Samantha puts her face 32.0 cm from a makeup mirror and notices that her image is magnified by 1

step 1 So in this problem, we're looking at a makeup mirror and we're asked to determine what kind of m

Samantha puts her face 32.0 cm from a makeup mirror and...

Samantha puts her face $32.0 \mathrm{cm}$ from a makeup mirror and notices that her image is magnified by 1.80 times.
(a) What kind of mirror is this? (b) Where is her face relative to the radius of curvature or focal length?
(c) What is the radius of curvature of the mirror?

Key Concepts

Radius of Curvature

For spherical mirrors, the focal length is directly related to the radius of curvature through the relation f = R/2. This relationship helps characterize the mirror’s geometry and is used alongside the mirror equation to compute unknown parameters. Understanding the radius of curvature is important because it not only influences the focal length but also plays a role in determining the image characteristics such as size and position.

Mirror Types

In geometrical optics, mirrors are classified as concave or convex based on the curvature of their reflecting surface. Concave mirrors (which curve inward) can produce magnified images when objects are placed within the mirror’s focal length, yielding virtual, upright images. Convex mirrors, by contrast, always produce reduced, virtual images. This concept is fundamental to understanding how the shape of a mirror affects the image properties.

Mirror Equation

The mirror equation, given by 1/f = 1/d? + 1/d?, is a central tool in analyzing image formation by spherical mirrors. It relates the object distance (d?), image distance (d?), and the focal length (f) of the mirror. This equation, along with proper sign conventions, allows one to determine the nature and location of the image, whether it is real or virtual.

Magnification

Magnification in mirror optics is defined by the ratio m = -d?/d?, where the sign indicates image orientation. A positive magnification indicates an upright image, while a negative one corresponds to an inverted image. The magnitude of m gives the size change, so a magnification greater than one implies that the image is enlarged relative to the object. This concept is essential in determining under what conditions a mirror produces an image that is bigger than the object.

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