Question Video: Recalling the Definition of the Focal Length to Calculate Its Value Using the Radius of Curvature Science

A large spherical mirror has a radius of curvature of 0.0099 km. 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.

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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.