Video Transcript
A large spherical mirror has a
radius of curvature of 0.0099 kilometers. What is the distance between the
point on the surface of the mirror that the optical axis passes through and the
focal point? Give your answer in meters.
This question is about a mirror’s
radius of curvature and how it relates to the focal length. If we look at the second sentence
in the question, we see that it contains the definition for the focal length of a
mirror.
To help us better understand this
definition, let’s draw a ray diagram for a convex mirror. We can consider some parallel light
rays incident on the mirror surface. When these rays are reflected from
the mirror, then if we trace the reflected rays back behind the surface, we see that
they would all meet at a particular point on the optical axis. This point is the focal point of
the lens.
The focal length is the distance
between this focal point and this point on the surface of the mirror where the
optical axis passes through it. Let’s call this focal length
𝑓. The focal length of any spherical
mirror is exactly one-half the radius of curvature, which we’ve labeled 𝑅 subscript
c. To understand what the radius of
curvature means, let’s recall that if we traced out the curve of a spherical mirror,
we would find that the mirror is a small part of a larger sphere. The distances shown here are all
equal. They are all the radius of the
sphere, which is the same in all directions. This radius is the radius of
curvature of the mirror.
Now, in this case, we’ve been told
the value of the radius of curvature and we want to work out the focal length. To do this, we can use this
relation between the two quantities. Note though that we’re asked for
the focal length in meters. So, before we substitute our value
for the radius of curvature into the equation for the focal length, we need to
convert this value from kilometers into meters. There are 1000 meters in a
kilometer. So, in meters, the radius of
curvature is equal to 0.0099 kilometers multiplied by 1000 meters per kilometer,
which is 9.9 meters.
Remember, we know that the focal
length, or the distance between the point on the surface of the mirror that the
optical axis passes through and the focal point, is equal to half this radius of
curvature. Dividing 9.9 meters by two, we get
a result of 4.95 meters. Our final answer then is 4.95
meters.
