In her job as a dental hygienist, Kathryn uses a convex...

In her job as a dental hygienist, Kathryn uses a convex mirror to see the back of her patient's teeth. When the mirror is $1.20 \mathrm{~cm}$ from a tooth, the image is upright and $3.00$ times as large as the tooth. What are the focal length and radius of curvature of the mirror?

Key Concepts

Sign Conventions

In optics, particularly with mirrors, sign conventions are used to assign positive or negative values to distances and focal lengths. For example, in the case of convex mirrors, the focal length is taken as negative, and the image produced is virtual, upright, and also given a negative image distance. These conventions are essential for correctly applying the mirror formula and interpreting the results.

Focal Length and Radius of Curvature Relationship

The focal length (f) of a spherical mirror is directly related to its radius of curvature (R) by the relationship f = R/2. This means that once the focal length is determined using the mirror equation, the radius of curvature can be easily calculated. This relationship is key in understanding the geometry and optical properties of spherical mirrors.

Mirror Equation

The mirror equation, given by 1/f = 1/do + 1/di, is fundamental to image formation by mirrors. It relates the object distance (do), the image distance (di), and the focal length (f) of the mirror. This equation is pivotal in solving problems where one needs to find any of the three quantities, given the other two.

Magnification

Magnification in mirror problems is defined by the ratio m = -di/do, which gives information about the image's size relative to the object and its orientation. A magnification greater than one indicates that the image is larger than the object, while the sign of the magnification helps determine if the image is inverted or upright.

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