Lesson Video: Concave Mirrors Science

In this video, we will learn how to describe the reflection of light rays from a concave mirror.

11:02

Video Transcript

In this video, we will learn how to describe the reflection of light rays from a concave mirror. Before we think about those reflected light rays though, let’s consider what a concave mirror looks like. One way to picture a concave mirror is to imagine that we’re standing in front of a shape like this. If we were to take a laser pointer and shine it onto this reflective surface, this mirror, then the beam of light would be reflected off of the mirror, something like this. If we were to look at all this from a top-down perspective, then our concave mirror would look this way.

For a concave mirror, light always approaches the mirror from this side, we could say that the light goes inside the curve of the mirror. A concave mirror has a special shape to it; it’s part of a circle. The center of that circle, the point right here, is called the center of curvature of the mirror. The distance between the center of curvature and any point on the surface of the mirror is the same. That distance is called the radius of curvature. Notice this distance is just the radius of the circle overall. We give it the special name, radius of curvature, because it describes the radius of our curving mirror.

Now let’s say we clear away everything except our concave mirror and our center of curvature. We know that mirrors are designed to reflect light rays. So if a ray of light comes in and lands on the mirror rather than passing through or being absorbed by the mirror, this ray of light is reflected. The direction of the reflected ray depends on which part of the mirror the original ray lands on. Here’s something interesting though. Say that we have another ray of light that’s parallel to the first ray that landed on our mirror. This second ray of light will also be reflected. And when it is, it will cross over the first reflected ray.

Things get even more interesting if we add yet another ray that’s parallel to the first two. And we find that this reflected ray passes through the same point where the first two crossed over. If we keep adding parallel incoming rays and watching where they reflect, we find they all pass through this same point right here. We can say that it’s at this point that all of these reflected rays come to a focus. For that reason, we call this the focal point of our mirror. Whenever parallel incoming rays of light land on a concave mirror, the mirror reflects all those rays to and then through its focal point.

Clearing away our rays of light, we see we now have two points of interest for our mirror. And let’s remember that this distance here is called the radius of curvature of our mirror. This distance is measured from the center of curvature to the center of the surface of our mirror. It turns out that this distance here, from the focal point to the center of the mirror’s surface, has its own name. It’s called the focal length. There’s a relationship between the focal length and the radius of curvature of a concave mirror. Writing it as a mathematical equation, it looks like this. The focal length of the mirror is equal to one-half times the radius of curvature.

Another way to say this is that the radius of curvature is twice as large as the focal length. This relationship is always true for a concave mirror. That’s very handy because it means if we know either the focal length or the radius of curvature, we can solve for the other distance. Knowing all this about concave mirrors, let’s look now at a few examples.

Which of the mirrors in the diagram below is a concave mirror? Incident light is shown by the yellow lines.

Here we see two mirrors, Mirror 1 and Mirror 2. The difference between them is the way that they’re curved with respect to the incoming light. Mirror 1, we see, is curved outward, bulging towards the incoming light, we could say. Opposite to this is mirror 2, which we see is curved inward. We want to know which of these two mirrors is a concave mirror. One way we can remember the shape of a concave mirror is to think about the word cave in this name. Just like a cave is something that we travel into, a concave mirror, we could say, is something that light travels into inside this curving arc. Because the light reaching Mirror 2 is entering into the curve of the mirror, we know that this mirror is concave.

Of these two mirrors then, Mirror 2 is the concave mirror.

Let’s look now at another example.

For any spherical mirror, the distance between the center of curvature and the center of the surface is just the radius of that sphere. Which one of the following sentences correctly describes the location of the center of curvature of a concave mirror? (A) The center of curvature of a concave mirror will always be on the side opposite to the observer. (B) The center of curvature of a concave mirror will always be on the same side as the observer. (C) Depending on the path of the light rays, the center of curvature of a concave mirror can sometimes be on the side opposite to the observer and can sometimes be on the same side as the observer.

To start figuring out which of these three answer options is correct, let’s sketch a curved mirror. And we’ll say that this is a concave mirror. A concave mirror is one where light always enters into the curved portion of the mirror, here on this side. This is the opposite of light being incident on the mirror from this side. All of our possible answers talk about which side of a concave mirror an observer will be on. In order to observe or see the way a concave mirror reflects light, an observer always needs to be on the same side of the mirror as the incoming rays of light. An observer of a concave mirror then is always facing this curve of the mirror shape.

We can recall that this shape is part of a larger circle. The center of that circle, located here, is called the center of curvature of the mirror. For a concave mirror then, the center of curvature of the mirror and an observer are always on the same side of the mirror. What we’re finding is that answer option (B) is the correct description of a concave mirror. The center of curvature of a concave mirror will always be on the same side as the observer.

Let’s look now at another example.

Below is a ray diagram for a concave mirror. Which one of the five locations along the optical axis represents the center of curvature of the spherical mirror?

Here we see a concave spherical mirror. There are two parallel rays of light that are incident on the mirror. And along with this, we see this line parallel to those rays and passing through the center of the surface of the mirror. This line is called the optical axis. For our purposes, the important thing is these five locations marked out along the optical axis. We want to identify which one represents the center of curvature of the spherical mirror. Looking at our diagram, we might first think that it’s point 2. But we must be careful. For a concave mirror, the point where parallel incoming rays of light are reflected to a focus is actually called the focal point. This is different from the center of curvature, so we won’t choose point 2 for our answer.

Knowing that point 2 is the focal point though helps us find our answer. That’s because of this statement here. The distance from the center of a concave mirror’s surface to the focal point is one-half the distance from the center of the mirror’s surface to the center of curvature. That’s a long statement. But here’s the idea. If we start here at point 3, the center of our mirror’s surface, and we go from point 3 to point 2, our focal point, then that total distance we’ve traveled, which by the way is called the focal length of our mirror, is one-half the distance from the center of our mirror’s surface to the center of curvature.

In other words, if we double the length of this line here, we’ll go from point 3, the center of our mirror’s surface, to the center of curvature of the mirror. Doubling the length of our line gets us out to point 1. The curve of our concave mirror is actually part of a larger circle. The name for the center of this circle is the center of curvature. On our diagram, that point is labeled as point 1. This then is our answer. It’s at point 1 along the optical axis where the center of curvature of the spherical mirror is located.

Let’s look now at one last example.

Which of the following sentences is a correct description of what will happen to parallel rays incident on a concave mirror? (A) They will continue undisturbed. (B) They will be focused at the center of curvature. (C) They will be focused at the focal point. (D) They will not be focused at a point at all.

To see which of these answer options is correct, let’s sketch in a concave mirror. And because this mirror is concave, we know that incoming rays of light will approach it from this side. Now our question specifically talks about parallel rays of incoming light. So let’s say we have two such rays like this. Answer option (A) says that when these rays reach the mirror, they will continue on undisturbed. If that happened, the path of the rays would look like these dashed lines. We know though that one property of a mirror is to reflect light. The incoming rays then won’t follow the original paths they were on. We can eliminate answer option (A) from consideration.

Options (B) and (C) talk about these incoming rays being focused. That means when the rays are reflected, they cross at a point. Answer option (B) says that this focusing happens at a point called the center of curvature. Let’s now recall that our mirror is actually part of a larger spherical surface. The center of that sphere, which is right here, is at the point called the center of curvature. For our parallel incoming rays of light, we see that this is not the point where the rays are brought to a focus. We can cross out answer option (B).

Option (C) says that the rays are focused at what’s called the focal point. This indeed is a correct description of what happens to these incoming parallel rays. The point where the reflected rays are brought to a focus is called the focal point. We see then that option (C) is correct. And in the process, we can tell that option (D) is not correct. We’ve seen that these parallel incoming rays really are brought to a focus after being reflected. For our answer, we choose option (C). When parallel rays of light are incident on a concave mirror, they will be focused at the focal point.

Let’s finish our lesson now by reminding ourselves of a few key points. In this video, we studied concave spherical mirrors. We saw that the surface of these mirrors are part of a larger circle. The center of this circle is called the center of curvature of the mirror. The distance between the center of curvature and the center of the surface of the mirror is called the radius of curvature. If parallel rays of light are incident on the concave mirror, they reflect off the mirror through a common point. The name for that point is the focal point. And we learned that the distance between the center of the surface of the mirror and the focal point is called the focal length. Lastly, we saw that the focal length is equal to one-half the radius of curvature of the mirror. This is a summary of concave mirrors.

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