An anti-shoplifting mirror

Question: Convenience stores often install a diverging mirror so that the clerk has a view of the whole store and can catch shoplifters. Use a ray diagram to show that the image is reduced, bringing more into the clerk's field of view. If the focal length of the mirror is 3.0 m, and the mirror is 7.0 m from the farthest wall, how deep is the image of the store?

Solution:

As shown in ray diagram (a), d _i is less than d _o. The magnification, M = d_i/ d _o, will be less than one, i.e. the image is actually reduced rather than magnified.

We now apply the method outlined above for determining the plus and minus signs. Step 1: The object is the point on the opposite wall. As an experiment, (b), we try making the object closer. I did these drawings using illustration software, but if you were doing them by hand, you'd want to make the scale much larger for greater accuracy. Also, although I did figures (a) and (b) as separate drawings in order to make them easier to understand, you're less likely to make a mistake if you do them on top of each other.

The two angles at the mirror fan out from the normal. Increasing θ_o has clearly made θ_i larger as well. (All four angles got bigger.) There must be a cancellation of the effects of changing the two terms on the right in the same way, and the only way to get such a cancellation is if the two terms in the angle equation have opposite signs:

θ_f = + θ_i - θ_o

or

θ_f = -θ_i + θ_o .

Step 2: Now which is the positive term and which is negative? Since the image angle is bigger than the object angle, the angle equation must be

θ_f = θ_i - θ_o ,

in order to give a positive result for the focal angle. The signs of the distance equation behave the same way:

An outbending mirror in the shape of a sphere. The image is reduced (minified), and is also distorted because the mirror's curve is not shallow.

Solving for d _i, we find

The image of the store is reduced by a factor of 2.1/7.0=0.3, i.e. it is smaller by 70%.