Thin Lenses — Question 7

Question 7

Two sharp lens positions determine focal length without locating either image directly

Use the thin-lens relations $1/p+1/q=1/f$ and $m=-q/p$ in one surrounding medium. Distances p and q are positive for real objects and real images; f is positive for a converging lens. A fixed object and screen are separated by L=100 cm. A positive thin lens forms a sharp image at two positions separated by d=60 cm.

1. Derive the focal length from L and d.

2. Find the two object distances and corresponding signed magnifications.

Question 7 - Solution

Two sharp lens positions determine focal length without locating either image directly

(a) Method and calculation

$$p+q=L,\quad|p-q|=d\Rightarrow pq=(L^2-d^2)/4.$$$$\boxed{f=pq/L=(L^2-d^2)/(4L)=16\,\mathrm{cm}}.$$

(b) Interpretation and further calculation

$$\boxed{p_1=(L-d)/2=20\,\mathrm{cm},\quad p_2=(L+d)/2=80\,\mathrm{cm}}.$$$$\boxed{m_1=-80/20=-4,\quad m_2=-20/80=-1/4}.$$

Physical check

The two lens positions exchange p and q; the method requires L greater than 4f for two distinct positions.