Spherical Mirrors — Question 7

Question 7

The focal point of a convex mirror belongs to backward extensions

A convex mirror has signed radius R=-60 cm. Let s measure physical distance away from the mirror into the reflected half-space. A paraxial parallel incident ray hits at height h. Its outgoing ray obeys $y(s)=h(1+s/F)$, where F is the positive focal-distance magnitude.

1. Find F and the signed mirror focal length.

2. Locate the ray intersection by continuation in s and find the separation of rays hitting at heights plus and minus 3 mm when s=30 cm.

Question 7 - Solution

The focal point of a convex mirror belongs to backward extensions

(a) Method and calculation

$$\boxed{f=R/2=-30\,\mathrm{cm},\quad F=30\,\mathrm{cm}}.$$$$y=0\Rightarrow s=-F=\boxed{-30\,\mathrm{cm}}\quad\text{behind the mirror}.$$

(b) Interpretation and further calculation

$$\Delta y(s)=2h(1+s/F).$$$$\boxed{\Delta y(30\,\mathrm{cm})=2(3\,\mathrm{mm})(2)=12\,\mathrm{mm}}.$$

Physical check

The physical reflected rays diverge for s positive; their backward intersection is a virtual focus.