The two convex lenses shown have different focusing powers. Which is the more powerful lens? How many times is the more powerful lens more powerful?
Okay, so in this question, we can see that we’ve got a diagram showing two different lenses, the first one and the second one here. We can see that both of these lenses are convex lenses, as we’ve been told in the question. And we’ve been told that each one of these lenses has a different focusing power. Now, based on the diagrams we need to find which one is the more powerful lens. So to answer this, we first need to recall what the power of a lens actually means. So let’s recall that the focusing power of a lens, 𝑃, is defined as one divided by the focal length of the lens, 𝑓. Where the focal length of the lens 𝑓 is defined as the distance between the plane of the lens and the pointer, which all of the light from the lens is focused. Assuming the light rays on the other side of the lens are all parallel to each other and also moving perpendicular to the plane of the lens.
Now that sounds a little bit complicated. So firstly, let’s recall that this is the plane of the lens. In other words, if we were to approximate the lens as just a flat surface, then this would be the flat surface where we’re looking at that surface from this side. Now if, in this case, the incoming rays are parallel to each other, which in this case they are because they’re all moving in the same direction, and each one of those parallel rays is perpendicular to the plane of the lens, which in this case they are once again. Then we can say that the focal length or focal distance of the lens is the distance between the plane of the lens and the point at which all of the light is focused, which in this case is this point here. And we have been given the focal length for both lenses.
For the first lens, we know it’s four centimeters. And for the second lens, we know it’s 10 centimeters. So let’s start by labeling our diagrams. Let’s say that the focal length of the first lens, which we will call 𝑓 one, is equal to four centimeters, and the focal length of the second lens, 𝑓 two, is equal to 10 centimeters. Then we can use this equation in both cases to work out the focusing powers of each of these lenses. So starting with lens one, we can say that the focusing power of the first lens is equal to one divided by the focal distance or focal length of the first lens, 𝑓 one. And then we can substitute 𝑓 one with four centimeters. And similarly, for the second lens, we can say that the focusing power of the second lens is equal to one divided by 10 centimeters.
Now, at this point, we can realize that we could have converted these quantities the focal distances into meters and then written our powers in terms of diopters, the most commonly used unit for lens focusing power. However, in this particular case, because both focal distances are given to us in centimeters, we’re gonna keep our focal distances in centimeters and then write the power in units of one divided by centimeters. So in the case of the first lens, we can see that the focusing power is equal to 0.25 percent meter. And in the case of the second lens, the focusing power is equal to 0.1 percent meter.
And hence, we can see that the lens with the larger focusing power is the first lens. So if we label these two lenses as lens 𝐴 and lens 𝐵, where everything with a subscript one refers to a lens 𝐴 and everything with a subscript two refers to lens 𝐵. Then we can say that lens 𝐴 is the more powerful lens, at which point we can move on to the second question.
How many times is the more powerful lens more powerful? In other words, we need to find out how many times more powerful this lens is compared to this lens. And to do this, we simply need to find the fraction 𝑃 one divided by 𝑃 two, because we could say that 𝑃 one divided by 𝑃 two is some random quantity 𝑛. We don’t know what 𝑛 is. But if we rearrange the equation by multiplying both sides by 𝑃 two, then we see that 𝑃 one, the power of the first lens — lens 𝐴, is equal to 𝑛 times 𝑃 two, the power of lens 𝐵. In other words, this equation tells us that 𝑃 one is 𝑛 times 𝑃 two. And in this question, that’s what we’re trying to find. We’re trying to find 𝑛.
How many times larger is the power of the first lens compared to the second lens? And so rearranging this equation back once again, we see the 𝑛 is equal to 𝑃 one divided by 𝑃 two. Then we simply need to substitute in our values to this equation. We can say that 𝑃 one divided by 𝑃 two is equal to 0.25 percent meter, that’s 𝑃 one, divided by 0.1 percent meter. That’s 𝑃 two. Then we see that the units of percent meter cancel in the numerator and denominator. And we find that 𝑃 one divided by 𝑃 two is simply 0.25 divided by 0.1. And that ends up being 2.5, which basically tells us that 𝑃 one is 2.5 times 𝑃 two. And hence, as the answer to a question, we can say that the more powerful lens, which is lens 𝐴, is 2.5 times more powerful than lens 𝐵.