A convex lens is used to produce a real image, as shown in the diagram. Which of the points A, B, C, D, and E is located at a distance from the lens that is equal to the lens’s focal length?
Okay, so, here in our diagram on either side of this convex lens, we see a blue bar, one bar on the left and one on the right. We can say that one of these bars represents a real physical object and the other bar represents the image of that object. It doesn’t make much difference whether we say the bar on the right is the object and the bar on the left is the image or vice versa, the bar on the left is the object and the bar on the right is the image.
By convention though, in diagrams like these, an object is typically positioned to the left of a lens or mirror. So, we’ll say that this blue bar here is our object, and that makes this blue bar here the image that is formed by the lens. To the right of the lens, along with the image, we see these five points A, B, C, D, and E. And our question asks, which of these points is located at a distance from the lens that is equal to the lens’s focal length?
To figure this out, let’s start by recalling what the focal length of a lens is. Say that we have a convex lens like the one in our diagram. If we were to send parallel rays of light into this lens, then the rays would be brought together so that they cross or intersect at a certain point. It’s the distance from that point to the center of the lens that’s known as the focal length.
So, going back to our diagram, to find the focal length of this particular lens, we can trace in rays of light coming from either end of our object that move parallel to this dashed line here, an imaginary line known as the optical axis. The point where those two lines intersect on the right side of our lens after they’ve been refracted by it will be at a distance from the lens that is equal to its focal length.
So, starting at the very top of our object, we’ll draw in a line parallel to the optic axis from here. This light ray would look like this. But then, the question becomes, how would it be bent or refracted by the lens? We actually know the answer to that question because we have this picture of the image of our object. This ray of light coming from our object will be refracted so that it passes through this lowermost point of the image.
We know that because this dashed line here tells us that this uppermost point in our object corresponds or correlates to this lowermost point in our image. So then, this incoming ray of light from our object will be bent by the lens. And it will be bent so that it moves like this. We know that because this part of our object, the part that’s above the optical axis on the left of the lens, creates this part of our image.
Now, let’s move on to consider a ray of light coming from this lowermost point of our object that, again, moves parallel to the optical axis. That ray would move along like this until it reaches the lens. And then, it would be refracted so that it passes through this point, the uppermost point of the image. So, that would look like this. We said earlier that the point where these two lines intersect right here is at a distance from the lens equal to the lens’s focal length. In other words, this distance here is the lens’s focal length.
And we can see that that distance starting from the center of the lens goes out to the point labeled B in our diagram. So, of the five points A, B, C, D, and E in the diagram, it’s point B which is located at a distance from the lens equal to the lens’s focal length.