At which of the points 𝐴, 𝐵, 𝐶, and 𝐷 shown in the diagram is the angle of refraction greatest?
We can see that the diagram shows seven parallel rays of light passing through an object. And as they pass through this object, we can see that the light rays change direction such that they’re spreading out. Specifically, we can see that, except for the one in the middle, each of the light rays changes direction twice, first as it enters the object and then as it leaves the object.
We can recall that the name given to the change of direction of light as it passes from one medium into another is refraction. So we can say that each of these light rays is refracting, once as they enter the object and again as they leave the object. And let’s just remind ourselves that, in this sentence, medium just refers to any material that the light travels through. One really important thing to note about this diagram is that the object the light rays are passing through is not just any old object. In fact, it’s a concave lens.
A couple of things give this away, firstly, its shape. If we look at a curved surface and see the middle of this surface is further away than the outer edges of this surface — that is, it curves away from us and then back towards us — then we can say that this surface is concave. But if we look at a surface and we see that the middle of it is closer to us than its outer edges — that is, it curves towards us and then away from us — then we can say that this surface is convex. So in this example, the surface on the left is concave and the surface on the right is convex.
One way to remember this is that if you had a big enough concave surface, you could walk towards it and go inside it like a cave. We can make lenses which are concave on both sides. And we call these concave lenses. Conversely, we can make lenses which are convex on both sides. And unsurprisingly, we call these convex lenses. We can see that the object in our diagram that the light rays pass through is the same shape as a concave lens.
The other thing that gives away the fact that the object in our diagram is in fact a concave lens is the way the light rays pass through it. For any concave or convex lens, we can define the optical axis. The optical axis of a lens is a straight line which passes through the center of the curved surfaces of the lens. The optical axis should also divide the lens symmetrically, meaning that the part of the lens above the optical axis is the mirror image of the part of the lens below the optical axis. That means that the optical axis here passes through the middle of these surfaces.
For any concave lens, any rays of light parallel to the optical axis passing through that lens will spread out or diverge, which is why you may hear a concave lens referred to as a diverging lens. We can see that the light rays passing through the object in the diagram we’ve been given are diverging as well. So we can conclude that this object is definitely a concave lens.
Now, the question asks us to find the point 𝐴, 𝐵, 𝐶, or 𝐷, at which the angle of refraction is greatest. So let’s start by reminding ourselves what the angle of refraction is. Let’s imagine a block of glass with a ray of light passing through it. We can see that this ray of light changes direction — it refracts — as it enters the block of glass and once again as it leaves. Now, whenever a ray of light diffracts, we can define two angles, the angle of incidence and the angle of refraction.
The angle of incidence is the angle between the incident ray of light and the normal to the surface. And the normal to the surface is the line which is at 90 degrees to the surface. So our angle of incidence is here. The angle of refraction is the angle between the refracted ray of light and the normal, which is here. In this example, we could define another angle of incidence and another angle of refraction when the light ray leaves the block of glass. So in this case, the angle of incidence is here and the angle of refraction is here.
In our question, each of the points 𝐴, 𝐵, 𝐶, and 𝐷 is where the light rays enter the lens. So we’re only interested in the angle of refraction as the light ray enters the lens. And we’re not interested in the angles of refraction as the light rays leave the lens. However, unfortunately, our diagram is not large enough for us to be able to actually measure the angles of refraction at each of the points 𝐴, 𝐵, 𝐶, and 𝐷. And potentially, the diagram is not accurate anyway. So to answer this question, we need to think more carefully about how the light will refract at each of these points.
In this example, we’re considering how a ray of light behaves as it passes from air into glass. Since generally lenses are made out of glass, we can assume that this is also the case in our question. What’s important about this is that air has a low refractive index, whereas glass has a high refractive index. When light passes from a medium with a low refractive index into a medium with a high refractive index, what we find is that it refracts toward the normal.
This dashed line represents the direction the light was initially traveling in. And we can see that it’s changed direction to get closer to the normal. In other words, when a ray of light passes from a medium with a low refractive index into a medium with a high refractive index, such as from air into glass, we see that the angle of incidence is always greater than the angle of refraction. Note that the opposite is true when light passes from a medium with a high refractive index into a medium with a low refractive index.
So when the light exits this block of glass, we can see that it refracts away from the normal. In other words, this angle of incidence is smaller than this angle of refraction. But since we’re interested in the refraction that occurs as the rays of light enter the glass lens, let’s stay focused on the light ray entering the glass block. Whenever a light ray refracts, we find that decreasing the angle of incidence will also decrease the angle of refraction. In other words, if we make this angle smaller, then this angle will also get smaller, like this.
And if we decrease the angle of incidence to its smallest possible size — that is, zero degrees — so that it’s actually going along the normal, we find that the refracted ray also goes along the normal. So the angle of incidence here and the angle of refraction are both zero. We can sum this up by saying the bigger the angle of incidence is, the bigger the angle of refraction will be. This fact is what we’ll use to find the answer to this question.
Let’s zoom in on each of the points 𝐴, 𝐵, 𝐶, and 𝐷. So in each of these boxes, we’ve drawn a slightly enlarged version of what’s happening at each of those points. We can see that even though the incident light rays are parallel to each other, they all strike the surface of the lens at different angles because it’s curved. In other words, they all have different angles of incidence. Looking at point 𝐴, we can see that where the light ray hits the surface, the surface is angled like this. If we draw the normal to the surface at right angles to this, we can see that the angle of incidence is this angle.
We can do the same at point 𝐵. At point 𝐵, the surface is angled like this, so it’s closer to vertical. Again, if we draw the normal, then we can see that the angle of incidence is this angle. We can see that because the incident ray is closer to the normal, the angle of incidence is smaller at point 𝐵 than it is at point 𝐴. Now, if we look at point 𝐶, we can see that the surface here is even closer to vertical. This means that the horizontal incident ray is really close to the normal. So the angle of incidence is really small.
And finally, at point 𝐷, we can see that the incident ray is actually at right angles to the surface at this point. So it’s traveling along the normal. Now, we just need to recall that the bigger the angle of incidence, the bigger the angle of refraction will be. And since we can see that at point 𝐴 the angle of incidence is greatest, then we have our answer. The angle of refraction will be greatest at point 𝐴.