# Question Video: Defining the Focal Length of a Convex Mirror Science

The radius of curvature of a convex mirror is 5 cm. Which one of the following sentences about the focal length is correct? [A] The focal length is 5 cm and is the distance from the center of the surface of the mirror to the focal point. [B] The focal length is 5 cm and is the distance from the center of the surface of the mirror to the center of curvature. [C] The focal length is 2.5 cm and is the distance from the center of the surface of the mirror to the center of curvature. [D] The focal length is 2.5 cm and is the distance from the center of the surface of the mirror to the focal point.

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### Video Transcript

The radius of curvature of a convex mirror is five centimeters. Which one of the following sentences about the focal length is correct? (A) The focal length is five centimeters and is the distance from the center of the surface of the mirror to the focal point. (B) The focal length is five centimeters and is the distance from the center of the surface of the mirror to the center of curvature. (C) The focal length is 2.5 centimeters and is the distance from the center of the surface of the mirror to the center of curvature. (D) The focal length is 2.5 centimeters and is the distance from the center of the surface of the mirror to the focal point.

Knowing that we have a convex mirror with a radius of curvature of five centimeters, we can clear space at the top of our screen and sketch this mirror. Let’s say that this is our convex mirror, and here is the mirror center of curvature. The distance between this point and the point at the center of the mirror here is called the radius of curvature. In this example, the radius of curvature is given as five centimeters. Now, all of our answer options describe the mirror’s focal length. This is different from the radius of curvature.

If we sketch it on our diagram, this mirror’s focal point would be a point here. It’s the point halfway between the center of curvature and the center of the surface of the mirror. Then, the distance between the center of the surface of the mirror and our focal point, that distance is called the focal length of the mirror. As an equation, we can say that focal length equals one-half times the radius of curvature. Since the radius of curvature of this mirror is five centimeters, then the focal length must be half of that, or 2.5 centimeters.

Only two of our answer options, (C) and (D), have the focal length at 2.5 centimeters. This means we can cross out options (A) and (B). The difference between our two remaining answer choices is that one says the focal length is the distance from the center of the surface of the mirror to the center of curvature, while the other has it as the distance from the center of the surface of the mirror to the focal point. To see the difference between these choices, let’s look at an up close view of our sketch.

Okay, in this view, we have our center of curvature, the focal point of the mirror, and then here is the center of the surface of the mirror. Answer option (C) says that the focal length of this mirror is measured as this distance. We can see, though, that this can’t be correct. That’s because this distance in pink is the radius of curvature, five centimeters. The focal length is one-half that distance. The true focal length is the distance from the center of the surface of the mirror to the focal point. This is described by option (D). The focal length is 2.5 centimeters and is the distance from the center of the surface of the mirror to the focal point.