### 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).