Images from Spherical Mirrors — Question 8

Question 8

The minimum sum of real conjugate distances occurs at unit magnification

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 concave mirror with f positive forms a real image of a real object. Define S=p+q as the sum of the two positive distances, not the physical separation between object and image.

1. Find the minimum possible S.

2. Find both possible p values for a specified S greater than 4f and relate their magnifications.

Question 8 - Solution

The minimum sum of real conjugate distances occurs at unit magnification

(a) Method and calculation

$$pq=f(p+q)=fS,\quad(p-q)^2=S^2-4pq=S(S-4f)\ge0.$$$$\boxed{S\ge4f},\quad S_{\min}=4f\ \text{at}\ p=q=2f.$$

(b) Interpretation and further calculation

$$\boxed{p_\pm=\frac{S\pm\sqrt{S^2-4fS}}2,\quad q_\pm=S-p_\pm}.$$$$\boxed{m_+m_-=(-q_+/p_+)(-q_-/p_-)=1}.$$

Physical check

The two solutions exchange object and image distances, demonstrating optical reversibility.