# Question Video: Relating the Focal Length of a Convex Lens to Its Radius of Curvature Science

Which of the following convex lenses has the shortest focal length? [A] Option A [B] Option B [C] Option C [D] Option D

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### Video Transcript

Which of the following convex lenses has the shortest focal length? (A), (B), (C), or (D)?

Before we start to tackle this question, let’s remind ourselves of what is meant by the focal length of a convex lens. When parallel rays of light reach a convex lens, the lens causes the rays of light to move towards each other, or converge. Once they’ve passed through the lens, they’re no longer parallel to each other. And the distance between each ray of light decreases, as the distance of the light from the lens increases. Since the rays are converging, they all meet at a single point. This is called the focal point of the lens. The focal length of a lens is defined as the distance between the focal point and the center of the lens.

In this question, we are asked to identify the convex lens that has the shortest focal length. Before we can do that, we need to understand why convex lenses with different shapes have different focal lengths. Convex lenses of different shapes have different focal lengths because they cause parallel light rays to converge by different amounts.

For example, let’s look at two sets of light rays, which have each been passed through different-shaped convex lenses and, hence, are converging by different amounts. If we compare these two sets of light rays, we see that the light rays on the right-hand side have converged much sooner than the light rays on the left. Let’s mark the focal points and focal lengths of each set of rays. If we compare these two diagrams, we see that the lens on the right has a much shorter focal length than the lens on the left. This is because this lens causes the light to converge more, so the light rays meet at a point that is much closer to the lens. So we have seen that the lens that causes the light rays to converge the most has the shorter focal length.

To answer the question we have been asked, we need to identify the lens that has the shortest focal length. This will be the lens that causes light rays to converge the most. At first glance, these lenses all appear very similar. All the lenses are convex and so have the same kind of shape. Each lens is thinner at the top and bottom than it is in the middle, and each lens has a smooth curve along either side. However, when we look closer, we see that the lenses do in fact have different shapes. The edges of the lenses curve by different amounts. This lens here has the most significant curve. And this lens here has the least significant curve.

So which of these lenses will cause light rays to converge the most and, hence, have the shortest focal length? Well, the curve along the edge of the lens is actually what causes the light rays to converge. So it makes sense that the lens with the most significant curve will cause the light rays to converge the most and, hence, will have the shortest focal length.

For example, if parallel light rays were to pass through a rectangular prism with no curved edges, the light rays would still be parallel after they had passed through the lens. They would not converge at all. When parallel light rays pass through a lens that is slightly convex, like option (D), the rays will converge slightly. The rays will meet at a point that is far away from the lens. When parallel rays pass through a lens that is very convex, like option (A), they will converge much more. This means this lens will have a much shorter focal length.

Out of the options we’re given, lens (A) has the most significant curvature. Hence, lens (A) will cause the greatest convergence of parallel light rays. This also means that lens (A) will have the shortest focal length. So the correct answer to this question is option (A).