An object 10.0 cm tall is placed at the zero mark of a meter...

An object 10.0 $\mathrm{cm}$ tall is placed at the zero mark of a meter stick. A spherical mirror located at some point on the meter stick creates an image of the object that is upright, 4.00 $\mathrm{cm}$ tall, and located at the 42.0 -cm mark of the meter stick. (a) Is the mirror convex or concave? (b) Where is the mirror? (c) What is the mirror's focal length?

Key Concepts

Sign Conventions in Geometrical Optics

Sign conventions provide a consistent method for assigning positive or negative values to quantities like object distance, image distance, and focal length based on their directions and positions relative to the mirror. They are crucial for correctly applying the mirror equation and understanding the nature of the image formed.

Mirror Equation

The mirror equation (1/do + 1/di = 1/f) relates the object distance (do), the image distance (di), and the focal length (f) of a spherical mirror. It is a fundamental tool in geometrical optics for solving problems involving image formation by mirrors.

Magnification

Magnification is the ratio of the image height to the object height and is also given by -di/do. This concept helps determine how large or small an image will be relative to its object and whether the image is inverted (negative magnification) or upright (positive magnification).

Real and Virtual Images

In mirror systems, real images are formed when light rays actually converge and can be projected onto a screen, whereas virtual images are formed when the light rays appear to come from a location behind the mirror and cannot be projected. Recognizing the nature of the image helps determine the correct signs for distances in the mirror equation.

Concave and Convex Mirrors

Concave and convex mirrors are types of spherical mirrors that differ in their curvature. Concave mirrors can produce both real and virtual images depending on the object's position relative to the focal point, while convex mirrors always produce virtual, diminished images. Understanding their characteristics is essential for deducing the mirror type based on the image properties.

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