A dedicated sports car enthusiast polishes the inside and...

A dedicated sports car enthusiast polishes the inside and outside surfaces of a hubcap that is a section of a sphere. When she looks into one side of the hubcap, she sees an image of her face 30.0 $\mathrm{cm}$ in back of the hubcap. She then flips the hubcap over and sees another image of her face 10.0 $\mathrm{cm}$ in back of the hubcap. (a) How far is her face from the hubcap? (b) What is the radius of curvature of the hubcap?

Key Concepts

Mirror Equation

The mirror equation, 1/object distance + 1/image distance = 1/focal length, is a fundamental relation in geometrical optics for spherical mirrors. This equation links the distances of the object and its image from the mirror with the mirror’s focal length. It is applicable to both concave and convex mirrors, though the sign conventions differ, and is crucial when determining unknown distances in mirror problems.

Focal Length and Radius of Curvature

For a spherical mirror, the focal length is directly related to its radius of curvature by the relation f = R/2. This relationship is important because it allows one to calculate the mirror’s curvature from measured focal properties. Understanding this connection is key to analyzing how a spherical surface forms images, whether it is used on its concave or convex side.

Concave and Convex Mirror Behavior

Concave mirrors (when used on their reflecting, inner side) and convex mirrors (when used on their reflecting, outer side) create images differently. Concave mirrors can produce real or virtual images depending on the object’s position relative to the focal point, while convex mirrors always produce virtual, diminished images. Recognizing this behavior helps in setting up the correct equations with appropriate focal lengths for each case.

Sign Conventions in Mirror Imaging

Proper use of sign conventions in mirror problems is essential to correctly applying the mirror equation. In optical calculations, distances measured in the direction of the incident light (typically in front of the mirror) are taken as positive, while distances measured behind the mirror (where virtual images form) are taken as negative. Applying these conventions accurately allows one to distinguish between real and virtual image formation in both concave and convex configurations.

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