Question Video: Recalling the Number of Foci of a Concave Lens Science

How many foci does a concave lens have?


Video Transcript

How many foci does a concave lens have?

In this question, we are asked to look at a concave lens and figure out how many foci it has. First, let’s remember that a concave lens is a lens that is thicker along the edges and thinner in the middle. We can think of this shape as a cylinder on its side with its circular edges along the horizontal axis like this, which is being overlapped by two spheres like shown here. If we were to remove the overlapped material from the cylinder, it would then have the same shape as a concave lens.

Notice that concave lenses are symmetrical, both horizontally and vertically. If we were to split this lens along the horizontal or vertical axis, we would have two identical halves. This symmetry will be important for our solution, as we will soon see. For now though, let’s consider these circles outside the lens.

These circles represent the curvature of our lens. Notice how their edges line up perfectly with the curve of our lens. Let’s mark the centers of these circles with points. These points are called the centers of curvature, and both are an equal distance from the lens. If we were to connect these two points with a line, it would pass directly through the center of our lens. This line is called the optical axis. If a ray of light is traveling along the optical axis and goes through the lens, its direction will not change. In fact, any ray of light that travels through the center point of the lens, which is also the halfway point between our centers of curvature along the optical axis, will pass through without having its direction change.

But now, let’s consider light rays that are traveling parallel to the optical axis. A concave lens is a diverging lens, which means that as light rays pass through it, they are directed away from each other.

Recall that foci are points at which light rays converge or meet. How can we have a focus, a convergence, with diverging rays? Let’s look closer at these light rays that are traveling through the lens. If we trace these diverging rays back along their newest path direction, they both appear to have started from the same point on the other side of the lens. This is the focal point of a concave lens. And its distance to the center of the lens is called the focal length.

Now, remember that concave lenses are symmetric about the vertical axis. The centers of curvature are the same distance from the center of the lens on both sides. This means if we have a focal point on one side of our lens, we can expect to have one on the other side as well due to symmetry.

We can prove this by tracing back the final direction of rays when they come from the other side of the lens. They also converge on a single point, which is the second focal point of this concave lens. This means that the answer to “How many foci a concave lens has?” is two, one on either side of the lens. This gives one focus for each of the two centers of curvature.

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