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Lesson Video: Drawing Ray Diagrams for Concave Mirrors Science

In this video, we will learn how to draw diagrams of light rays interacting with concave mirrors.

17:04

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.

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