Question Video: Relating the Focal Length of a Concave Lens to Its Radius of Curvature | Nagwa Question Video: Relating the Focal Length of a Concave Lens to Its Radius of Curvature | Nagwa

Question Video: Relating the Focal Length of a Concave Lens to Its Radius of Curvature Science • Third Year of Preparatory School

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Which of the following concave 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 concave 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 concave lens.

When parallel rays of light reach a concave lens, the lens causes the rays of light to spread out, or diverge. Once they’ve passed through the lens, they’re no longer parallel to each other, and the distance between each ray of light increases as the distance of the light from the lens increases. If we look at these diverging light rays, we might notice something interesting. If we trace each of these rays backwards, ignoring the lens, it appears as though all of the rays come from a single point. We call this point the focal point.

Note that the actual light rays don’t all physically pass through this point. It just appears that they come from this point, because of the way the lens has caused the rays to spread out. 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 concave lens that has the shortest focal length. Before we can do that, we need to understand why concave lenses with different shapes have different focal lengths. Concave lenses of different shapes have different focal lengths because they cause parallel light rays to diverge by different amounts. For example, let’s look at two sets of light rays which have each been passed through different-shaped concave lenses and hence are diverging by different amounts. If we compare these two sets of light rays, we see that the light rays on the right-hand side have been spread out much more than the light rays on the left.

Now, let’s trace the rays back and mark the focal points and focal lengths of each lens. 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 diverge more, so it appears that the light is coming from a point closer to the lens. So we have seen that the lens that causes the light rays to diverge 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 diverge the most. At first glance, these lenses all appear very similar. All of the lenses are concave and so have the same kind of shape. Each lens is thicker 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 edge 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 diverge 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 diverge. So it makes sense that the lens with the most significant curve will cause the light rays to spread out 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 diverge at all.

When parallel light rays pass through a lens that is slightly concave, like option (B), the rays will diverge slightly. When parallel rays pass through a lens that is very concave, like option (A), they will diverge 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 divergence 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).

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