Images from Spherical Mirrors — Question 10

Question 10

Two successive mirror images require resetting the object distance

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. Mirror 1 is at x=0, faces +x, and has f1=10 cm. An object is at x=30 cm. Mirror 2 is at x=40 cm, faces -x, and has f2=15 cm. Follow rays that reflect first from mirror 1 and then mirror 2. Ignore obstruction and further reflections.

1. Find the first image and the object distance for mirror 2.

2. Find the final laboratory image coordinate and total transverse magnification.

Question 10 - Solution

Two successive mirror images require resetting the object distance

(a) Method and calculation

$$q_1=10(30)/(30-10)=15\,\mathrm{cm}\Rightarrow\boxed{x_{I1}=15\,\mathrm{cm}}.$$$$\boxed{p_2=40-15=25\,\mathrm{cm}}.$$

(b) Interpretation and further calculation

$$q_2=15(25)/(25-15)=37.5\,\mathrm{cm}\Rightarrow\boxed{x_{I2}=40-37.5=2.5\,\mathrm{cm}}.$$$$\boxed{m_{\mathrm{total}}=(-15/30)(-37.5/25)=+0.75}.$$

Physical check

The first image is passed before rays reach mirror 2, so it is a real object for that mirror. Local positive image distances point in opposite laboratory directions.