Video Transcript
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.
