An object 1.50 cm high is held 3.00 cm from a person's cornea, and its reflected image is meas

step 1 Part A, magnification is given by height of the image divided by height of the object. The heigh

An object 1.50 cm high is held 3.00 cm from a person's...

An object $1.50 \mathrm{~cm}$ high is held $3.00 \mathrm{~cm}$ from a person's cornea, and its reflected image is measured to be $0.167 \mathrm{~cm}$ high. (a) What is the magnification? (b) Where is the image? (c) Find the radius of curvature of the convex mirror formed by the cornea. (Note that this technique is used by optometrists to measure the curvature of the cornea for contact lens fitting. The instrument used is called a keratometer, or curve measurer.)

Key Concepts

Sign Conventions in Mirror Optics

The sign conventions in mirror optics are critical for correctly interpreting the mirror equation and the associated parameters. For convex mirrors, the focal length is taken as negative, and the image distance is also negative when the image is virtual. These conventions help in ensuring consistency when calculating image positions, magnifications, and the radius of curvature.

Convex Mirror Properties

Convex mirrors are outward-curving mirrors that always produce virtual, upright, and reduced images regardless of the object's position relative to the mirror. They are widely used in applications requiring a wide field of view, such as in keratometers for measuring the curvature of the eye's cornea, ensuring that the images remain smaller and correctly oriented.

Mirror Equation

The mirror equation, which is given as 1/f = 1/do + 1/di, is a fundamental relationship in geometrical optics. It relates the focal length (f) of a mirror to the object distance (do) and the image distance (di). This equation is used to determine the position of the image produced by mirrors, including convex mirrors, by taking into account the proper sign conventions.

Magnification

Magnification in optics is the ratio of the image size to the object size, and can also be expressed as the ratio of the image distance to the object distance (with consideration for sign conventions). It provides a measure of how much larger or smaller the image is compared to the object, and its sign indicates whether the image is upright or inverted relative to the object.

Focal Length and Radius of Curvature

In mirror optics, particularly for spherical mirrors, the focal length (f) is connected to the radius of curvature (R) by the relation f = R/2. This relationship is essential in determining the shape and performance of the mirror. For convex mirrors, the focal length is considered negative, and this modification reflects the way images are formed by such mirrors.

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