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

step 1 There is only one type of mirror that creates a magnified image, an enlarged image. And that is

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 facerelative to the radius of curvature or focal length?
(c) What is the radius of curvature of the mirror?

Key Concepts

Concave Mirror

Concave mirrors are spherical mirrors that curve inward, causing parallel light rays to converge to a focal point. They are capable of producing both real and virtual images depending on the object's position relative to the focal length. In cases where an object is placed within the focal length, as is common with makeup mirrors to create enlarged, upright images, the mirror produces a virtual and magnified image.

Mirror Equation

The mirror equation, given by 1/f = 1/d_o + 1/d_i, relates the object distance (d_o), the image distance (d_i), and the focal length (f) of a spherical mirror. This equation is fundamental in predicting the position and type (real or virtual) of the image formed by a mirror.

Magnification and Image Characteristics

Magnification in mirror systems is defined by the ratio of the image height to the object height and can also be expressed as m = -d_i/d_o. The magnitude of magnification indicates the image size relative to the object, while its sign reveals whether the image is upright (positive) or inverted (negative), which in turn helps determine the image's nature, such as real or virtual.

Focal Length and Radius of Curvature

The focal length of a spherical mirror is directly related to its radius of curvature by the relation f = R/2. This relationship underscores the geometric basis of a mirror’s focusing ability and is essential for calculating one when the other is known.

Optical Sign Conventions

Optical sign conventions are critical in solving mirror problems as they dictate the correct use of positive and negative values for the object distance, image distance, and focal length. Understanding these conventions allows for accurate determination of image properties such as magnification, orientation, and whether the image is real or virtual.

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