Images from Spherical Mirrors — Question 1

Question 1

An object beyond the center of curvature gives a reduced real mirror image

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 12-mm arrow stands p=45 cm before a mirror of focal length f=15 cm.

1. Find the image distance and transverse magnification.

2. Find signed image height and locate the image relative to the focus and center of curvature.

Question 1 - Solution

An object beyond the center of curvature gives a reduced real mirror image

(a) Method and calculation

$$q=\frac{fp}{p-f}=\frac{15(45)}{45-15}=\boxed{22.5\,\mathrm{cm}}.$$$$\boxed{m=-q/p=-0.50}.$$

(b) Interpretation and further calculation

$$\boxed{h_I=mh_O=-6.0\,\mathrm{mm}}.$$$$f=15<q=22.5<2f=30\,\mathrm{cm}\Rightarrow\boxed{\text{real, inverted, between }F\text{ and }C}.$$

Physical check

The object lies beyond C, while its image lies between C and F; the negative magnification encodes inversion.