Images from Spherical Mirrors — Question 3

Question 3

An upright reduced image determines convex mirror curvature

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 real object is 60 cm in front of a mirror. Its upright image is one-quarter the object height.

1. Infer the signed image distance and focal length.

2. Find signed radius of curvature and the image location relative to vertex and focus.

Question 3 - Solution

An upright reduced image determines convex mirror curvature

(a) Method and calculation

$$m=+1/4=-q/p\Rightarrow\boxed{q=-15\,\mathrm{cm}}.$$$$\frac1f=\frac1{60}-\frac1{15}=-\frac1{20}\Rightarrow\boxed{f=-20\,\mathrm{cm}}.$$

(b) Interpretation and further calculation

$$\boxed{R=2f=-40\,\mathrm{cm}}.$$$$|q|=15<|f|=20\Rightarrow\boxed{\text{virtual image between vertex and virtual focus}}.$$

Physical check

For a real object, an upright reduced mirror image identifies a convex mirror in this paraxial model.