Apparent depth of a coin

Problem

You look at a coin on the bottom of a swimming pool. Does it look closer or farther than it really is? Explain in terms of refraction and the eye-brain’s geometric reconstruction.

Solution

Answer. The coin looks closer (shallower) than it really is.

Why. The light rays leaving the coin, exiting the water ($n \approx 1{,}33$) into the air ($n = 1$), pass from a denser to a less dense medium and refract away from the normal (angle of refraction greater than the angle of incidence).

Eye-brain reconstruction. The eye does not “know” that the rays have bent: it extends the rays it receives backwards in a straight line. These extensions meet at a depth shallower than the real one, so the brain places the image higher up.

Estimate. Looking almost vertically, the apparent depth is

$h_{\text{app}} \approx \frac{h}{n}$

$\ev{\text{Looks closer; apparent depth } \approx h/n.}$

For water, the bottom appears at about $75\%$ of the true depth: this is why people underestimate the depth of a pool.

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Topics: Ottica Concepts: Rifrazione e legge di Snell Skills: Analisi di casi limite e fantafisica