Spherical Refracting Surfaces — Question 10

Question 10

A ball lens requires two surface calculations with a virtual intermediate object

Use $n_1/p+n_2/q=(n_2-n_1)/R$ and $m=-n_1q/(n_2p)$ for paraxial refraction. Here p is positive on the incident side, q on the outgoing side, and R when the center lies on the outgoing side. A transparent sphere of radius 10 cm and index 1.50 is in air. A paraxial parallel beam enters from the left. The first vertex is x=0 and the second x=20 cm. Use signed radii +10 cm and -10 cm at the two surfaces.

1. Find the first surface image coordinate and the signed object distance for the second surface.

2. Find the final image distance from the rear surface and its laboratory coordinate.

Question 10 - Solution

A ball lens requires two surface calculations with a virtual intermediate object

(a) Method and calculation

$$p_1\to\infty:\quad1.50/q_1=0.50/10\Rightarrow\boxed{q_1=30\,\mathrm{cm}}.$$$$\boxed{p_2=20-q_1=-10\,\mathrm{cm}}\quad\text{virtual object for surface 2}.$$

(b) Interpretation and further calculation

$$\frac{1.50}{-10}+\frac1{q_2}=\frac{1-1.50}{-10}\Rightarrow\boxed{q_2=5\,\mathrm{cm}}.$$$$\boxed{x_I=20+5=25\,\mathrm{cm}}.$$

Physical check

The second surface intercepts the converging rays before the first surface alone would bring them to focus; its object distance must therefore be negative.