Spherical Mirrors — Question 2

Question 2

Paraxial focal signs distinguish converging and diverging mirrors

Two spherical mirrors have radius magnitudes 80 cm. One is concave and the other convex as viewed from the incident light. Use signed R and f positive for concave mirrors, negative for convex mirrors; in the paraxial limit f=R/2.

1. Find each signed focal length and locate its focus relative to the reflecting surface.

2. Find the signed reciprocal focal lengths in inverse meters and the radius needed for a concave focal length of 25 cm.

Question 2 - Solution

Paraxial focal signs distinguish converging and diverging mirrors

(a) Method and calculation

$$\boxed{R_c=+0.80\,\mathrm m,\ f_c=+0.40\,\mathrm m};\quad\text{real focus in front}.$$$$\boxed{R_v=-0.80\,\mathrm m,\ f_v=-0.40\,\mathrm m};\quad\text{virtual focus behind}.$$

(b) Interpretation and further calculation

$$\boxed{1/f_c=+2.50\,\mathrm{m^{-1}},\quad1/f_v=-2.50\,\mathrm{m^{-1}}}.$$$$\boxed{R=2f=+0.50\,\mathrm m}.$$

Physical check

The focal sign describes the reflected bundle: concave mirrors converge a paraxial parallel beam, while convex mirrors make it diverge.