Light from the object O at the bottom of the tank passes from denser medium (water) to rarer medium (air) to reach our eyes. It deviates away from the normal in the rarer medium at the point of incidence B. The refractive index of the denser medium is n1 and rarer medium is n2. Here, n1> n2. The angle of incidence in the denser medium is i and the angle of refraction in the rarer medium is r. The lines NN’ and OD are parallel. Thus angle ∠DIB is also r. The angles i and r are very small as the diverging light from O entering the eye is very narrow. The Snell’s law in product form for this refraction is,
n1sin i = n2sin r
As the angles i and r are small, we can approximate,
sin i ≈ tan i;
n1 tan i = n2 tan i
In triangles ∆DOB and ∆DIB,
DB is cancelled on both sides, DO is the actual depth d and DI is the apparent depth d’. Rearranging the above equation for the apparent depth d’ ,
d’ = \((\frac{n_2}{n_1})\)d
As the rarer medium is air and its refractive index n2 can be taken as 1, (n2 = 1). And the refractive index n1 of denser medium could then be taken as n, (n1 = n). In that case, the equation for apparent depth becomes,
d = \(\frac{d}{n}\)
