Question 23.4: The Face in the Mirror Goal Find a focal length from a magni...
The Face in the Mirror
Goal Find a focal length from a magnification and an object distance.
Problem When a woman stands with her face 40.0 \mathrm{~cm} from a cosmetic mirror, the upright image is twice as tall as her face. What is the focal length of the mirror?
Strategy To find f in this example, we must first find q, the image distance. Because the problem states that the image is upright, the magnification must be positive (in this case, M=+2 ), and because M=-q / p, we can determine q.
Obtain q from the magnification equation:
\begin{aligned} & M=-\frac{q}{p}=2 \\ & q=-2 p=-2(40.0 \mathrm{~cm})=-80.0 \mathrm{~cm} \end{aligned}
Because q is negative, the image is on the opposite side of the mirror and hence is virtual. Substitute q and p into the mirror equation and solve for f :
\begin{aligned} \frac{1}{40.0 \mathrm{~cm}}-\frac{1}{80.0 \mathrm{~cm}} & =\frac{1}{f} \\ f & =80.0 \mathrm{~cm} \end{aligned}
Remarks The positive sign for the focal length tells us the mirror is concave, a fact we already knew because the mirror magnified the object. (A convex mirror would have produced a smaller image.)