see that actually an
image does form. To see how it forms though, we’ll
need to trace these two reflected rays backward. If we do that, we find that these
dashed line traces do meet. They meet right here. This spot behind the mirror is
where the image of this point on our object forms. Now let’s figure out where the
image of the bottommost point on our object forms.
As before, we imagine these two
rays leaving this point. The ray parallel to the optical
axis reflects through the focal point, and the ray on line with the center of
curvature passes through that point. Tracing the reflected rays
backward, we see that they meet right about here. This then is about what the image
of our object looks like. It’s bigger than the object, so the
image is magnified. And then look at this. The image is not upside down
anymore, but it’s right side up. Images like this are called
upright.
Lastly, notice that the image is
now on the opposite side of the mirror as the object. Such images are not real. We can’t project them onto a screen
like a piece of paper. We can see this by imagining that
this concave mirror is, say, hanging on a wall. The space behind the mirror is
occupied. Rays of light can’t really get
there and intersect. Images like this that cannot be
projected onto a screen are called virtual. We can still see them. And to our eyes, they don’t seem
any different from real images. But nonetheless, there is a
difference between a virtual and a real image.
Let’s finish our lesson now by
reviewing a few key points. In this video, we learned that a
concave mirror is an object that reflects light, where the surface of the mirror is
part of a larger circular shape. The center of this circle is called
the center of curvature of the mirror. And a line that passes through this
point and the vertex of the mirror is called the optical or the principal axis. A ray of light that travels
parallel to the optical axis and lands on the mirror is reflected through a point
called the focal point. Similarly, whenever a ray of light
passes through the center of curvature on the way to the mirror, it reflects back
through the same point.
We learned further that when an
object is farther away from the mirror than the center of curvature, the image that
forms is reduced in size compared to the object, it’s inverted upside down, and it’s
real. However, if we move the object so
it’s in between the center of curvature and the focal point, then in that case the
image that forms is magnified, inverted, and real. And finally, if the object is
located between the focal point and the mirror, then the image that forms is
magnified, upright, and virtual. This is a summary of drawing ray
diagrams for concave mirrors.
