A spherical mirror forms an erect image three times the...

A spherical mirror forms an erect image three times the linear size of the object. If the distance between the object and the image is $80 \mathrm{~cm}$, What is the focal length of the mirror?
(A) $30 \mathrm{~cm}$
(B) $40 \mathrm{~cm}$
(C) $-15 \mathrm{~cm}$
(D) $15 \mathrm{~cm}$

Key Concepts

Virtual Image Formation

Virtual images are those formed by the apparent divergence of reflected rays; they are not formed by the actual convergence of light on the opposite side of the mirror. In spherical mirrors, an erect and magnified image that is virtual indicates that the object is placed such that the mirror forms the image behind it, influencing the sign of the image distance in the mirror equation.

Sign Conventions

Proper use of sign conventions in the analysis of spherical mirrors is crucial. These conventions determine the positive and negative directions for distances and heights. For instance, real images typically have positive or negative signs depending on whether the light rays actually converge or appear to diverge from the mirror, and these signs affect the application of the mirror equation and the interpretation of magnification.

Mirror Equation

The mirror equation, given by 1/f = 1/u + 1/v, is fundamental in geometrical optics. It relates the focal length of a spherical mirror (f) to the object distance (u) and the image distance (v), allowing one to determine the position of the image for a given object placement. This equation is widely used in problems involving image formation by spherical mirrors.

Magnification

Magnification in mirror optics is defined as the ratio of the image height to the object height and is also expressed as m = -v/u, where v is the image distance and u is the object distance. The sign and magnitude of the magnification indicate whether the image is erect or inverted and whether it is enlarged or diminished relative to the object.

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