Question Video: Recalling the Definition for the Focal Length of a Concave Mirror and How It Relates to the Radius of Curvature Science

The radius of curvature of a concave mirror is 5 centimeters. Which one of the following sentences correctly describes the focal length? [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 2.5 cm and is the distance from the center of the surface of the mirror to the focal point. [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 5 cm and is the distance from the center of the surface of the mirror to the center of curvature.

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Video Transcript

The radius of curvature of a concave mirror is five centimeters. Which one of the following sentences correctly describes the focal length? (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 2.5 centimeters and is the distance from the center of the surface of the mirror to the focal point. (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 five centimeters and is the distance from the center of the surface of the mirror to the center of curvature.

In this question, we are being asked about the focal length of a concave mirror. Let’s recall that a concave mirror looks like this so that the reflective surface that light is directed at is concave in shape. This dashed line we’ve drawn on our diagram is the optical axis of the mirror.

Now, let’s recall how focal length is defined. The focal length of a concave mirror is the distance from the center of the surface of the mirror, that’s this point on the surface of the mirror where the optical axis passes through it, to the mirror’s focal point.

In this case, we’re not told anything about the focal point of this concave mirror directly. However, we are told that the radius of curvature of the mirror is five centimeters. Recall that the radius of curvature is the distance from the center of the surface of the mirror to the mirror’s center of curvature. So, if this point here is the center of curvature of the mirror, then we know that this distance is the mirror’s radius of curvature, which in this case has a value of five centimeters.

We can use this information to work out the mirror’s focal length. Let’s recall that the focal length 𝑓 is equal to half the radius of curvature, 𝑅 subscript 𝑐. Since the focal length is the distance from the center of the mirror surface to the focal point, then this means that the focal point of the mirror is on the optical axis and is exactly halfway between the mirror’s surface and the center of curvature.

We know that for this mirror, the radius of curvature is five centimeters. This tells us that the focal length is equal to five centimeters divided by two, which is 2.5 centimeters. This value for the focal length matches the value given in answer options (B) and (C). We can therefore reject options (A) and (D) as these show an incorrect value for the focal length.

Option (C) claims that the focal length is the distance from the center of the surface of the mirror to the center of curvature. But we know that this is not the case. This distance is the radius of curvature of the mirror, not the focal length. Option (B) states that the focal length is the distance from the center of the mirror to the focal point. We can identify this as the correct definition.

We have therefore found that the correct answer is option (B). The focal length is 2.5 centimeters and is the distance from the center of the surface of the mirror to the focal point.