Problem 13.3.17

Solution

The magnification is a result of the planoconvex water lens in front of an observer. Using the refractive index of $4/3$ for water, this lens has the focal length of $f=R/(4/3-1)=3R$, which is behind the aquarium. As a result, the magnification is largest when the fish is closest to the focal point, i.e. at Point $A$ farthest from the observer. The water between the fish and the lens serves only to bring the first image of the fish to Point $B$ with the apparent distance of $u=2R/(4/3)=3R/2$ from the lens. Thus, the magnification by the lens is

$$m=\frac{f}{u-f}=\frac{3R}{3R/2-3R}=-2.$$

Note that $v=mu=-3R$, so the (virtual) image of the fish at Point $C$ is in fact where the focal point of the lens is.