Spherical Refracting Surfaces — Question 4

Question 4

A plane interface changes apparent depth while preserving transverse size

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 small object is p=12 cm below a plane water-air surface. Light travels from water n1=4/3 to air n2=1. Observe close to the normal, so the paraxial model applies.

1. Take the infinite-radius limit to find the signed image distance and apparent depth.

2. Calculate the transverse magnification and explain the distinct depth and height factors.

Question 4 - Solution

A plane interface changes apparent depth while preserving transverse size

(a) Method and calculation

$$R\to\infty:\quad n_1/p+n_2/q=0\Rightarrow\boxed{q=-(n_2/n_1)p}.$$$$\boxed{q=-9\,\mathrm{cm},\quad d_{\mathrm{apparent}}=9\,\mathrm{cm}}.$$

(b) Interpretation and further calculation

$$m=-\frac{n_1}{n_2p}\left(-\frac{n_2p}{n_1}\right)=\boxed{1}.$$$$\boxed{d_{\mathrm{apparent}}/d_{\mathrm{actual}}=3/4,\quad h_I/h_O=1}.$$

Physical check

The object appears shallower without a first-order change of transverse size; these are different directions of magnification.