The origin of a coordinate system is placed at the center of...

The origin of a coordinate system is placed at the center of curvature of a spherical mirror with radius of curvature $R=58.1 \mathrm{~cm}$ (see the figure). An object is placed at $x_{\mathrm{o}}=19.7 \mathrm{~cm} .$ What is the $x$ -coordinate of the image?

Key Concepts

Spherical Mirrors

Spherical mirrors are reflective surfaces that form part of a sphere. They use the geometrical properties of a circle to focus or spread light, with their reflecting surfaces being either concave (converging) or convex (diverging). The behavior of light rays reflecting off these mirrors is determined by the mirror’s curvature, and the mirror equation describes how object and image distances relate to the focal length.

Mirror Equation

The mirror equation is a fundamental relation in optics that connects the object distance, image distance, and the focal length of a mirror. This equation is typically written as 1/f = 1/s_o + 1/s_i, where f is the focal length, s_o is the object distance, and s_i is the image distance. It is essential for determining where an image will form relative to the mirror.

Focal Length

The focal length of a spherical mirror is the distance from the mirror to the point where parallel light rays converge (for a concave mirror) or appear to diverge from (for a convex mirror). For spherical mirrors, the focal length is directly related to the radius of curvature by the relation f = R/2. This relation is critical in linking the geometry of the mirror with its optical properties.

Sign Conventions in Geometrical Optics

Sign conventions are rules that determine the positive and negative directions for distances and focal lengths in mirror and lens equations. They ensure that calculations correctly reflect the physical reality of image formation, such as distinguishing between real and virtual images. Proper application of sign conventions is necessary to correctly solve problems in mirror optics.

Coordinate Systems in Optics

Coordinate systems in optics provide a framework for describing the positions and directions of objects and images relative to a reference point, such as the center of curvature or the vertex of a mirror. The choice of origin and the associated coordinate axes directly affect the application of equations and sign conventions when analyzing optical systems.

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