Lesson Video: Drawing Ray Diagrams for Convex Lenses Science • Third Year of Preparatory School

Video Transcript

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

Let’s begin by recalling what we mean by a convex lens. A lens is a piece of transparent material, that is, material that lets light pass through it. Lenses have particular shapes that can be used to change the direction of light rays. A convex lens is a type of lens with a shape like this. It’s thinner at the edges and thicker in the middle. It is important to point out that this shape is how the lens looks when we view it from the side.

Typically, if we were to draw a diagram of light rays passing through a convex lens, then we would have our lens like this. And we would have our light rays coming in from the side like this. The dashed horizontal line that we’ve drawn, which passes through the center of the lens, is known as the optical axis. Again, it’s worth being clear that this is all a view from the side. If instead we were to look along the direction that the light rays are traveling, then the convex lens would look like a circle. When we’re drawing ray diagrams, we’ll stick to this side-on view because it provides a clear way of seeing the direction of the light rays.

Speaking of the direction of light rays, we know that when light rays are just traveling through air, they move in straight lines. However, we have also said that lenses can change the direction of these light rays. Now, it’s not all that much use knowing that a lens can change the direction of a light ray unless we also know how that direction is going to change. So the question that we want to answer is, just how does a convex lens affect the light rays passing through it?

It turns out that there are a few simple rules for how this works. So let’s clear some space and look at what these rules tell us. The first rule is that any light ray that passes through the center of a convex lens does not change direction. When we talk about the center of a convex lens, we mean this point here, which is on the optical axis and also an equal distance away from the front surface and the back surface of the lens. If instead we were to look along the direction of the optical axis so that the lens looks like a circle, then the center of the lens would be this point in the center of the circle, which is on the optical axis.

What this first rule is saying is that whatever direction a light ray is initially traveling in, if it goes through this center of the lens, then the light ray keeps traveling in that same direction. So a light ray traveling horizontally along the optical axis goes through this center of the lens and continues traveling horizontally. A ray through the center initially traveling in this direction will keep traveling in this same direction.

At this point, it’s worth mentioning that in reality it’s a little bit more complicated than this. A light ray traveling through a convex lens experiences refraction when it enters the lens through the front surface and again when it leaves the lens through the back surface. In this diagram here, we have drawn a particularly wide convex lens, which we can use to help us see the effects of the refraction more clearly.

Let’s consider the path of this light ray here. In the absence of any refraction, this ray would pass through the center of the lens. But in reality, refraction at this front surface means that the ray’s direction bends by a small amount when it enters the lens. Likewise, refraction at the back surface of the lens causes the direction of the ray to get bent back again so that it’s pretty much the same as the ray’s direction before it entered the lens. If instead we had ignored these refraction effects, then the light ray would have followed the path shown by this green line here.

We should make it clear that in this diagram we have used a deliberately wide lens to exaggerate the effect. For very thick lenses, this effect is pretty strong. However, for very thin lenses, the effect is negligible. This means that for very thin lenses, the effect is so small that we can ignore it and continue to use rule one.

Now, let’s look at the second rule. This rule applies to light rays which are parallel to the optical axis. That means any light rays which before they reach the lens are traveling parallel or side by side with this dashed line here. Now, if we have a ray traveling exactly along this optical axis, then this ray passes through the center of the lens. And so we know from the first rule that this ray keeps traveling in the same direction. So this second rule is going to be useful for rays which are parallel to this line but not exactly along the line itself, for example, rays like these two shown here.

The second rule says that any light ray which is initially traveling parallel to the optical axis has its direction changed by the lens so that it ends up going through a position known as the lens’s focal point. The focal point of a lens is a point on the optical axis which is a horizontal distance of one focal length away from the center of the lens. The focal length distance is a property of the lens itself. Different lenses will in general have different focal lengths. These light rays change direction as a result of refraction at both the front surface and the back surface of the lens.

The second rule tells us that the combined effect of these two direction changes is that both the rays will end up going through this focal point. So the direction of the rays will change a bit as they enter the lens. And then it will change a bit more as they leave the lens so that the rays end up passing through the focal point. Every light ray which is initially parallel to these ones before it enters the lens will also end up going through this focal point. We can say that the rays are focused or converged to this point by the convex lens.

The third and final rule is kind of like the reverse of the second rule. To understand this rule, it’s important to recall that the lens has a focal point on both sides of it. In each case, the distance from the lens to the focal point is one focal length.

The third rule says that any ray which passes through the focal point before going through the lens has its direction changed so that it ends up parallel to the optical axis. Let’s consider this light ray here. We can see that this ray passes through the focal point before it gets to the lens. So we know from the third rule that the lens is going to change the direction of this ray so that it ends up parallel to the optical axis. This change in direction happens in two stages. The ray’s direction changes once at the front surface of the lens and a second time at the back surface of the lens. The overall effect of these two direction changes is that the ray comes out from the lens parallel to the optical axis, like this.

The third rule applies to any light ray which goes through this focal point as long as that ray then also goes through the lens. So for this light ray here, after two direction changes, it will come out from the lens looking like this. We could also imagine any number more light rays all passing through this initial focal point.

These three rules for light rays passing through a convex lens can also be used to help us figure out what happens to light rays coming from an object. Let’s clear some space so we can see how this works. We’ve kept the diagrams which illustrate each of the three rules on the screen, because we’ll find that these diagrams are going to be useful.

Let’s suppose that we have a convex lens and that some distance away we have an object. We have drawn this object in a position such that the bottom of the object is on the optical axis and the top is some distance above this axis. The object that we have drawn looks like a straight vertical line. But it’s important to remember that we’re looking at this situation from the side. Perhaps this object is actually a piece of paper with a drawing on it. And we just can’t see that drawing because we’re looking at the paper along its edge.

We want to know what effect this convex lens has on the light rays coming from this object. Let’s suppose that the lens has focal points here and here. We will begin by considering some light rays coming from the top of the object. From the first rule, we know that the light ray which passes through the center of the lens keeps traveling in the same direction. The second rule tells us that the light ray which is initially parallel to the optical axis gets its direction changed by the lens so that it ends up going through the lens’s focal point. The third rule is about the light ray which passes through the focal point on the near side of the lens. In this case, when we draw that ray in, we see that it doesn’t actually pass through the convex lens. So this light ray isn’t going to be any help to us.

But it turns out that that’s actually okay and that we only need these two light rays in order to figure out what’s going on. The two light rays come from the same point on the top of the object. When these two light rays meet each other again on the far side of the lens, they form an image point which corresponds to this point on the object. So let’s mark this position on our diagram. We can do exactly the same thing for any other point on the object.

Let’s now consider the bottom of the object. The first rule tells us that the light ray which goes through the center of the lens keeps traveling in the same direction. The second rule is about the ray which is initially parallel to the optical axis. And the third rule is about the ray which goes through the focal point on the near side of the lens. In this case, these are both the same as the ray that we have already drawn.

Let’s look at one more point on the object halfway between the bottom and the top. Since the diagram is getting a little busy, we’ll draw the light rays from this point in a different color. Using the first rule, we know that the light ray through the center of the lens keeps going in the same direction. Using the second rule, we know that the ray which starts out parallel to the optical axis gets its direction changed so it ends up going through the focal point. The position at which these two rays cross each other on the far side of the lens is the image point corresponding to the point on the object where these two rays came from.

We could go on and do this for every single possible point on the object. And we would find that they all have a corresponding image point. Of course, in practice, we don’t actually need to do that. It turns out that if we trace the rays from the top of the object, we can find the position of the top of the image. And this is enough to draw in what the image looks like. In this way, the convex lens has produced an actual image of the object. This image formed here can be viewed with the human eye or projected onto a screen. We call it a real image.

We can see that this image is inverted. That is, it’s upside down compared to the object. We can also see that the image and the object are not the same size. The image is bigger than the object. If we’re looking along the direction of the optical axis so that we can see this drawing on the object, then in the image everything in this drawing will be bigger and the whole thing will be upside down.

Let’s illustrate that with a much simpler drawing consisting of an arrow pointing directly upward. The image of the paper is bigger than the object. And so the drawing of the arrow in this image is also bigger. Since the image is inverted relative to the object, the arrow in the image now points down instead of up.

Let’s now quickly consider what happens when we move the object further away from the convex lens. We know that the light ray through the center of the lens continues on in the same direction. And we know that the ray parallel to the optical axis gets its direction changed so that it goes through the focal point. These two light rays came from a point on the top of the object. So the position on the far side of the lens where they cross corresponds to the same point on the image.

We can use this information to draw in the image. We can see that in this case the image is still inverted compared to the object. But now the image is smaller than it was before. And in fact it’s actually smaller than the object. So the size of the image depends on how far away the object is from the convex lens. By moving the object further away from the convex lens, we make the image smaller. And it turns out that the image can be either bigger or smaller than the object depending on this distance.

Let’s now finish up by summarizing what we have learnt in this video. First, we saw that a convex lens is a lens that has a shape like this when we look at it from the side. Then we learnt that there are three rules for what happens to light rays when they pass through a convex lens.

The first rule is that any ray which passes through the center of the lens continues traveling in the same direction. The second rule tells us that any light ray which is initially parallel to the optical axis, so that’s this blue dashed line here, has its direction changed by the lens so that it goes through a position on the optical axis known as the lens’s focal point. In our diagram, that’s this green dot here. The third rule says that any ray that goes through the focal point before passing through the lens has its direction changed so that it comes out from the lens parallel to the optical axis.

Finally, we learnt that a convex lens can be used to form an image of an object. We saw how we could use our rules for light rays passing through a convex lens to work out where the image would be formed and what it will look like. We discovered that the size of the image formed depends on the distance between the object and the convex lens.