Images from Spherical Mirrors — Question 4

Question 4

A required real magnification and separation fix the mirror design

Use the paraxial mirror relations $1/p+1/q=1/f$ and $m=-q/p$, with p positive for a real object, q positive for a real image, and f positive for a concave mirror. A concave mirror must form an inverted image three times the height of a real object. Object and real image are both in front of the mirror, and their physical separation is 60 cm.

1. Find the object and image distances.

2. Find the required focal length and radius.

Question 4 - Solution

A required real magnification and separation fix the mirror design

(a) Method and calculation

$$m=-3=-q/p\Rightarrow q=3p.$$$$q-p=60\,\mathrm{cm}\Rightarrow\boxed{p=30\,\mathrm{cm},\quad q=90\,\mathrm{cm}}.$$

(b) Interpretation and further calculation

$$f=pq/(p+q)=30(90)/120=\boxed{22.5\,\mathrm{cm}}.$$$$\boxed{R=45\,\mathrm{cm}}.$$

Physical check

The physical object-image separation is q-p here, not p+q, because both lie on the same side of the mirror.