Spherical Refracting Surfaces — Question 9

Question 9

Measured conjugates can reveal an unknown refractive index

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. Light enters an unknown transparent medium from air through a surface with R=+10 cm. A real object at p=20 cm produces a real image at q=60 cm.

1. Solve for the unknown index n2 without assuming magnification first.

2. Find the transverse magnification and image height for a 6-mm object.

Question 9 - Solution

Measured conjugates can reveal an unknown refractive index

(a) Method and calculation

$$\frac1{20}+\frac{n_2}{60}=\frac{n_2-1}{10}\Rightarrow3+n_2=6n_2-6.$$$$\boxed{n_2=9/5=1.80}.$$

(b) Interpretation and further calculation

$$\boxed{m=-\frac{1(60)}{1.80(20)}=-5/3}.$$$$\boxed{h_I=(-5/3)(6)=-10\,\mathrm{mm}}.$$

Physical check

An index inferred from image positions must also predict the measured size; the magnification offers an independent consistency check.