Three Proofs — Question 5

Question 5

Adding two surface equations proves the thin-lens formula

A thin lens of index nl is surrounded by the same index n0 on both sides. Surface radii R1 and R2 are positive for centers to the right. Let q1 be the first surface image distance. In the zero-thickness limit, the second surface object distance is p2=-q1.

1. Write the two spherical-surface equations and add them, showing cancellation of the intermediate image.

2. Derive both the thin-lens relation and lensmaker power.

Question 5 - Solution

Adding two surface equations proves the thin-lens formula

(a) Method and calculation

$$\frac{n_0}{p}+\frac{n_l}{q_1}=\frac{n_l-n_0}{R_1},\quad\frac{n_l}{-q_1}+\frac{n_0}{q}=\frac{n_0-n_l}{R_2}.$$$$n_0(1/p+1/q)=(n_l-n_0)(1/R_1-1/R_2).$$

(b) Interpretation and further calculation

$$\boxed{1/p+1/q=1/f}.$$$$\boxed{1/f=(n_l/n_0-1)(1/R_1-1/R_2)}.$$

Physical check

Cancellation requires negligible thickness and identical exterior indices; silently keeping a finite separation would invalidate this particular proof.