Video: Convex Lenses

In this video, we will learn how to define a convex lens, describe the paths of light rays refracted through these lenses, and explain how rays are focused by such lenses.

11:36

Video Transcript

In this video, we’re going to be discussing convex lenses. We will see what it means for a lens to be convex and how it affects light passing through it.

So let’s start by understanding that the word “convex” refers to an object that curves outward like the outside of a circle or sphere. Now this is a difficult thing to visualize, so let’s draw a diagram. Let’s say that we’re looking at an object from this perspective. Let’s say that this is our eye looking at the object that we’re about to draw. The object that we’re looking at is shaped like this. This shape is convex from the direction in which the eyeball is looking at it because it curves toward the observer in the middle and then back away from the observer just like the outside of a circle. So from this perspective, the object that we’ve drawn is convex.

But interestingly, this particular object could be seen as concave as well if the observer is observing from this side because from this side it appears to the observer that the object curves away in the middle and then back toward the observer. So the observer on the left sees the object as convex, and the observer on the right sees the object as concave, which is essentially the opposite of convex. So now that we understand what it means for an object to be convex, let’s take a look at convex lenses specifically.

A convex lens, just like any other lens, is designed to very specifically manipulate the direction of light rays passing through it. Now, technically, this lens that we’ve drawn here should be known as a biconvex lens. The word bi- means two because this shape that we’ve drawn here when viewed from the left side is convex. It curves towards the observer in the middle and then back away from the observer. And when viewed from the right, by another observer say, it’s also convex. It curves toward the observer in the middle and then back away from them, which is why this shape should be called biconvex. Bi- because it’s convex on both sides. However, often a lens cut into this shape is just referred to as a convex lens. We drop the bi- prefix.

Now to understand the effect of a convex lens on light entering it, we first need to understand that lenses are often made from materials such as glass, in other words, materials with a different refractive index to the air in which the light will be traveling initially. If we assume that this lens is made of glass, then we’ll see that glass specifically has a higher refractive index than air. And when light rays travel from a medium with a low refractive index, in this case air, into a medium with a higher refractive index, in this case glass, they undergo refraction.

Similarly, when they leave a medium with a high refractive index and enter a medium with a lower refractive index, they also undergo refraction. In other words, when light rays move from air to glass, they would change direction due to refraction. And when they move from glass back to air, they would change direction once again due to refraction at the boundary between the glass and the air.

Let’s recall how this works. Let’s quickly recall what would happen if a light ray meets a rectangular glass block rather than a convex lens. So here’s our rectangular glass block. And it’s surrounded by air on either side. Now let’s also imagine that a light ray travels from this region in the air to the glass. Specifically, we’ve got a light ray travelling, let’s say, in this direction, which means it meets the glass block at this point here. Now, in order to understand what happens when the light ray meets the boundary between the air and the glass, we can draw a dotted imaginary line that is normal to or at right angles to the boundary between the air and the glass at the point at which the light ray hits this boundary. This line is known as the normal line because it’s at 90 degrees to the boundary.

And the reason we draw this line is because we can now define an angle, which is the angle between the normal line and the incoming ray of light, known as the angle of incidence. And we can recall that when light leaves a medium of a lower refractive index, which in this case is air, and enters a medium with a higher refractive index, which in this case is glass, it refracts or changes direction. So if the incoming light ray had continued to travel in the same direction as before, this is the direction in which it would have gone.

However, we can see the actual light beam in the glass moves closer to the normal line. In other words, the angle between the refracted ray and the dotted normal line, we’ll call this the angle of refraction, is smaller than the angle of incidence. So that’s what happens when light leaves a lower-refractive-index material and enters a higher-refractive-index material.

However, as light continues to travel through the glass block and eventually arrives at the boundary between glass and air this time, we see the opposite effect. Once again, we can draw a dotted normal line at the point at which the light ray hits the boundary. We can then define a new angle of incidence, which is the angle between the incoming ray, which is now this ray, and the normal line, which is this dotted line. And we see that this time, the light ray will bend away from the normal. So if it’d continued to travel in the same direction as before, this is where it would’ve gone. But we can see that it’s bent away from the normal line, and hence the angle of refraction is larger than the angle of incidence.

So to recap, when light enters a material with a higher refractive index, the light ray refracts towards the normal, whereas when leaving a material with a higher refractive index, the light bends away from the normal.

So with all of that in mind, let’s come back to our convex lens. Now for our own convenience, let’s define something known as the plane of the lens. This is a flat surface, which we’ll represent it by the dotted line. That goes right down the middle of the lens dividing it into two symmetrical sections. We’re representing it as a dotted line here because we’re looking at it side-on. But remember that this is a plane or a flat surface, and it goes in and out of the screen.

Additionally, let’s also define the optical axis of the lens. This is an imaginary line that goes straight through the center point of the lens, the very middle of the lens, and is also perpendicular to or at right angles to the plane of the lens.

Now, the reason that we define the plane of the lens and the optical axis is to more easily help us describe light rays passing through our lens. So let’s do that now. Let’s imagine that we’ve got a ray of light moving towards our lens that is parallel to the optical axis. And so, in this case, the ray of light is moving from left to right. And we can zoom in to this part of our diagram to see what will happen when our ray of light meets the convex lens.

So here’s our zoomed-in diagram of the green box. And we can see our ray of light moving from left to right, initially moving in air and then meeting the glass that forms the convex lens. In this case, the ray of light is meeting this boundary here, which means even though the ray of light is traveling parallel to our optical axis, it’s actually meeting the boundary between air and glass at an angle, which means we can draw a dotted line that is normal to the boundary. And we can define our angle of incidence. That’s this angle here.

Now, remember that when light leaves a medium with a lower refractive index and enters a medium with a higher refractive index, it refracts toward the normal, which means that if it were to continue in the same direction as before, this is the direction in which it would travel. But, in reality, it bends toward the normal, meaning this is the angle of refraction. In this diagram, we’ve drawn it very small. And so coming back to our larger diagram, we can see that the light ray will not continue to travel parallel to the optical axis but rather will move off in this direction, at which point our light ray meets once again the boundary between glass and air. So let’s now work out what happens at this boundary.

So this is a zoomed-in version of this green box here. And we can see that the light ray is initially traveling in this direction moving through glass and is leaving the glass and entering the air, in other words, leaving a medium with a higher refractive index and entering one with a lower refractive index. Now, the boundary that the light ray meets is this boundary here. So we start once again by drawing our normal line. And then we recall that light when leaving a medium of high refractive index and entering one with a lower refractive index bends away from the normal. Therefore, instead of continuing on in this direction, the light ray will go off in this direction, clearly bent away from the normal.

And so coming back to our larger diagram, we can draw the light ray going in, say, this direction. And so at this point, we can draw multiple different light rays initially traveling parallel to the optical axis and see how they behave as they pass through the convex lens. When we do this, we can notice a couple of things. Firstly, all of the light rays that were initially traveling parallel to the optical axis have now been refracted by the convex lens so that they all meet at this point here on the other side of the lens. This point is a special point for this particular lens known as its focus, or sometimes known as the focal point. And, specifically, the focus is the point at which a lens will refract rays of light initially traveling parallel to its optical axis so that they all meet or converge at the focus.

And because all these light rays are converging, they’re all meeting at one point, a convex lens is also sometimes known as a converging lens. Now it’s important to note that a convex lens will only focus rays of light at its focus or focal point if initially they were traveling parallel to the optical axis. This is not true in general for any ray of light. Say we had a ray of light traveling in this direction, the convex lens will not necessarily refract it so it goes through the focal point.

There’s also something else we can notice here. Let’s take a closer look at the ray of light traveling along the optical axis. We see that as it enters the lens, it meets a boundary that it’s traveling exactly perpendicular to. And because of this, the ray of light does not change direction; it continues to travel in the same direction. And the same is true for the boundary on the other side. It’s perpendicular to the direction that the ray of light is traveling in. So the ray of light along the optical axis is not refracted; it continues to move straight on. This is uniquely true for the ray of light traveling specifically along the optical axis.

However, we often make an approximation that this is true for any ray of light traveling through the very center of the lens. So let’s say we’ve got this convex lens here. And let’s say we’ve got a ray of light moving in this direction. Now this ray of light would actually travel straight through the very center of the lens, the center point of the lens. And so, to a very good approximation, we can say that this ray of light is not actually refracted by the lens at all.

This is not really true. The ray of light will be slightly refracted by the lens. But we make an approximation that any ray of light passing through the very center is not refracted at all because for such rays, the refraction is actually quite small. And making this approximation makes our life a lot easier when considering how rays of light from a particular object pass through a convex lens.

So these are a couple of important points to remember: firstly, that any ray of light passing through the very center of the lens, regardless of what direction it’s traveling in, can be approximated as traveling straight on without being refracted. The only exception is the ray of light traveling along the optical axis because that one actually isn’t refracted. So if we imagine that it’s traveling straight on, this is actually what’s happening; we’re not making an approximation. And secondly, if we have rays of light that are initially moving parallel to our optical axis entering a converging or convex lens, these rays of light are refracted so they all meet at the focus or focal point.

So now that we’ve learnt a bit about convex lenses, let’s summarize what we’ve talked about in this lesson. We firstly saw that the word “convex” refers to an object that curves outward like the outside of a circle or sphere. So, for example, if we have an observer here, this object is convex because it curves toward them and then back away from them. Secondly, we saw that a convex lens is also known as a converging lens. Light rays entering parallel to the optical axis are refracted so they converge at the focus. And finally, we saw that light rays traveling through the very center of the lens in any direction can be approximated as traveling straight through without being refracted.

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