Video Transcript
In this video, we will be looking at image production using lenses. We will look at how light passing through a lens can produce an image and what this even means. Additionally, we will learn about a set of rules that will allow us to determine how light rays will behave, when traveling through lenses. So let’s begin by looking at two different types of lens, a convex lens and a concave lens. Now, convex and concave lenses are basically lenses shaped in very particular ways.
A convex lens has surfaces that are convex on either side, which is why it’s known as a convex lens, which in other words means that to an observer, say on the left side, if this is their eyeball, then they see the lens bulging outwards in the middle and away from them at the edges. And the same is true for an observer on this side. If this is their eye ball. They see the lens bulging towards them in the middle and away from them at the edges, whereas the concave lens has two concave surfaces on either side. And hence this type of lens is known as a concave lens.
Now as well as having a drawing of a convex lens and a concave lens, lets draw in the optical axis for each one of these lenses. Let’s recall that the optical axis of a lens is the imaginary line that goes straight through the center of the lens and is perpendicular to the surface that cuts the lens right down the middle. In other words, if we took this pink dotted line as the surface that’s right down the middle of the lens, then our optical axis, the orange dotted line, is perpendicular to the surface. And the same applies to the surface that cuts the concave lens down the middle. Our optical axis is perpendicular to this. So anyway, this is the optical axis of the convex lens we’ve drawn in the diagram, and this is the optical axis of the concave lens.
Now, let’s recall what happens when we send in light rays to both these lenses parallel to their optical axes. So firstly, if we send light into the convex lens such that each ray of light is parallel to the optical axis, then we can recall that each ray of light gets bent or refracted, which is the technical term, so that all of the rays of light meet at a very special point known as the focal point or focus of the lens. And that is how specifically light behaves on passing through a convex lens. However, if we pass rays of light through a concave lens such that those rays of light are parallel to the optical axis, then those rays of light actually diverge. They get spread out by the concave lands.
However, if we trace each ray of light backwards, then to an observer on this side of the lens, so this is their eyeball, we’ll see all of these rays of light. And to them it will seem like the rays of light originated at this special point here, also known as the focal point or the focus of the concave lens. Because remember, to this observer, they can only see the rays of light coming out of the concave lens. They cannot see the actual rays of light moving parallel to the optical axis on the other side of the concave lands. And hence to them, it appears as if the rays of light that are coming out of the concave lens are originating at the focus that we’ve drawn in.
Now, it’s worth noting that every single lens does not just have one focus. Rather, it has two, one on either side. So the convex lens has another focus here. And the concave lens has another focus here. Both the focuses of each lens or, to use the correct term, both the foci of each lens are the same distance away from the lens on each side. In other words, if the distance between the lens and, let’s say, this focus for the convex lens, which by the way is known as the focal length or focal distance, is five centimeters, then the distance between the lens and the other focus is also five centimeters; it’s the same distance. And the reason that every lens has two foci becomes apparent if we instead send in light rays from the right as we’ve drawn the diagram. If we took the same convex lenses above and sent in light rays from right to left, then we would see that these light rays converge at the focal point on the left-hand side of the lens.
But anyway, so we’ve had a quick look at the behavior of convex and concave lenses. And we’ve specifically considered light coming in parallel to the optical axis for both of these lenses. But where is this light even coming from? Well, let’s imagine what would happen if we had light being reflected off an object, an everyday object, and then going through the lens. What would happen to that light coming from the object? Let’s first take a look at how a convex lens would affect the light coming from an object. So here’s our convex lens. Here’s the optical axis of the convex lens. Here are the focal points of the convex lens. And here’s the object which is going to be reflecting light into our lens.
Let’s imagine it’s an arrow-shaped wooden block. And we’ve placed it so that the base of the arrow is on the optical axis of the lens. Now, the object itself will be reflecting light in all directions. In other words, light from the light source, let’s say the sun, will be falling on it. And the subject will be reflecting that light in all directions. And that’s what allows us to actually see the object, the reflected light coming off the object. And so we will be focusing on the light that is reflected off the object and then ends up passing through the lens.
So let’s start by specifically considering the light reflected off the tip of the arrow. Now, we know that light will be reflected in this direction, in this direction, in this direction, in this direction, and so on and so forth. However, in reality, we only need to consider two of the beams of light coming from the tip of the arrow in order to work out what the lens does to the light coming from the arrow. The two beams of light that we need to consider are, firstly, the ray of light traveling parallel to the optical axis of the lens and, secondly, the ray of light that will be moving straight through the center of the lens. So let’s extend those two rays of light until they reached the lens itself. And then let’s try and work out what the lens will do to these rays of light.
Now let’s start by thinking about the ray of light going straight through the center of the lens. Let us recall that for convex lenses and actually also for concave lenses and all other kinds of lens, if the lens itself is thin enough, then the ray of light passing through the center of the lens will not be refracted at all. In other words, it will continue traveling in the same direction as before. Its direction will not change. And this is because, for a thin lens, the ray of light passing through the center doesn’t experience a huge amount of refraction. And so the thinner the lens, the better our approximation that the ray of light just goes straight through.
But because we mostly deal with thin lenses, we will make this assumption for all rays of light traveling through the center of any lens that we work with that they simply go straight through without being bent in anyway. And then if we look at the other ray of light, the one traveling parallel to the optical axis, then we can work out what happens to that ray by looking at this diagram here. All the rays of light travelling parallel to the optical axis were refracted or bent so that they passed through the focal point of the lens. And hence this ray of light will now be bent so that it passes through the focal point. And now, we can see that the two rays of light that we’ve drawn in actually intersect on this side of the lens at this point here. And that is the point at which an image of the tip of our arrow shaped object will form.
And actually, if we were to consider light rays, coming from all of the rest of the arrow to raise from every single point along the arrow where one of them moves parallel to the optical axis and the other one goes straight through the center of the lens, then we would see an image of the arrow formed, which would look like this. And so at this point, if we labeled the original object from which we’re considering the light rays and we label the image formed by the lens of this object, then we can make a few comparisons between the two. Firstly, the most obvious thing we could see is that the image of the arrow is upside down compared to the object itself. The technical term for this is that the image is inverted, which means flipped upside down.
Secondly, we can see, although perhaps not so clearly, that the size of the object is slightly larger than the size of the image. Well in this case, when we say size, we’re talking about the length of the arrow, which is the object, and the length of the arrow, which is the image. To make a comparison between these two, we can introduce a term called magnification, where the magnification of the lens is defined as the ratio between the size of the image produced by the lens and the size of the object itself. And we use the broad term size because it is unlikely that objects will simply be one long straight line like an arrow. And so sometimes we’ll have to compare their widths as well as their depths.
But anyway, so magnification is often represented simply by the capital letter 𝑀. And if we just used the size of the image and the size of the object to calculate the magnification, then the value of the magnification, the value of 𝑀 can be any value greater than or equal to zero. Sometimes a convention is used such that a negative value of magnification suggests an inverted image, which in this case we have but we won’t be using that convention. Instead, we’ll simply use the size of the image and the size of the object. Now, if the value of the magnification is anywhere between zero and one, then we say that the image is diminished. And the reason for this is that a value for the magnification between zero and one is given because the size of the image is less than the size of the object. In other words, the image is smaller than the object. And hence, the image is said to be diminished.
If the value of the magnification is greater than one, then the image is said to be magnified. And the reason for this is that a magnification of greater than one suggests that the size of the image is larger than the size of the object. And so the image is enlarged compared to the object. Or it’s magnified compared to the object. And of course, it’s worth mentioning that if 𝑀 is equal to one, then the image is neither diminished nor magnified because it’s the same size as the object itself. But anyway, so coming back to our set up here, we see that the image is smaller than the object itself. And, therefore, it’s going to have a magnification of somewhere between zero and one, although, of course, it’s not going to be zero itself because the image does have some length.
And hence, this fraction is not zero for this image. And, of course, the magnification is not going to be one, because that would suggest an image that’s the same size as the object which we’ve seen is not the case. So the magnification for this image is somewhere between zero and one. And so at this point, we’ve used two words to describe this image here. One is that it’s inverted, and the second is that it’s diminished. A third word we can use to describe the image is the word real. In other words, this is a real image. And a real image is one which is formed by the actual convergence or meeting of light rays, which we’ve already seen as the case. We’ve seen that these two light rays are meeting at the point here, which is, therefore, producing an image of the tip of the arrow that was our object.
And a real image is important because if we were to take a screen and place it at this position, then we would actually see a physical image of the object itself on that screen. Very soon we will see images that aren’t real images. But for now, let’s realize that we’ve placed an object here at this position. And the distance from the object to the lens can be measured in terms of the focal length or focal distance of the lens. We know that the focal distance is the distance between the focal point and the center of the lens. So this distance that we’ve just drawn is one focal length. This arrow is a second focal length. And then the object itself is placed at a slightly larger distance than two focal lengths away from the lens. And so placing an object more than two focal lengths away from the lens produces a diminished inverted real image.
Let’s see what happens if we move the object slightly closer to the lens, in other words, between one and two focal lengths away from the lens. So let’s say that our object is placed here now. So, like we said, between one and two focal lengths away from the lens. And let’s once again consider the rays of light coming from the tip of the object itself. The first ray of light is the one that’s parallel to the optical axis. And the second ray of light is the one going through the center of the lens.
Now, the second ray of light, the one through the center of the lens, will continue to travel in the same direction as before. Whereas the first ray of light, the one that’s parallel to the optical axis, will once again be refracted through the focal point or the focus of the lens. And so at this point, we see that the two rays of light now meet at this point here. And therefore, that’s where the image of the tip of the object is going to form. And if we once again considered rays of light from the entire object, we would see that the full image of the object forms looking something like this. And so we labeled the image at which point we try and use the words that we’ve already talked about to describe this image.
The first thing that we can say about it is that it is inverted once again. It’s upside down relative to the object itself. The second thing we can say is that the image is a real image because, once again, it’s formed by the convergence of two light rays at a point. In other words, light rays do actually meet to form this image. And then the third word that we would use will be based on the size of the image itself, compared to the size of the object. We can see in this case that the image is larger than the object is itself. And therefore, we can see that if we calculate the magnification, that is going to be greater than one. In other words, the image is said to be magnified here. And so to recap, placing our object between one and two focal lengths away from the lens produces an inverted real magnified image, whereas we saw earlier, placing our object more than two focal lengths away from the lens produced an inverted real diminished image.
Now, let’s quickly see what happens when we move our object even closer to the lens, in other words, less than one focal length away from the lens. So now, here is our object placed less than one focal length away from the lens. In other words, it’s closer to the lens than the focus is on this side of the lens. So let’s once again think about the two rays of light coming from the tip of the object. The first one is a ray moving parallel to the optical axis. And the second is the ray of light moving straight through the center of the lens. Well, like we saw earlier, the ray of light moving through the center of the lens continues to move in the same direction, whereas the other ray of light, the one parallel to the optical axis, is going to be refracted through the focus.
And at this point, we can see that the two rays of light coming out of the lens are actually spreading apart from each other. They are not going to meet like we saw earlier. However, if we trace the two rays of light back through to the left side of the lens, then we see that to an observer on this side of the lens, so if this is their eye, it will appear as if the two rays in question are being emitted from this point here, which is where they intersect. And so the observer on this side of the lens will see an image that looks something like this. And the image is seen through the lens, because once again the observer can only see these two rays of light. They cannot see that the original rays of light are these two rays. And so they will only be able to trace back the rays of light to the point at which they meet and see an image of the object through the lens.
In other words, then they will not see the object itself with this length and this distance away from the lens. But what they’ll instead see is the image of the object, which they’ll see this distance away from the lens and with this size. Now, this kind of image, firstly we can say, is not inverted because look, the arrow is pointing the same way as the object itself; it’s the right way up. And secondly, we can say with regards to the magnification that the size of the image is larger than the size of the object. And therefore, a magnification ratio is going to be larger than one. Therefore, the object itself is magnified.
And lastly, we can say about this image that it’s not a real image. And the reason for this is that it’s not formed by the convergence of light rays, by the meeting of light rays, but instead is formed by the apparent meeting of light rays, as seen by the observer on this side of the lens. Because to them it appears as if those two light rays are coming from this point here, so once again, not an actual meeting of light rays but the apparent meeting of light rays when traced backward. And this kind of image is not known as a real image, instead it’s called a virtual image at which point we can realize that depending on how far away we place the object from the lens itself, whether it’s less than one focal distance away or between one and two focal distances away or more than two focal distances away from the lens. The image that will be produced can either be diminished or magnified. It can either be virtual or real. And it could be either inverted or the right way up.
So a convex lens can produce lots of different kinds of image, whereas a concave lens can only produce one kind of image. To see this, let’s draw a concave lens with the optical axis and the two focal points here. And let’s once again take our object, our arrow shaped object, and see what happens to the rays of light coming from the tip of the arrow. The first ray of light is the one parallel to the optical axis, and the second one is going straight for the center. So that’s this ray of light here. Now, with a concave lens, the ray of light going through the center of the lens behaves in the same way. It just continues to travel in the same direction.
However, the other ray of light, the one parallel to the optical axis, actually ends up diverging, so that it goes in this direction. And the reason is because if we trace that ray of light back, then it appears to an observer on the right side of the lens as we’ve drawn it. So here’s our observer and they will see the ray of light going in this direction. And if they trace that back, it would appear to be coming from the focus of the lens. So in this ray diagram where we’re reforming the image of our object well, once again, the image will form where either the rays of light converge, which we can see in this case will not converge because they’re going away from each other. And so the other option is to form an image where they appear to converge to an observer looking through the lens.
And to the observer, if they were to trace the two rays of light back that they can see on their side of the lens, then to them, it would appear as if they were converging at this point here. In other words, then the image of the arrow object that we had is going to be this. Now, we can see that the image is smaller than the object itself. And so the size of the image is less than the size of the object, as the magnification is going to be between zero and one. And the image is diminished. Secondly, we see the image is pointing in the same direction as the object itself. And hence, the image is not inverted, it’s the right way up.
And finally, because this image is not formed by the actual convergence of light rays, rather it’s formed by the apparent convergence of light rays, as seen by an observer on this side of the lens, the image itself is a virtual image. And interestingly, a concave lens can only produce a diminished virtual image the right way up, regardless of how far we place our object from the lens itself. Try it for yourself. Try moving the object around and drawing the ray diagrams in each case.
But anyway, so now that we’ve seen this, let’s summarize what we’ve talked about in this lesson. Firstly, we saw that rays of light through the center of the lens do not change direction, whereas rays of light parallel to the optical axis refract through a focus. Secondly, we saw that magnification was defined as the size of the image divided by the size of the object. And finally, we looked at real and virtual images.
