Example 13.2: Convex mirrors

Question: How far must an object be placed in front of a convex mirror of radius of curvature $R=50$cm in order to ensure that the size of the image is ten times less than the size of the object? How far behind the mirror is the image located?
 
Answer: The focal length $f$ of a convex mirror is minus half of its radius of curvature (taking the sign convention for the focal lengths of convex mirrors into account). Thus, $f=-25$cm. If the image is ten times smaller than the object then the magnification is $M=0.1$. We can be sure that $M=+0.1$, as opposed to $-0.1$, because we know that images formed in convex mirrors are always virtual and upright. According to Eq. (352), the image distance $q$ is given by

$$q = - M\,p,$$

where $p$ is the object distance. This can be combined with Eq. (358) to give

$$p = f\left(1-\frac{1}{M}\right) = -(25)\,(1-10) = 225\,{\rm cm}.$$

Thus, the object must be placed $225$cm in front of the mirror. The image distance is given by

$$q = -M\,p = -(0.1)\,(225) = -22.5\,{\rm cm}.$$

Thus, the image is located $22.5$cm behind the mirror.