Video: Convex Lenses

Video Transcript

In this video, we will be looking at what a convex lens is. And how a convex lens can refract rays of light, changing the direction in which the rays travel in very specific ways. Now, a convex lens is a particular type of lens. So we should first look at what a lens actually is.

A lens is a device made of a transparent material, such as glass. That exploits the refraction of light in order to focus a beam of light, passing through it, or to disperse it. For example, in this video, we will be focusing on convex lenses, well accurately known as biconvex lenses. And biconvex lenses have this particular shape and are made from a transparent material such as glass or perspex. Which allows them to bend rays of light going through them in a very specific way.

Now the shape of the lens itself, as well as the material that it’s made from, dictates just how light behaves when it passes through the lens. For lenses, all made from the same material, the important thing is the shape of the lens that will determine how light rays behave. For example, another type of lens, a biconcave lens, will actually cause light rays passing through it to bend outwards rather than inwards like we saw in the biconvex lens. And this is all down to the shape of the biconcave lens, as opposed to the biconvex lens’s shape.

So now that we’ve seen what a lens is, specifically what is a convex lens. Well, the word convex refers to anything that has a surface curved like the outside of a sphere. In other words then, if we take a sphere and we look at it from the outside. So if we place our eyeball here, then we see that the surface of the sphere curves towards us, towards the observer. And that is what we mean when we say convex. And this is most easily understood in context when we look at the opposite of convex, concave. Concave refers to any surface that is curved, like the inside of a sphere. In other words, if we now place our observer’s eye inside the sphere as well as the rest of the observer ideally so they can actually use the eye. Then the observer will see the surface starting near them and then curving away from them and then coming back towards them as we move downward.

And an easy way to remember this is that if the observer were to walk in this direction inside the sphere. Then it would be like they’re walking into a cave. And hence this type of shape is known as concave. Whereas looking at the sphere from outside, we see it as convex. Now, that’s all well and good. But we still haven’t answered the question. What is a convex lens? Well, a convex lens is, for example, a piece of glass or another piece of transparent material shaped in this very particular way. It’s very thin at the edges, so top and bottom in this case, and thick in the center. Now it’s worth noting that we’re looking at a cross section of this convex lens. In other words, we’re looking at it side-on. So that we’re going to be shining light rays into it from this direction, from the left.

Now, as we’ve already mentioned, this kind of lens is called a convex lens. But if we’re going to be more technical about it, then it’s actually called a biconvex lens. The reason is because the word “bi” means two. And a biconvex lens is convex on the outside from both sides. In other words, if we place our observer, let’s say on the left side of the lens. Then to them, the lens appears convex from this side. Remember, it curves towards them in the middle like the outside of a sphere. And if we were to place an observer on the other side, then, to this observer, the lens would appear convex as well. It’s curving towards them in the middle and away from them at the edges.

Now there are other kinds of lenses that are convex on one side, but, for example, are plane, are flat surface on the other side. And so we could call this lens a convex lens because it’s convex on one side. But it’s not a biconvex lens. And in fact, we can even have lenses that are convex on one side. So our observer over here would see the lens curving towards them in the middle and away from them at the edges. But an observer on the other side would actually see this lens as concave because it curves towards them at the edges and away from them in the middle. And hence the technical term for the lens that we’ll be looking at today is biconvex. But most often, this kind of lens is just called a convex lens. And when we hear convex lens, we have to realize that it means convex on both sides, not just one side.

So now that we’ve seen what a convex lens or biconvex lens is, let’s take a look at how it can be used to bend light. To do this, we first need to recall that light bends or refracts when traveling through two different media of different refractive index. So what do we mean by this? Well, let’s start by imagining that we have a glass block here. And the glass block has flat sides on the left and on the right. We’re not talking about a biconvex lens yet. Now, let’s recall what happens when we first shine light on this block. And it’s coming perpendicular to the first surface of the block. Well, in that situation, the light simply continues to travel through without being bent.

However, things get interesting when we send in a beam of light at a slight angle to this surface. So let’s say we send a beam of light moving in this direction towards the glass block. Now, initially, the beam of light is traveling in air. But then, it arrives at a boundary between air and glass because here is the air. And here is the glass. And at that point, the light will have to travel through a different medium. It’s now gonna have to travel through glass, which has a higher refractive index.

Now what this means in practice is that if we first draw a line that is normal to the air glass boundary. In other words, this dotted line which is perpendicular to the surface that we’re talking about. Then we see that as the light enters the medium with a higher refractive index, the beam of light slows down and bends towards the normal line. And we can see this clearly if we measure, firstly, this angle here. The angle between the dotted normal line and the beam of light initially moving into the glass block, which we will call 𝜃 sub i. 𝜃 representing an angle and i representing the angle of incidence. And we then also measure this angle. The angle between the beam of light moving in the glass block and the same dotted line, the same normal line as earlier. We will call that angle 𝜃 sub 𝑟. Once again, 𝜃 representing an angle and 𝑟 representing the angle of refraction.

And we can see that the angle of incidence is quite large. But the angle of refraction is quite small. In other words, the beam of light as it goes towards the glass block is bending towards the normal line. But then, as the beam of light reaches the other glass surface, to work out what happens at this point, when it enters the air again. We have to draw yet another normal line to the point where the beam of light hits the side of the glass block.

So now the beam of light is entering a medium that has a lower refractive index than the medium it was already in. Because remember it’s going back into air. And therefore, the beam of light will bend away from the normal. In other words, if we now say that this here is the angle of incidence. And this here is the angle of refraction. We see that the angle of refraction is larger than the angle of incidence. So the whole point of this is to realize that when light enters a medium with a larger refractive index, such as glass, coming from a medium with smaller refractive index, such as air. Then the light bends towards the normal line and vice versa. When the light leaves the glass block and enters air, it bends away from the normal line.

So now that we’ve seen this, let’s take a look at a convex lens. So here’s our convex lens, made up of glass again. And let’s first see what happens to a ray of light passing straight through the center of the convex lens. Well, we see the point at which the ray of light hits the convex lens. The surface is this surface here. And so the line that is normal to the surface is this line here. And hence we see that the ray of light is traveling along the normal line. Which means that the ray of light is not going to bend either way. Hence, it will simply continue to travel straight and go straight through the other side of the lens.

And by the way, if the lens here is thin enough, then this is true for all light rays passing in any direction but ones which go through the center of the lens. In other words, if we imagine the lens to be very, very thin, then even a light ray going this way as long as it is going through the center of the lens. We can approximate as just going straight through the lens. It’s not going to bend in any direction. And this is a good approximation to make in most cases, because the lenses that we work with are generally quite thin. And this approximation allows us to work many things out by making our life slightly easier when drawing ray diagrams.

So we’ve seen what happens to rays of light traveling straight through the center of the lens. What about, say, a ray of light coming from here and meeting the lens quite far away from the center? Well, in that situation, the ray of light is meeting this glass surface here. In which case the normal dotted line is this line that we need to draw here. Remember that dotted line must be at right angles to the surface of the glass itself. And we can extend out that normal line to the other side. So that we can see the ray of light coming in this direction is going to bend towards the normal. And so it’s going to continue in this direction now.

And remember, the reason it bends towards the normal is because it’s going from a region of low refractive index to a region of high refractive index, from air to glass. And then we see that beam of light arriving at this surface here. Which means the normal dotted line, the line at 90 degrees to that surface, is this dotted line here. And so if this is the beam of light coming in, then as it goes out, it must bend away from the normal. Remember, once again, this angle of incidence must be smaller than this angle of refraction. And so overall, what’s happening is that this beam of light is now traveling in this direction when initially it was traveling left to right. And if we see what happens to a beam of light coming towards this part of the lens, then we see the same sort of refraction behavior.

And actually, if we draw in a few more rays of light, we see a very interesting pattern emerging. Specifically, if all the light rays coming in towards the lens are firstly parallel with each other, which in this case they are. They’re all moving in the same direction, left to right. And secondly, they’re perpendicular to the surface that runs straight down the middle of the lens, which once again we can see they are. If we were to draw that surface a little bit to the left, we can see that each one of those rays is at right angles to this surface. Then we see that all of those right rays get focused onto a single point over here.

Now, in reality, that point is very small, especially if the lens is very good. But for the sake of drawing a diagram, we’ve drawn it fairly large. Now this particular point is known as the focus of the lens. And the distance between the center of the lens and the point at which the focus exists is known as the focal length, or focal distance. Also, it’s worth noting that the beam of light going straight through the center of the lens, that is also perpendicular to the plane of the lens itself. Is moving along a line, known as the optical axis of the lens, which is basically a straight line that runs straight through the center of the lens and is perpendicular to the plane of the lens. Which, remember, is the flat surface right down the middle of the lens.

Now it’s worth remembering that all of the light rays only bent towards the focus in a distance known as the focal distance. If the light rays coming in are perpendicular to the plane of the lens and are parallel to each other. We could have rays of light that are parallel to each other coming toward the lens. But they’re not parallel to the optical axis of the lens. In which case all of the rays of light would be bent towards some other point, which is not the focus. And even though all the light rays do converge or meet at the same point, that point is not the focus of the lens. And this distance is not the focal distance of the lens.

So, anyway, this is the behavior of a convex lens. It’s sometimes also known as a converging lens because all the light rays that are perpendicular to the plane of the lens that pass through it converge on or meet at the focus. Now it’s all well and good discussing what happens to light rays coming in towards the lens. But let’s think about a real-life situation where the light rays are actually coming from a particular object. Or at least being reflected off of that particular object.

Let’s start with another clean diagram and with a new representation of a convex lens. Basically a line with outward pointing arrows can be used to describe a convex lens. Because if we were to join the tips of the arrows up, then we would see that the shape matches that oval convex lens. But anyway, so here is our convex lens. And let’s start by drawing in the dotted line that goes straight through the middle of the convex lens, known as the optical axis. So the optical axis goes through the center of the lens and is also perpendicular to the flat surface. Going right down the middle of the lens, which is this surface here. And we’re looking at that surface from the side. So we only see one dimension of it. But anyway, so the optical axis is going to be this line here, going straight to the middle of the lens and perpendicular to the plane of the lens. And we can see this is true because the angle is 90 degrees here.

Now let’s say we take an object. Let’s say a wooden block shaped like a triangle and place it here. We will imaginatively call this object our object. Now we know that light from other sources will be bouncing off our object. And the object itself will be reflecting some of these rays of light back. Because that’s how we can actually see the object. And crucially, if we consider just the tip of the object, we know that that part of the object will be reflecting like back in all directions. And some of these light rays will actually go through our lens. So we can track how these rays of light are going to move and see what happens to the light coming from the object when it passes through the lens.

Now there are a couple of things to note here. Firstly, we can very much simplify how the light rays behave inside a lens. Recall from earlier that we saw that light rays refract first when they go into the lens and then refract again when they go out of the lens. We can massively simplify this down in order to make our diagrams easy to draw and the situation easier for us to understand. By simply saying that the light ray going in goes to the center of the lens. And then just gets bent towards the direction in which it was traveling once it leaves the lens. So we don’t need to worry about two refractions, one at the first surface, one at the other surface. Instead, we just think of it as one refraction and leave it at that.

Now, coming back to our object here. We said that the tip of the object would be reflecting light in all directions. And the reason we know this is because we can still see the tip of the object, regardless of where we place our eyeball. It’s not like at some angles we don’t see the tip of the object at all. Unless, of course, we’re looking at it from here. But that’s a different story. But anyway, to see what the lens does to the rays of light coming from the object, we only need to consider two rays coming from the object. And once again, in this case, we’re just considering the tip of the object. Now the two rays that we need to consider are, firstly, the ray of light travelling straight through the center of the lens. And, secondly, the ray of light travelling parallel to the optical axis.

So let’s see what happens to these two rays of light. Now recall from earlier we said that any ray of light passing through the center of a lens can be approximated to just travel straight in the same direction. It’s not going to change direction or bend any other way. And secondly, we said that any rays passing parallel to the optical axis going through the lens will be bent. So they meet at the focus, which we can arbitrarily say is here for this particular lens. In other words then, we can draw that ray of light passing through the focus. And it’s gonna continue on in this direction.

Now at this point we see that the two rays of light that we’ve drawn intersect at this particular point here. And hence, that’s where the image of the tip of the object is going to be. In other words, the tip of the object is going to appear here on the other side of the lens. And hence the image of the object will look something like this. And in this particular case, we can see that the image of the object is inverted. It’s upside down. And we could actually play some sort of screen at this position. And we would see an image of the object. So now that we’ve had a look at the behavior of a convex lens, let’s take a look at an example question.

Which of the following statements correctly describes a convex lens? A) A convex lens is thicker than a concave lens. B) A convex lens increases in thickness from its center to its edges. C) A convex lens decreases in thickness from its center to its edges. D) A convex lens has uniform thickness.

So to answer this question, we’re trying to recall what a convex lens actually is. And to answer this, we need to know that the word convex refers to any surface that curves the same way as the outside of a sphere. In other words then, if this is our sphere and we place an observer on the outside of the sphere. So this is the eye of the observer. Then we see that the surface of the sphere starts far away from the observer. Then comes towards the observer and then bends away from the observer again. And a convex lens simply has that same convex behavior on both sides of the lens. In other words, a convex lens is convex on one side. And it’s convex on the other as well. Sometimes, therefore, it’s known as a biconvex lens, bi meaning two showing that it’s convex on the left and on the right.

So looking at the first possible answer, option A. This says that a convex lens is thicker than a concave lens. Now that’s not necessarily true. Firstly, we recall that concave is any surface that bends like the inside of a sphere. So if we place our observer here, then they see the inside of the sphere as starting near them at the top. Then curving away from them and then curving back towards them. And so a concave lens, or biconcave lens, is one which is concave on one side and concave on the other. And the way that we’ve drawn this concave lens it is thinner than the convex lens. However, this is also a concave or, rather, a biconcave lens. And yet it’s much thicker than our convex lens. So it’s not necessary for a convex lens to be thicker than a concave lens. Hence, option A is not what we’re looking for.

Option B then, a convex lens increases in thickness from its center to its edges. Okay, so if we start at the center, we see that it’s got a fairly large thickness. And then as we go towards the edges, or as we go higher up or lower down, we see that the thickness actually decreases. And hence, option B is not the answer we’re looking for. Option C says the opposite. A convex lens decreases in thickness from its center to its edges. And we’ve just seen that this is true. If we start at the center, the thickness is large. And as we go further towards the edges, the thickness decreases. So it looks like option C may be our answer. But if we quickly look at option D, this one says that a convex lens has uniform thickness. And we’ve clearly seen that that’s not true. Once again, the convex lens is thick in the middle. And it’s thin at the edges. Therefore, it cannot have a uniform or same thickness all the way along its length.

Hence, we’ve found our answer. A convex lens decreases in thickness from its center to its edges.

So to summarize the main points of this lesson, we’ve seen that convex lenses make incoming rays of light converge onto a point. And that rays of light through the center of the lens do not bend.