Physics · Ch 6 — Optics

Apparent Depth

6.4.5

Apparent Depth

When viewed from nearly directly above, an object at the bottom of a denser medium (e.g. water) appears closer to the surface than it really is, because light refracting from the denser medium into the rarer air bends away from the normal and so appears, on backward extension, to originate from a shallower point. For near-normal viewing, the apparent depth is $d'=\dfrac{n_2}{n_1}d$ , where $d$ is the real (actual) depth in the denser medium (index $n_1$ ) and $n_2$ is the index of the medium the observer is viewing from. Viewed from air ( $n_2=1$ ) into a medium of index $n$ ( $n_1=n$ ), this simplifies to $\boxed{d'=d/n}$ , so the apparent depth is always less than the real depth, with the bottom appearing elevated by $d-d'=d(1-1/n)$ . For several immiscible transparent layers stacked on top of one another, the total apparent depth is the sum of each layer's own individu …

Figure 6.20Apparent depth

What this figure shows. Three linked panels showing a coin (or point object O) at the real depth on the bottom of a water tank, viewed by an eye above the surface. Panel (a) shows the everyday effect -- the bottom of the tank looks raised. Panels (b) and (c) show the actual ray diagram: light from O refracts away from the normal on leaving the denser water for the rarer air, so the diverging rays reaching the eye appear, when extended backward, to originate from a shallower point I -- the apparen …

Misc Example 6.8Apparent depth of a coin under three stacked immiscible liquids

Worked out. A coin at the bottom of a trough is covered by three immiscible liquid layers, of refractive indices 1.3, 1.4 and 1.5 and actual thicknesses 30 cm, 16 cm and 20 cm respectively (real total depth d = 66 cm). Because each layer refracts near-normal viewing rays independently, the total apparent depth is the sum of each layer's own contribution, d(i)/n(i): 30/1.3 + 16/1.4 + 20/1.5 = 23.1 + 11.4 + 13.3 = 47.8 cm. So even though the coin genuinely sits 66 cm down, an observer looking straight down from air sees it appearing to be only 47.8 cm deep -- each layer shrinks its own share of the depth by its own refractive index before the next l …

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